Computational Ecology and Software, 2015, Vol. 5, Iss. 2

Computational Ecology and Software

Vol. 5, No. 2, 1 June 2015

International Academy of Ecology and Environmental Sciences

Computational Ecology and Software ISSN 2220-721X Volume 5, Number 2, 1 June 2015 Editor-in-Chief WenJun Zhang

Sun Yat-sen University, China

International Academy of Ecology and Environmental Sciences, Hong Kong

E-mail: [email protected], [email protected]

Editorial Board Ronaldo Angelini (The Federal University of Rio Grande do Norte, Brazil)

Andre Bianconi (Sao Paulo State University (Unesp), Brazil)

Bin Chen (Beijing Normal University, China)

Daniela Cianelli (University of Naples Parthenope, Italy)

Alessandro Ferrarini (University of Parma, Italy)

Yanbo Huang (USDA-ARS Crop Production Systems Research Unit, USA)

Istvan Karsai (East Tennessee State University, USA)

Vladimir Krivtsov (Heriot-Watt University, UK)

Lev V. Nedorezov (University of Nova Gorica, Slovenia)

Fivos Papadimitriou (Environmental and Land Use Consultancies, Greece)

George P. Petropoulos (Institute of Applied and Computational Mathematics, Greece)

Vikas Rai (Jazan University, Saudi Arabia)

Santanu Ray (Visva Bharati University, India)

Kalle Remm (University of Tartu, Estonia)

Rick Stafford (University of Bedfordshire, UK)

Luciano Telesca (Institute of Methodologies for Environmental Analysis, Italy)

Bulent Tutmez (Inonu University, Turkey)

Ranjit Kumar Upadhyay (Indian School of Mines, India)

Ezio Venturino (Universita’ di Torino, Italy)

Michael John Watts (The University of Adelaide, Australia)

Peter A. Whigham (University of Otago, New Zealand)

ZhiGuo Zhang (Sun Yat-sen University, China)

Editorial Office: [email protected]

Publisher: International Academy of Ecology and Environmental Sciences

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E-mail: [email protected]

Computational Ecology and Software, 2015, 5(2): 119-129

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Article

Fluctuating asymmetry and developmental instability in Protoreaster

nodosus (Chocolate Chip Sea Star) as a biomarker for environmental

stress

D.J. V. Trono, R. Dacar, L. Quinones, S. R. M. Tabugo

Department of Biological Sciences, College of Science and Mathematics, Mindanao State University-Iligan Institute of

Technology, Philippines


Received 12 December 2014; Accepted 20 January 2015; Published online 1 June 2015

Abstract

Fluctuating asymmetry (FA), pertains to small and random departures from perfect symmetry of an organism’s

bilateral traits and has been used as a measurement of developmental instability and as a potential indicator of

stress in populations. It measures the variations from symmetry of a symmetrical structure whose sides are said

to be genetically identical, with similar history of gene activity and experiencing the same environment.

Symmetries are potentially the basis for studies on FA. Hence, this study assessed the potential of FA as a

reliable developmental instability and environmental stress indicator in five-fold dihedral symmetrical

Protoreaster nodosus (Chocolate chip sea fish) from three (3) different sites (Linamon, Lanao del Norte; Initao,

Misamis Oriental and Jasaan, Misamis Oriental). FA for each population from every site was measured for

comparison. In this study, anatomical landmarks were subjected to Procrustes superimposition and Principal

Component Analysis (PCA) using “Symmetry and Asymmetry in Geometric Data” (SAGE) program. Results

showed highly significant FA and significant DA for population from Jasaan and Linamon where habitat

disturbance due to anthropogenic activities were prevalent. Thus, experienced more stress compared to the

other populations, suggesting that significant variation in size or left-right side of each individual could be a

product of genotype-environment interaction. Moreover, insignificant FA and high DA was obtained from

Initao (protected seascape area) which indicated that variation among individual genotypes and asymmetry in

phenotypes is mostly induced by genetics under less stressful environment. Significant FA and increase FA

present inability of species to buffer stress in its developmental pathways and have implications on species

fitness. Hypothesis assumes that fluctuating asymmetry has costs, reflects the quality of individuals and the

level of genetic and environmental stress experienced by individuals or populations during development. Here,

FA proved to be efficient when applied to five-fold dihedral symmetrical organisms.

Keywords fluctuating asymmetry; developmental instability; biomarker; Protoreaster nodosus; environmental

stress.

Computational Ecology and Software ISSN 2220721X URL: http://www.iaees.org/publications/journals/ces/onlineversion.asp RSS: http://www.iaees.org/publications/journals/ces/rss.xml Email: [email protected] EditorinChief: WenJun Zhang Publisher: International Academy of Ecology and Environmental Sciences


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1 Introduction

Monitoring the impacts of a wide range of environmental stressors on ecosystem health is of interest to

conservation biology and sustainability (Beasley et al., 2013). Stressors of interest include pollutants, changes

in natural environmental parameters such as temperature, humidity, density, and shifts in resource use induced

by global warming, habitat defragmentation and habitat loss, that often leads to detrimental consequences or

loss of inhabitant organisms (Whiteman and Loganathan, 2001). Thus, there is a need to find reliable and

suitable early-warning biological indicators of such stress for conservation programmes.

Along this line, in marine environment sea stars play an ecological role as keystone species (Paine et al.,

1985). Wherein, they are not necessarily abundant in the marine community however, they exert strong control

on community structure not by numerical might but according to their pivotal ecological roles or niches.

Ecological data can be utilized from ecologically important species of sea stars (e.g. Protoreaster nodosus and

Pisaster ochraceus) because they have a long life span, with a maximum of 34 years (Menge, 1975; Bos et al.,

2008). Hence, are suitable for long term monitoring. With this, is the incessant search for easily measured

biomarkers which resulted in the investigation of asymmetry of morphological characters as a possible

biomarker for stress and the most widely used measure of asymmetry is Fluctuating Asymmetry (FA).

Symmetry is a major trait of life and it has been suggested that more symmetrical individuals have higher

developmental stability (DS), reproductive success and survival rate. Developmental stability is defined as the

ability of an organism to moderate its development against genetic or environmental conditions and produce

the genetically determined phenotype (Daloso, 2014; Galbo and Tabugo, 2014; Carpentero and Tabugo, 2014

and Trotta et al., 2005). DS has been used to monitor the effects of anthropogenic and natural factors of stress

on living organisms (Albarra´n-Lara, 2010). Under normal conditions, development follows a genetically

determined pathway, and minor perturbations are controlled by developmental stability mechanisms. Under

stressful conditions (e.g., increased toxicants) and tolerance limits have exceeded, the stress leads to

developmental instability such that development cannot be restored to the original pathway causing increase

phenotypic variations of the organism, reduce the homeostasis of a biological system, or generate symmetry

deviations in an organ or an organism’s relative symmetry. In this context, stress identified at morphological

level generally means that the physiological and molecular plasticity were unable to buffer the stress

(Whiteman and Loganathan, 2001). The most common tool for measuring DS is FA. Generally, FA is defined

as fine and random deviations from perfect symmetry of organism’s morphology. Also, considered a reliable

factor for measuring developmental stability because it reflects both genetic and environmental stress and this

has been an important theory in evolutionary biology for decades (Palmer, 1994). In this respect, it is perceived

that FA measures the capacity of the organism to buffer its developmental pathways against any

environmentally derived and genetic stressors. It is believed that the presence of either of the said stressors

during ontogeny may impair the effectiveness of these buffering mechanisms. This may affect normal

developmental process and could be manifested as increase levels of FA of an otherwise bilaterally

symmetrical character on organisms (Mpho et al., 2000). In this context, it is perceived that there is a direct

relationship between FA and developmental instability (Graham et al., 2010).

In this regard, this study was done to investigate the potential of FA as a biomarker of environmental

stress and determine developmental instability in the sea star (Protoreaster nodosus). It assessed the difference

in the FA indexes of Protoreaster nodosus from three (3) different sites (Linamon, Lanao del Norte; Initao,

Misamis Oriental and Jasaan, Misamis Oriental) portraying different environmental conditions and determine

the possibility of FA as a tool to determine ecological stress and its efficiency when applied to five-fold

dihedral symmetrical organisms. Fluctuating asymmetry is also a useful potential indicator of an organism’s

health and welfare.

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2 Materials and Methods

2.1 Study sites and specimen collection

Three study sites were surveyed within the coast of Northern Mindanao region: Bobotan, Initao; Linamon, and

Aplaya, Jasaan. The sites chosen differ with their proximity to human settlements and predicted to differ in

terms of environmental conditions. Bobotan, Initao is a protected seascape under Initao-Libertad Protected

Landscape and Seascape declared last 2002. There were more less 15 households residing near the area which

take part of the local government unit (LGU) activities in protecting and preserving the area. Linamon and

Aplaya were chosen as sites which differ in anthropogenic disturbance. Survey and sampling procedures were

done during low tide. Samples were collected in a 30 meter by 2 meter line transect (10 m by 2 m line transect

per sampling replicate) running parallel with the shore and placed randomly at each site within the low

intertidal zone (see Fig. 1).

2.2 Physico-chemical parameters

Measurement of ecological variables could serve as indicator for pollution or water disturbance in a particular

community. The following variables were recorded at each site: water and air temperature, hydrogen ion

concentration (pH) and salinity. These parameters were measured in situ conducted by triplicate in the 30

meter by 2 meter line transect.

2.3 Digital imaging preparation and measurement of Fluctuating Asymmetry (FA)

Thirty (30) individuals of P. nodosus were photographed for each site using a standard procedure. Samples

were carefully removed from the water, photographed and then returned to their original nest such that no

animal was harmed during the process. The digital images of the sea stars were processed and landmarked

assignment was done using tpsUti1 and tpsDig2 softwares. Landmarking was done in triplicates to quantify

and minimize measurement error. For morphometric analysis, forty-one (41) landmark points were assigned

for each individual. Arm opposite to the madreporite was designated as Arm 1, and the others follow

clockwise successively in aboral view (see Fig. 2) based on Ji et al., (2012) study. Fig. 3 shows the location of

the landmarks used in the sea star. Descriptions of identified landmarks are presented in Table 1.

Fig. 1 Transect line used for the collection of P. nodosus.

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Fig. 2 Aboral view of P. nodosus showing the anterior arm, opposite to the madreporite, which serves as Arm 1 and other arms follow clockwise successively.

Fig. 3 Location of forty-one (41) landmarks in P. nodosus sea star used for fluctuating asymmetry (FA) analysis.

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Table 1 Description of assigned landmarks on P. nodosus sea star.

Landmark

points Anatomical Description

Landmark

points Anatomical Description

1 Eyespot of anterior arm

(Arm 1) 30, 34 Distal end of Arm 3

2,3 Distal end of anterior arm 32 Eyespot of Arm 3

4,5 Middle of anterior arm 40 Interambulacrum between bivium Arms

6,7 Proximal end of anterior arm 27, 39 Proximal end of Arm 4

8 Interambulacrum between Arm 1

and Arm 2 29, 37 Middle of Arm 4


and Arm 5 31, 35 Distal end of Arm 4

10, 22 Proximal end of Arm 2 33 Eyespot of Arm 4

12, 20 Middle of Arm 2 25 Interambulacrum between Arm 4 and 5

14, 18 Distal end of Arm 2 11, 23 Promixal end of Arm 5

16 Topmost/ eyespot of Arm 2 13, 21 Middle region of Arm 5


and 3 15, 19 Distal end of Arm 5

26, 38 Proximal end of Arm 3 17 Eyespot of Arm 5

28, 36 Middle of Arm 3 41 Center of central disk

Individual levels of FA were obtained using the SAGE (Symmetry and Asymmetry in Geometric Data)

program. This software analysed the x- and y-coordinates, using a configuration protocol that divided both

sides of the sea star body. Object symmetry was applied in this case as sea stars have five-fold dihedral

symmetry. The FA theory has mostly been applied to bilaterally symmetrical organisms with only a few

published studies on organisms having five-fold dihedral symmetry (most echinoderms including starfish).

Herewith, breaking dihedral symmetry produces a bilaterally symmetrical object having just one reflective axis

of symmetry.

Procrustes methods were used to analyze shape by superimposing configurations of landmarks into two

or more specimens to achieve an overall best fit (see Fig. 4). The squared average of Procrustes distances for

all specimens is the individual contribution to the FA component of variation within a sample. To detect the

components of variances and deviations, a two-way, mixed model ANOVA with three replicates was used.

(Marquez, 2006; Klingenberg et al., 1998).

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The effect called ‘sides’ is the variation between the two sides; it is a measure of directional asymmetry.

The effect called ‘individuals’ is the variation among individual genotypes (size and shape variation). The

individual’s mean square is a measure of total phenotypic variation and it is random. The ‘individual by sides’

interaction is the failure of the effect of individuals to be the same from side to side; it is a measure of

fluctuating asymmetry and anti-symmetry; variations could be dependent to both environmental and genetic

conditions (Graham et al., 2010).

3 Results and Discussion

There were many assumptions behind fluctuating asymmetry (FA) and developmental stability (DS) and the

nature of the factors behind developmentally unstable phenotypes is not yet well understood. However, it is

noted that developmental stability is the situation wherein an organism has adequately buffered itself against

epigenetic disturbances hence, displaying its developmentally programmed phenotype. At the instance, that an

organism fails to buffer such disturbances, it may display signs of developmental instability. The origin of the

disturbance is assumed to be genetic, environmental or the product of genotype-environment interaction

(Markow, 1995). Fluctuating asymmetry (FA), pertains to small and random departures from perfect symmetry

of an organism’s bilateral traits and has been used as a measurement of developmental instability and as a

potential indicator of stress in populations. Thus, investigating the link between FA and DS in ecologically

important natural populations of sea stars shed light on the quest for morphological characters as a possible

biomarker for stress and knowledge on gene-environment interaction (Daloso, 2014; Galbo and Tabugo, 2014;

Trotta et al., 2005).

Herewith, Table 2 shows the Procrustes two-way, mixed model ANOVA table with expected mean

squares. The effect called “individuals” is the variation among individual animals and can be interpreted as a

size/shape variation; the “individuals” mean square is a measure of total phenotypic variation and is random.

The effect called “sides” is the variation between the two sides; it is a measure of directional asymmetry (DA).

The “individual by side interaction” is the failure of the effect of individuals to be the same from side to side; it

is a measure of FA and antisymmetry. It is a mixed effect. The error term is the “measurement error”; it is a

random effect (Parés-Casanova and Kucherova, 2013).

Fig. 4 Procrustes fitted image of P. nodosus done by SAGE software.

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Results show not significant FA value for Initao site (protected seascape) yet highly significant DA value,

which indicated that variation among individual genotypes and asymmetry in phenotypes is mostly induced by

genetics under less stressful environment. Meanwhile, the F values of “individual x sides” suggested highly

significant FA for the other two sites, Linamon and Aplaya, Jasaan as indicated by low mean square value of

measurement error compared to the individual by sides mean square values. Chocolate chip sea star

populations in Linamon and Aplaya, Jasaan have also shown significant scores on “individual” and “side”

effects. The effect called “sides” which refer to the variation between the two sides, a measure of directional

asymmetry (DA) were significant for both populations and were of the same level. A high FA and significant

DA leads to generation of phenotypes interacting with the perturbed ambient. Thus, may indicate interplay of

genotype and environment under more stressful environment. Noteworthy, was that Linamon and Aplaya,

Jasaan sampling sites displayed some level of environmental disturbance based on the ocular site inspection.

Anthropogenic disturbance were prevalent in Linamon sampling site due to human settlements along the shore

such that sewage and canal run offs go directly to the bay and various litters (e.g. plastics, diapers) were often

found scattered in the shoreline. While, Jasaan sampling area was situated near two industrial plants,

Philippine Sinter Corporation (PSC) and Pilipinas Kao, Inc., that produce sintered ore and biodegradable

chemicals, respectively. Yet, there were no official report on heavy metals or toxic contamination in the site.

Results coincide with the study of Utayopas (2001) on Trichopsi vittatus (croaking gourami) with highest

mean asymmetries were detected from the most polluted site in almost all characters. This suggests that

significant variation in size or left-right side of each individual could be a product of genotype-environment

interaction. Thus, P. nodosus individuals in these areas may have developmental instability during ontogeny

which could be due to exposure to environmental or genetic stressors. The individual’s inability to buffer the

stress leads to deviation in its relative symmetry. In this context, it is perceived that there is a direct

relationship between FA and developmental instability (Graham et al., 2010).

Table 2 Procrustes two-way, mixed model anova results of body symmetry of p. Nodosus.

*Significant, P< 0.05.

Effect/Site SS dF MS P F

Linamon

Individuals 0.13662 1131 0.0001208 0 2.4497******

Sides 0.030035 39 0.00077013 0 15.6182******

Individuals x sides 0.055769 1131 4.9309e-005 0.0020071 1.1418***

Measurement error 0.20211 4680 4.3186e-005 --

Initao

Individuals 0.14522 1131 0.0001284 0 2.4965******

Sides 0.049487 39 0.0012689 0 24.6727******

Individuals x sides 0.058167 1131 5.143e-005 0.76773 0.96575

Measurement error 0.2492 4680 5.3253e-005 --

Jasaan

Individuals 0.18584 1131 0.00016432 8.6159e-009 1.3995******

Sides 0.069279 39 0.0017764 0 15.1301******

Individuals x sides 0.13279 1131 0.00011741 0 2.285******

Measurement error 0.24046 4680 5.1381e-005

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In addition, principal component analysis from Procustes analysis may serve as reliable tool in

visualization of variations in landmarks (Galbo and Tabugo, 2014). The percentage values of PCA represent

the total variation in FA (see Table 3 and Fig. 5). Based on the results exhibited by PC 1 and PC2, population

from Initao express less variation compared to Linamon and Aplaya, Jasaan. Reversely, higher FA was

exhibited by the individuals in Linamon and Aplaya, Jasaan compared to Initao. Generally, PC 1 accounts for

most of the variation. Highest variation (PC 1= 70%) is exhibited in Jasaan followed by Linamon (PC 1= 37%).

Thus, explaining the symmetry deviation observed. This could have been attributed by both genetic and,

largely, of environmental stressors.

Table 3 Variance explained by first two principal components of Protoreaster nodosus in three sites.

Moreover, using Canonical Correspondence Analysis (CCA) physico-chemical parameters of each

sampling sites were correlated with PC values. Results revealed that individual variation in each samples were

more likely related to pH stress in P. nodosus (Fig. 6). Study of Dupont et al. (2008) show that at low pH

larvae of the ecological keystone brittlestar, Ophiothrix fragilis, either were abnormal, had altered skeletal

proportions and asymmetry during skeletogenesis and there was a delay in development. The exposure of

larvae to elevated CO2 (high pH) treatment takes longer to reach the same developmental stage. Herewith, FA

has costs and reflects the degree of environmental stress, health and quality of individuals.

Sites PC 1 (%) PC2 (%) Overall (%)

Linamon 37.19 19.81 57

Initao 34.14 19.28 53.42

Jasaan 69.66 11.35 81.01

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(a) Linamon PC1 (37.19%) Linamon PC2 (19.81%)

(b) Initao PC1 (34.14%) Initao PC2 (19.28%)

(c) Jasaan PC1 (69.66%) Jasaan PC2 (11.35%) Fig. 5 Deformation grid of fluctuating asymmetry for PC1 and PC2 of (a) Linamon, (b) Initao, and (c) Jasaan P. nodosus

populations.

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4 Conclusion

Under stressful conditions, the genotypes may show some deviations in their perfect bilateral symmetry, which

can be identified through the analysis of fluctuating asymmetry (FA). Results show that sampling areas which

are environmentally disturbed (Linamon and Jasaan) due to various anthropogenic activities such as industrial

or residential pollutants exhibited high FA and significant DS. This suggests that significant variation in size or

left-right side of each individual could be a product of genotype-environment interaction. Meanwhile, Initao

(protected seascape) population have exhibited insignificant FA level and high DS, such that variation among

individual genotypes and asymmetry in phenotypes is mostly induced by genetics under less stressful

environment. Hence, P. nodosus found in Linamon and Aplaya, Jasaan could be considered as

developmentally unstable and its inability to buffer the environmental and genetic stressors above tolerance

limit have led to deviation of its relative symmetry. It is perceived that there is a direct relationship between

FA and developmental instability. Thus, the study demonstrates the potential of FA as a biomarker for

environmental stress in five-fold dihedral symmetry of sea stars and a tool in detecting developmental

instability. Moreover, this tool should be applied to other similar organisms and wide range of physico-

chemical parameter should be included to fully assess the health of a certain habitat.

Acknowledgment

The authors would like to thank their families and DOST.

References

Albarrán-Lara AL, Mendoza-Cuenca L, Valencia-Ávalos S, González-Rodríguez A, Oyama K. 2010. Leaf

fluctuating asymmetry increases with hybridization and introgression between Quercus magnoliifolia and

Quercus resinosa (Fagaceae) along an altitudinal gradient in Mexico. International Journal of Plant

Fig. 6 Canonical Correspondence Analysis (CCA) of PCA scores and physico-chemical parameters.

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IAEES www.iaees.org

Sciences, 171(3): 310-322

Beasley DA, Bonisoli-Alquati A, Mousseau T. 2013. The use of fluctuating asymmetry as a measure of

environmentally induced developmental instability: A meta-analysis. Ecological Indicators, 30: 218-226

Bos AR, Gumanao GS, Alipoyo JCE, Cardona LT. 2008. Population dynamics, reproduction and growth of the

Indo-Pacific horned sea star, Protoreaster nodosus (Echinodermata; Asteroidea). Marine Biology, 156:55-

63

Carpentero ER, Tabugo SRM. 2014. Determining developmental instability via fluctuating asymmetry in the

shell shape of Arctica islandica Linn. 1767 (ocean quahog). European Journal of Zoological Research,

3(3): 1-7

Daloso DM. 2014. The ecological context of bilateral symmetry of organ and organisms. Natural Science, 6(4):

184-190

Dupont S, Havenhand J, Thorndyke W, Peck L, Thorndyke M. 2008. Near-future level of CO2-driven ocean

acidification radically affects larval survival and development in the brittlestar Ophiothrix fragilis. Marine

Ecology Progress Series, 373: 285-294

Galbo K, Tabugo SRM. 2014. Fluctuating asymmetry in the wings of Culex quinquefasciatus (Say) (Diptera:

Culicidae) from selected barangays in Iligan City, Philippines. AACL Bioflux, 7(5): 357-264

Graham JH, Raz S, Hel-Or H, Nevo E. 2010. Fluctuating asymmetry: Methods, theory, and applications.

Symmetry, 2(2): 466-540

Ji C, Wu L, Zhao W, Wang S, Lv J. 2012. Echinoderms have bilateral tendencies. PLoS ONE 7(1): e28978

Klingenberg CP, McIntyre GS, Zaklan SD. 1998. Left-right asymmetry of fly wings and the evolution of body

axes. Proceedings of the Royal Society of London B, Biological Sciences, 265: 1255-1259

Markow T. 1995. Evolutionary ecology and developmental instability. Annual Review of Entomology, 40:

105-120

Marquez E. 2006. Sage: symmetry and asymmetry in geometric data. Ver 1.0.

http://www.personal.umich.edu/~emarquez/morph/

Menge BA. 1975. Brood or broadcast? The adaptive significance of different reproductive strategies in the two

intertidal sea stars Leptasterias hexactis and Pisaster ochraceus. Marine Biology, 31: 87-100

Mpho M, Holloway GJ, Callaghan A. 2000. The effect of larval density on life history and wing asymmetry in

the mosquito Culex pipiens. Bulletin of Entomological Research, 90: 279-283

Paine RT, Castillo JC, Cancino J. 1985. Perturbation and recovery patterns of starfish-dominated intertidal

assemblages in Chile, New Zealand and Washington State. American Naturalist, 125: 679-691

Palmer AR. 1994. Fluctuating asymmetry analysis: a primer. In: Developmental Instability: Its Origins and

Evolutionary Implications (Markow TA, ed). Kluwer Academic, London, UK

Parés-Casanova PM, Kucherova I. 2013. Horn asymmetry in a local goat population. International Journal of

Research in Agriculture and Food Sciences, 1(2): 12-17

Trotta V, Corrado F, Calboli F, Garoia F, et al. 2005. Fluctuating asymmetry as a measure of ecological stress

in Drosophila melanogaster (Diptera: Drosophilidae). European Journal of Entomology, 102: 195-200

Utayopas P, 2001. Fluctuating Asymmetry in Fishes Inhabiting Potluted and Unpolluted Bodies of Water in

Thailand. Thammasat International Journal of Science and Technology, 6(2): 10-20

Vishalakshi C, Singh BN. 2008. Effect of environmental stress on fluctuating asymmetry in certain

morphological traits in Drosophila ananassae: nutrition and larval crowding Canadian Journal of

Zoology, 86(5): 427-437

Whiteman HH, Loganathan BG. 2001. Developmental stability in amphibians as a biological indicator of

chemical contamination and other environmental stressors. Kentucky EPA/EPSCoR, USA

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IAEES www.iaees.org

Article

Distinguishing niche and neutral processes: Issues in variation

partitioning statistical methods and further perspectives

Youhua Chen Department of Renewable Resources, University of Alberta, Edmonton, T6G 2H1, Canada


Received 30 September 2014; Accepted 6 November 2014; Published online 1 June 2015

Abstract

Variance partitioning methods, which are built upon multivariate statistics, have been widely applied in

different taxa and habitats in community ecology. Here, I performed a literature review on the development

and application of the methods, and then discussed the limitation of available methods and the difficulties

involved in sampling schemes. The central goal of the work is then to propose some potential practical

methods that might help to overcome different issues of traditional least-square-based regression modeling. A

variety of regression models has been considered for comparison. In initial simulations, I identified that

generalized additive model (GAM) has the highest accuracy to predict variation components. Therefore, I

argued that other advanced regression techniques, including the GAM and related models, could be utilized in

variation partitioning for better quantifying the aggregation scenarios of species distribution.

Keywords multivariate ordination; regression models; general additive models; dispersal limitation;

environmental filtering.

1 Introduction

1.1 Variation decomposition in community ecology

It is quite often not only one process regulating and determining community structure. Typically, the

combination of multiple processes and their interactions will have profound impacts on resultant community

structure. So, it is natural to ask which kinds of processes are dominant, while others are auxiliary. Thus, the

variance in response variables can be separated into several parts, and by employing statistical methods, we

can identify the contribution and relative importance of different ecological mechanisms.

Fig. 1 depicts the methods for performing variation decomposition at different data levels. The methods

range from simple linear regression, to multiple regression models, to multivariate regression model, and other

multivariate statistical methods.



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Fig. 1 Different methods, ranging from a simple regression to multivariate statistics, have been used for variation partitioning to distinguish environmental and spatial filtering. Solid arrow denotes that the method can be used for performing variation partitioning tests, while dashed arrow denotes that that method (pointed by the head of arrow) can be generalized or deduced from another method.

1.2 How is variation partitioning related to the debate between niche and neutral processes?

Two mechanisms significantly affect species diversity patterns: neutral and deterministic factors.

Environmental descriptors, defining the niche of species, are deterministic; while spatial descriptors, defining

the dispersal ability of species, are neutral.

Since Hubbell’s neutral theory (Hubbell, 2001), a great amount of works tried to predict the power of

neutral theory in empirical data. However, most of them failed to support neutral theory (McGill, 2003; Gotelli

and McGill, 2006). There are several ways to test neutral theory. One is to generate individual predictions and

test them by regression based on neutral theory. For example, the distance decay of species composition

(Gilbert and Lechowicz, 2004); the priority effect of juvenile co-occurrence reduction (Gilbert et al., 2008).

Since the repellence of pure neutral theory in empirical test, the mainstream nowadays is to combine both

niche and neutrality to explain community structure, and test the relative importance of each component.

Therefore, variation partitioning is introduced (Borcard et al., 1992) to detect the contribution from each part

of variations. In a work of Tuomisto and Ruokolainen (Tuomisto and Ruokolainen, 2006), they suggested that

dispersal limitation derived from neutral process can only be tested by using distance-based methods.

1.3 How to perform variation partitioning?

To character species composition and environmental variables is a major topic in current ecological research.

Redundancy analysis (suited for linear relationships between species composition and environmental variables)

and Canonical correspondence analysis (handling nonlinear species-environment relationship) are the two

widely used methods to investigate the relationship of environmental variables and species diversity

information (Fig. 1). Variation partitioning can be used to test and determine the possibilities of individual

predictors in influencing species distribution and abundance (Peres-Neto et al., 2006).

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Variation partitioning can be divided into four parts: pure environmental variation, pure space variation,

mixed environmental and space variation, and unexplained variation (Borcard et al., 1992). Fig. 2 showed the

schematic map of the variation components which were often complied in previous literature.

Fig. 2 Schematic map showing different components and fractions that are related to variation partitioning. a-pure environmental variation; b-mixed environmental and space variation; c-pure space variation; d-unexplained variation. a+b+c+d=total variance involved in the community data.

In principle, I use the spatial coordinates as the basic spatial descriptors. I can use the eigenvectors derived

from the principal components of spatial coordinates, which has been used in some previous works (Dray et al.,

2006). Or, I can get the Moran’s eigenvector maps (Dray et al., 2006; Sattler et al., 2010), which is a general

form of principal coordinates of neighbour matrices (Borcard and Legendre, 2002). Both methods use the

eigenfunctions of spatial connectivity matrices, thus they are scale-independent.

When setting spatial descriptors as covariables, I can know the proportion of pure environmental variation.

In contrast, when setting environmental variables as covariables, I can deduce the proportion of pure space

variation. The mixed environmental and spatial variation can be derived from the subtraction of total known

variation-pure environmental variation-pure spatial variation.

It is quite simple to perform comparative studies in ecological data by implementing variation partitioning

in regression results. Basically, all kinds of statistical software and tools can implement variation partitioning,

as long as they can perform multiple regression analysis. As I stated above, it needs only three times of

running regressions, each of which should comprise environmental variables only, spatial variables only and

both spatial and environmental variables together. From a convenience perspective, there is a commercial

statistical software called Canoco (ter Braak, 1986), which is designed for constrained community ordination

analysis. Variation partitioning can be implemented using partial CCA method in the package. Besides that, I

can perform variation partitioning by using some open-source packages in R software. For example, the

command “varpart” can be recalled to perform variation partitioning using “vegan” package (Oksanen et al.,

2012).

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1.4 Literature searching and study classification

To search the available publications relevant to separate the effects of niche processes and neutrality, I used the

following databases: Google Scholar, Web of Science, Springlink, Wiley-Blackwell, and Elsevier publishers.

The keywords used for query include “variation partition”, “redundancy analysis”, “niche and neutral

processes”, “dispersal limitation and habitat filtering”, “spatial and environmental descriptors”.

Based on the query results, I summarized two basic categories for the subject, which are 1) theoretical and

methodological development; and 2) applications on different taxa and habitats. Table 1 summarized all the

relevant work on the subject.

Table 1 A literature review on the methodology and applications of variation partitioning in community ecology.

Sub-discipline Description Literature

1,

Methodological

background and

development

Advocating or

criticizing the

variance

partitioning and

developing

relevant tools

1, the original paper describing variation partitioning:

Borcard et al. (Borcard et al., 1992)

2, comparison of different methods on performing

variation partitioning. For example, Mantel test, multiple

regression model, canonical correspondence analysis and

so on (Legendre et al., 2005, 2008)

3, ecological questions that can be addressed by variation

partitioning: beta diversity (Legendre et al., 2005, 2009);

neutrality versus niche (Smith and Lundholm, 2010;

Tuomisto et al., 2012);

4, rebuttal to variation partitioning methods and relevant

technical aspects:

Tuomisto and Ruokolainen (Tuomisto and Ruokolainen,

2006) suggested that distance method (like Mantel test)

should be the only choice to test neutral hypothesis.

Gilbert and Bennett (Gilbert and Bennett, 2010) found that

traditional variation partitioning methods have a very

restricted power to correctly quantify each part of

variances involved in the simulated data.

Diniz-Filho et al. (Diniz-Filho et al., 2012) suggested that

spatial autocorrelation test (Moran’s I index) can be linked

to niche and neutrality partitioning.

2, Applications

on various

taxonomic

hierarchy and

spatial scales

Applying

variation

partitioning to

different

community

assemblages

across different

ecosystems,

areas and taxa.

1, terrestrial ecosystems:

Oribatid mites (Borcard and Legendre, 1994; Lindo and

Winchester, 2009);

Forest birds (Cushman and McGarigal, 2002; Pearman,

2002);

Pteridophyte plants (Jones et al., 2008)

2, marine and aquatic ecosystems:

Stream fish community (Steward-Koster et al., 2007);

Pelagic fish assemblages (Peltonen et al., 2007)

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From the table, it seemed that most of applications of variation partitioning are on terrestrial ecosystems,

and the studied taxa varied from birds to plants. Interestingly, there are only a few relevant works on marine

and aquatic ecosystems, with only focus on fish assemblages.

1.5 A review of current methods and possible problems

There are many possible methods by applying nonlinear regression techniques to reveal the correlation of

environmental and spatial variables against species distribution-composition matrix. The possibilities of

introducing advanced nonlinear regression model, including local regression methods, general additive models

and least partial square methods, can be beneficial to overcome the challenges.

Gilbert and Bennett (Gilbert and Bennett, 2010) performed a simulation comparison for analyzing the

powers and differences among a variety of variation partitioning methods, most of which are widely used.

Typically, the most prevailing methods are the regression on distance matrices (e.g., Mantel test), canonical

correspondence analysis and redundancy analysis. Moreover, it is suggested to better retain spatial information

by using some kinds of transformation called principal components of neighbour matrices (or Moran’s

eigenvector maps).

Despite their wide applications, Gilbert and Bennett (Gilbert and Bennett, 2010) found out that all kinds of

tools have greatly underestimated each part of variances. For example, they found out that canonical ordination

under-fitted the environmental variation, which was simulated in a high amount.

1.6 Potential methods

As found, the under-fit problem of different variation components by traditional ordination methods is largely

due to the disability of traditional regression models. This is because all the available methods are built on the

basis of least-square estimation of regression coefficients. All the conventional methods have the implicit

assumption of homoscedasticity involved in the dataset more or less. Thus, as long as the data were composed

of inherent heteroscedasticity, the power of least-square regression was questioned. Simple linear or nonlinear

(e.g., polynomial regression and general linear models) fitting will be not possible to remove the impact of

shifting data variance in the data set.

Fig. 3 showed the impacts of heteroscedasticity are hard to remove when plotting fitted residuals after

conventional regression models. Thus, it sounds that a promising method to overcome the under-fit problem

identified by Gilbert and Bennett (Gilbert and Bennett, 2010) is to adopt advanced regression models. Hence,

in the following section of our synthesis is to propose advanced regression tools.

1.7 General additive model (GAM) and relevant nonlinear smoothing methods

The regression models in this category are of course nonlinear, however, another important feature is that they

employed completely different ways aiming to solve the problem of heteroscedasticity. General additive model

(GAM) typically has the power to remove the problem of variance heterogeneity, with the cost of difficult

ecological explanation.

Here I generated a simple relationship between species abundance and one environmental variable with

increasing variance across the landscape. Then, I applied different regression models to fit this heterogeneous

variance case. The resultant residuals after fitting were as showed in Fig. 3, the simple linear model have only

R2-adjusted=0.4171, in contrast, GAM returned an R2-adjusted= 0.542. Polynomial regression model is no

more than a simple linear model, with R2-adjusted=0.417. Moreover, when checking the regression residuals, I

can find out that resultant residuals have no heterogeneity. In contrast, residuals from linear models (the same

applied to quadratic nonlinear models, but not showed here) still have variance heterogeneity, and the case

becomes worse at both lateral sides of the points.

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0 20 40 60 80 100

-20

-10

01

02

0

Linear model

Re

sid

ual

s

0 20 40 60 80 100

-10

01

02

0

Additive model

Re

sid

ual

s

Fig. 3 The problem of variance heterogeneity and the effectiveness of additive models compared to linear models to remove variance heterogeneity (left figure is the residuals fitted by simple linear model, while the residuals derived from general additive fitting with Gaussian family is showed in the right figure). It is clear a humped pattern (non-homogenous residuals) still occurs in the left graph.

2 Implications and Further Perspectives

2.1 Spatial heterogeneity and distributional aggregation may reduce the power

Typically geographic coordinates are our only choices to measure spatial patterns and drivers of the

community structure. However, sometimes it is hard to extract enough spatial information from simple

geographic coordinates. In the case of multi-dimensional folding and transformation, Euclidean distances of

geographic locations might not be sufficient to capture the variation caused by spatial distances. For example,

species distributions typically show the aggregated, rather than random, patterns across different taxa. The

driving reasons are usually dispersal-limited colonization and the constraint of habitat heterogeneity, and also

biotic interactions, e.g., inter- and intra- specific competition.

2.2 Moran’s spatial scales and edge effects

Sampling of different variables at different locations and scales might typically encounter the scale problem.

The inconsistence of scales for different variables may lead to bias prediction on disentangling niche and

neutrality processes. In such a case, the relative contribution of neutrality and niche processes driving the

community structure may be misleading. Fig. 4 (upper graph) illustrated the scale issue when doing sampling

in fields. Insufficient sampling across the region can give us a rough estimation of spatial gradient, but which

is largely departed from the true gradient caused by middle-degree Moran’s process. In this case, spatial

variation should be overestimated. This issue can happen of course for environmental variables as well.

Edge effects may also inflate the possible separation of environmental filtering and spatial limitation. As

showed in the same Fig. 4 (lower graph), if the sampling effect is focused on the transitional boundary areas of

an environmental variable, the resultant explanation can be that the signaling of environmental filtering is not

strong. It is easy to avoid the edge problem for one environmental variable. However, for the case of multiple

variables, as it should be hard to predict their transitional boundaries, the sampling scheme can be always

coupled with edge effects.

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Fig. 4 Sampling biases caused by Moran’s spatial scales and edge effects. Two rectangular areas indicate two distinct levels of an environmental variable (e.g., precipitation, elevation, temperature and so on). Square and triangle points with red and blue colors indicated two species. Ellipse circles represent density of species population. Transparent gray squares represent sampling plots across the region. In principle, the community bounded by the large square is structured by environmental filtering. The upper graph illustrates the insufficient sampling case, which make the wrong conclusion that environmental filtering is not important to capture the beta-diversity. The lower graph illustrate the edge sampling case, which make the wrong conclusion that spatial filtering is much more important than environmental filtering to structure the community.

At another side, as known that, both processes can have similar predictions on many facets of community

structures. For example, niche process can generate the same distance-decaying pattern as that of neutrality

process. In the case like that, typically the resultant community structure is co-dominated by both mechanisms

and hard to separate without additional information about the community. Thus, it might be not effective to use

partial regression techniques to separate niche and neutrality processes.

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2.3 Variable selection process

Maybe variable selection is an improved way to better capture the most correlated variation information for

spatial and environmental drivers. In tradition, backward or forward variable selection procedure is applied to

choose the optimal subsets of variables. In the case of variation partitioning, I can do the variable selection

separately, then obtain the most correlated spatial and environmental predictors to perform variation

decomposition. This method did provide the most significant variables, but a small change of variable subset

will lead to a great amount of changes in the resulting variation. Thus, the discrete process may reduce the

prediction accuracy when new variables are included (Tibshirani, 1996). Fortunately, I have other advanced

model selection methods, for example, Lasso and ridge regressions can be good alternatives for choosing good

candidates in variation partitioning.

2.4 An integrative way to partition and understand ecological communities

The explanation is the most challenging thing for general additive models, although it has a higher appealing

prediction power than linear models. The comprising manner is to use simple linear or nonlinear regression to

fit the data when obtaining the possible curve pattern inspired by GAM models. Then, combining the fitted

adjusted R2 from GAM and fitted line from either linear or simple nonlinear regression models, I can clearly

and easily explain the possible relationships and mechanisms that might dominate the community data.

References

Borcard D, Legendre P. 2002. All-scale spatial analysis of ecological data by means of principal coordinates of

neighbor matrices. Ecological Modelling, 153: 51-68

Borcard D, Legendre P. 1994. Environmental control and spatial structure in ecological communities: an

example using oribatid mites (Acari, Oribatei). Environmental and Ecological Statistics, 1: 37-61

Borcard D, Legendre P, Drapeau P. 1992. Partialling out the Spatial Component of Ecological Variation.

Ecology, 73: 1045

Ter Braak, CJF. 1986. Canonical correspondence analysis : a new eigenvector technique for multivariate direct

gradient analysis. Ecology, 67: 1167-1179

Cushman S, McGarigal K. 2002. Hierarchical, multi-scale decomposition ofspecies-environment relationships.

Landscape Ecology, 17: 637-646

Diniz-Filho J, Siqueira T, Padial A, Rangel T, Landeiro V, Bini L. 2012. Spatial atucorrelation analysis allows

disentangling the balance between neutral and niche processes in metacommunities. Oikos, 121: 201-210.

Dray S, Legendre P, Peres-Neto PR. 2006. Spatial modelling: a comprehensive framework for principal

coordinate analysis of neighbour matrices (PCNM). Ecological Modelling, 196: 483-493

Gilbert B, Bennett JR. 2010. Partitioning variation in ecological communities: do the numbers add up? Journal

of Applied Ecology, 47: 1071-1082

Gilbert B, Lechowicz M. 2004. Neutrality, niches, and dispersal in a temperate forest understory. Proceedings

of the National Academy of Sciences, 101: 7651-7656

Gilbert B, Srivastava DS, Kirby KR. 2008. Niche partitioning at multiple scales facilitates coexistence among

mosquito larvae. Oikos, 117: 944-950

Gotelli N, McGill B. 2006. Null versus neutral models: what’s the difference? Ecography, 29: 793-800

Hubbell SP. 2001. The Unified Neutral Theory of Biodiversity and Biogeography (MPB-32) (Monographs in

Population Biology). Princeton University Press, USA

Jones M, Tuomisto H, Borcard D, Legendre P, Clark D, Olivas P. 2008. Explaining variation in tropical plant

community composition: influence of environmental and spatial data quality. Oecologica, 155: 593-604

137


IAEES www.iaees.org

Legendre P, Borcard D, Peres-Neto P. 2005. Analyzing beta diversity: Partitioning the spatial variation of

community composition data. Ecological Monographs, 75: 435-450

Legendre P, Borcard D, Peres-Neto P. 2008. Analyzing or explaining beta diversity : Comment. Ecology, 89:

3238-3244

Legendre P, Mi X, Ren H, Ma K, Yu M, Sun IF, He F. 2009. Partitioning beta diversity in a subtropical broad-

leaved forest of China. Ecology, 90: 663-674

Lindo Z, Winchester NN. 2009. Spatial and environmental factors contributing to patterns in arboreal and

terrestrial oribatid mite diversity across spatial scales. Oecologia, 160: 817-825

McGill B. 2003. A test of the unified neutral theory of biodiversity. Nature, 422: 881-885

Oksanen J, Blanchet F, Kindt R, Legendre P, Minchin P, O’Hara R, Simpson G, Solymos P, Stevens M,

Wagner H. 2012. vegan: Community Ecology Package. R package version 2.0-4. http://CRAN.R-

project.org/package=vegan.

Pearman P. 2002. The scale of community structure: habitat variation and avian guids in tropical forest

understory. Ecological Monographs, 72: 19-39

Peltonen H, Luoto M, Pakkonen J, Karjalainen M, Tuomaala A, Poni J, Viitasalo M. 2007. Pelagic fish

abundance in relation to regional environmental variation in the Gulf of Finland, northern Baltic Sea.

ICES Journal of Marine Science, 64: 487-495

Peres-Neto P, Legendre P, Dray S, Borcard D. 2006. Variation partitioning of species data matrices: estimation

and comparison of fractions. Ecology, 87: 2614-2625

Sattler T, Borcard D, Arlettaz R, Bontadina F, Legendre P, Obrist MK, Moretti M. 2010. Spider, bee, and bird

communities in cities are shaped by environmental control and high stochasticity. Ecology, 91: 3343-

3353

Smith TW, Lundholm JT. 2010. Variation partitioning as a tool to distinguish between niche and neutral

processes. Ecography, 33: 648-655

Steward-Koster B, Kennard M, Harch B, Sheldon F, Arthington A, Pusey B. 2007. Partitioning the variation in

stream fish assemblages within a spatio-temporal hierarchy. Marine and Freshwater Research, 58: 675-

686

Tibshirani R. 1996. Regression shrinkage and selection via the Lasso. Journal of Royal Statistics Society B, 58:

267-288

Tuomisto H, Ruokolainen K. 2006. Analyzing or explaining beta diversity? Understanding the targets of

different methods of analysis. Ecology, 87: 2697-708

Tuomisto H, Ruokolainen L, Ruokolainen K. 2012. Modelling niche and neutral dynamics: on the ecological

interpretation of variation partitioning results. Ecography, 35: 961-971

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Article

Application of homotopy perturbation method to the Navier-Stokes

equations in cylindrical coordinates

H. A. Wahab1, Anwar Jamal1, Saira Bhatti2, Muhammad Naeem3, Muhammad Shahzad1, Sajjad

Hussain1

1Department of Mathematics, Hazara University, Manshera, Pakistan 2Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 3Department of Information Technology, Hazara University, Manshera, Pakistan


Received 30 September 2014; Accepted 8 February 2015; Published online 1 June 2015

Abstract

This paper deals with the approximate analytical solution of the Navier-Stokes equations in cylindrical

coordinates. The homotopy perturbation method is used to get the analytical approximation. Depending upon

different available choices for the linear operator, we also have the advantage to choose different initial

approximations to start our analysis. The analysis is done without calculating the Adomian’s polynomials.

Keywords Navier-Stokes equations; homotopy perturbation method; iterative approximation; infinite series

solution.

1 Introduction

The Navier-Stokes equations describe the motion of fluids that is a substance which can be flow and it arises

from Newton 2nd law applying to the fluid motion (Square, 1952). The Navier-Stokes equations are widely

used in physics, they are used for modeling of weather and seas currents, designing of aircrafts and cars, for

motions of stars, they are used in video games, flow of water in a pipe, blood circulations, analysis of power

stations, and study of populations (Thorpe, 1997).

In fluids mechanics, the dynamics of a flowing fluid is governed and represented by the Navier- Stokes

equations which are nonlinear partial differential equations. Here our case of interest is to approximate the

governing equations of the flow field in a tube, since it is nonlinear in character and it is impossible to solve

these equations analytically to get the exact solution. To solve these equations, we are led to adopt some

restrictive assumptions and some simplifications, which involve the suppositions of weak non linearity to

apply traditional perturbation methods, small parameter assumptions which restrict the wide applications of the



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perturbation techniques, linearization which is certainly a handy task, discretization to apply numerical

techniques etc. In using the traditional numerical methods for the numerical solution of the Navier- Stokes

equations are very difficult and it is due to mixing of different length scales involving in the fluid flow which

results in massive out prints.

Our objective here is to find the continuous analytical solution to the governing equation in cylindrical

coordinates without massive outsprints and restrictive suppositions as discussed above, which change physical

problem into a mathematical problem. K. Haldar (Haldar, 1995) used Adomian’s decomposition method

(Adomian, 1996; Adomian, 1989) for the analytical approximation of the problem which is most transparent

method for the solutions of the Navier-Stokes equation in cylindrical coordinates. However the limitations of

this method involve a handy task of the calculations of the Adomian polynomials, which proved to be too

difficult and cause to slow down the application. To overcome this shortcoming, we make use of the homotopy

perturbation method to get analytical approximations for different choices of linear operators and the initial

guesses available. Recently, the homotopy perturbation method being a powerful technique was developed by

He (He, 1999, 2005) .The main advantage of this technique is to overcome the difficulties arising in the

process of calculations for the nonlinear terms arising in the problem. This gives analytical approximation to

the different classes of the nonlinear differential equations, system of differential equations, integral and

integro-differential equation and systems of such equations. Haldar applied the Adomian’s decomposition

method to the Navier- stokes equations in cylindrical coordinates for two dimensional irrotational fluid flow in

a tube (Hardar, 1997). Our present analysis gives the application of homotopy perturbation method without

any restrictive assumptions and handy calculations of the Adomian polynomials to the Navier Stoke Equations

in cylindrical coordinates, in which the steady two dimensional irrotational flow of fluid in a tube of non-

uniform circular cross section can be studied.

2 The Governing Equations

Consider the governing equations of motion for the two dimensional flow field for a viscous fluid in a tube

which are described by the cylindrical coordinate transformation of the Navier-Stokes equations read as;

2 2

2 2

1 1,

u u u u u Pu v

z r r r zr z

(1)

2 2 2

2 2 2

1 1.

v v v v v v Pu v

z r r r rr r z

(2)

It is suggested that the rotational motion of the fluid is negligible. Then the equation of continuity reads is

10

urv

r r z

(3)

Where ,u is fluid velocity components in the axial x coordinate and v is in the radial coordinate r , and

the fluid pressure is described by P , the fluid density by , and the kinematic viscosity by for the fluid.

Introducing and labeling the stream function as , then we may have,

1,u

r r

and 1

vr z

, (4)

The equation of continuity is satisfied identically. The dynamical equation of motion in term of the stream

function are obtained by eliminating P between (2) and (3), and making us of the relation (4), it is read as;

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22

,1 2

, zr r r

(5)

Introducing as a linear operator which is defined as;

2 2

2 2

1,

r rr z

(6)

and the Jacobean defined as

, ( ) ( )

., z

L r rr r z r z

z z

(7)

Now, we here mainly discuss to forms of the linear operator defined by equation (5). We will split the

linear operator in two parts and discuss the two cases. It is to note that in the homotopy perturbation

method we are free to choose the linear operator. This mainly depends upon the given form of the initial or

boundary condition and the problem under investigation. Therefore, depending upon our choices and the

possibilities for the appearance of the auxiliary linear form of operator in the problem we consider two cases

here.

Case 1:

The first form of the linear operator extracted from equation (5) for the possible form of the linear operator is

supposed to be;

2

1 2

1.

r rr

(8)

Then the operator becomes 2

1 2z

, which implies that

22 2 2

1 1 122

z

then

22 2 2

1 1 122 .

z

(9)

Using (9) in (5), the equation (5) takes the following form

2 2 42 21 12 2 4

1 ( , ) 22 ( ) ,

(r, z)r zr z z

Taking 1 both sides

2 42 111 2 4 2

( ) 1 ( , ) 22 .

(r, z)r zz z r

In order to apply the proposed homotopy perturbation method to the given problem, we need to define the

nonlinear term appearing in the governing equations. Therefore, we define the nonlinear term as N in the

above equation which is given as;

2

1 ( , ) 2.

(r, z)N

r zr

Then we get the following nonlinear form of equation for our analysis,

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2 42 1 1

1 2 2 4

( )1 ( , ) 22

(r, z)r r z z z

Operating 11

on both sides of the above equation,

2 41 1 1

1 1 2 2 4

1 ( , ) 22 ,

(r, z)r zr z z

(10)

Using homotopy perturbation method (HPM) proposed by J. H. He (He, 2006), we construct a homotopy for

equation (10) as; (r, z; ) : [0,1] This satisfies

0, 1 0,H v v u A v f r and here 0,1 is designed to be an

embedding parameter,

41 1 1 11

1 1 0 1 1 12 2 4

,1 21 , 2 0.

,

w w w

r r z zr z z

(11)

Suppose the solution of (11) is of the form of

20 1 2, ; ,r z (12)

Using (12) in (11) we get

0 1 0 11 1 11 0 1 1 0 1 1

1 2 40 1 1 0 1 0 11 1 1

1 0 1 12 2 4

, ,1, +

r, z

, , ,2, 2 0.

owr

zr z z

(13)

Now we simplify the quantities enclosed in brackets,

0 1 0 1

0 1 0 1

0 1 0 1

, , ,

r, zr r

z z

1

2 2 2 20 2 0 1 2 0 1 2 0 1 2

r z z r

2 30 0 0 02 2 2 2 1 1 1 1( ) ( ) ( ),r z z r r z z r r z z r

0, 10 0 1 0 2 0 0 2 1 11 2, , , , ,

, , , , , ,o

r z r z r z r z r z r z

(14)

The calculations made in (14) is according to the definition of the Jacobean, and

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1

0 1 1 0 10 0 10 1 0 1 0

2 0 1 22 1 0 ,

z z z z

z z z

(15)

Combining the terms containing the equal powers of in equation (14) and (15)

0 00 1 01 1 02

,1 2,

,C

r r z zr

(15a)

1 0 0 11 0 12 1 02 2

, , 2 2,

, ,C

r z r z z zr r

(15b)

2 0 0 2 1 12 0 1 23 2 1 22 2 2

, , , 2 2 2,

, , ,C

r z r z r z z z zr r r

(15c)

and so on. Now

2 2 2 2 20 1 2 0 1 20 1 2

2 2 2 2z z z z

(16)

4 0 1 2 4 4 40 1 2 0 1 20 1 2

4 4 4 4z z z z

(17)

Now combining the terms containing the equal powers of in equations (16) and (17)

2 41 0 0

1 2 42 ,D

z z

(17a)

2 41 1 1

2 2 42 ,D

z z

(17b)

2 41 2 2

3 2 42 ,D

z z

(17c)

and so on. Using equations (15a), (15b), (15c), (17a), (17b), (17c) in equation (13). We get from equation (13),

0 1 11 0 1 1 1 0 1 0

1 1 0 1 2 0 1 21 1 2 3 1 2 3 0.C C C D D D

(18)

Equating the coefficients of equal powers of we have the zeroth order problem as:

Zeroth Order problem: 1 0 1 0 0, which implies 0 0 ,r z

(19)

Here 0 is defined as the solution of homogenous equation 21 0 0, (20)

subject to the pre-prescribed boundary conditions. Now to find the approximation for 1 for which we first

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find the inverse operator 21 and for it we consider equation (3), 1 0 (21)

We now define 2

2& ,rr r rr

then operator 1 takes the following form as

1

1,rr rr

1

1.rr rr

Using equation (21) we get 10rr rr

Solving for rr and r that is for

linear terms

1,rr rr

(22)

,r rrr (23)

Operating 1r on (24) and 1

rr on (24) we get

11

1,rr rr

(24)

12 ,r rrr (25)

1 and 2 are the solutions of two homogenous equations 0,rr and 0,r respectively. The

inverse linear operators 1rr and 1

r are defined as

1

1

,

.

rr

r

drdr

dr

(26)

Adding (25) and (26) we get 1 11 2

12 ,rr r r rrr

r

and dividing both

sides by 2, to get 1 11 2 1,

2 rr r r rrrr

(27)

1 1

0

1 1,

2 rr r r rrrr

(28)

where 1 20 2

, then

1 11 0

1 12 1

1 11

1 1,

2

1 1,

2

1 1.

2

rr r r rr

rr r r rr

n rr r r rr n

rr

rr

rr

(29)

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Let the quantity in brackets is denoted by

1 0

22 1 0 02

11 01

1

21 1 1 1

,2 2 2 2

1

2n

n n

(30)

1 1

0 00 0

1 1 1

2 2

n

nrr r r rrn n

n n

rr

(31)

Now the inverse of the linear operator is defined 2

21

1rr rr

as

2 1 11

0

1 1

2

n

rr r r rrnn

rr

(32)

Now we come to equation (19) and define the zeroth order problem as, 0 0. And the 1st order problem

as 1 11 1 1 0 1 1 1 .C D

Substituting values from equation (15a) (17a) we get,

2 40 01 1 0 1 0 0

1 1 1 0 1 0 2 2 4

,1 22 .

,r r z zr z z

Operating 1 on both sides of the above equation yields,

2 40 02 2 1 0 1 0 0

1 1 1 0 0 2 2 4

,1 22 .

,r r z zr z z

Making use of 0 0 , for the initial guess of HPM methodology, 21 0 0,

2 40 02 1 0 1 0 0

1 1 0 2 2 4

,1 22 .

,r r z zr z z

Operating with 21 on both sides of the above expression,

2 40 01 2 2 20 1 0 0

1 1 0 1 12 2 4

,1 22 ,

,r r z zr z z

(33)

where 21 is given in equation (32). The 2nd order problem is given as, 1 1

1 2 1 2 2 .C D Using

the values of 2C and 2D from equations (15b) and (17b), we get

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2 41 0 0 11 1 0 1 1 1 1

1 2 1 1 02 2 2 4

, ,1 2 22 .

, ,r r z r z z zr r z z

Operating with 11 both sides of the above equation to get,

1 0 0 11 2 0 12 1 1 02 2

2 42 21 1 1

1 12 4

, ,1 2 1 2

, ,

2 ,

r r z z r r z zr r

z z

(34)

Now since in the methodology of HPM, we suppose the following expression for the approximate solution of

the problem,

0 1 2 31

lim ,

(35)

where the components of the series solution are defined to be as; 0 0 ,

2 40 01 2 2 20 1 0 0

1 1 0 1 12 2 4

,1 22 ,

,r r z zr z z

1 0 0 11 2 0 12 1 1 02 2

2 42 21 1 1

1 12 4

, ,1 2 1 2

, ,

2 ,

r r z z r r z zr r

z z

and so on. If once 0 is obtained, which can be easily obtained by constructing the homotopy for the given

problem and equating the coefficients for the zeroth order problem, then we can find 1 in terms of 0 and

in the similar fashion 2 can be evaluated in terms of 1 and 0 .The other higher order components can

be easily obtained having the all other lower order values. Thus all the components of can be calculated.

The series solution 0

,nn

thus can be given the following form,

2 40 01 2 2 20 1 0 0

0 1 0 1 12 2 4

1 0 0 11 2 0 11 1 02 2

2 42 21 1 1

1 12 4

,1 22

,

, ,1 2 1 2

, ,

2

r r z zr z z

r r z z r r z zr r

z z

(36)

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Case 2:

We may have the other available or chosen form of the linear operator as;

1.rr r zzr

Whereas ,r r

2

2rr r

and

2

2,zz z

then from equation (5) we have,

22

,1 2.

, zr r r

Taking 1 both sides of the above expression,

2 12

,1 2.

, zr r zr

2 12

,1 20.

, zr r zr

(37)

Using the methodology of HPM, we may construct a homotopy for equation (37) as;

, ;q : 0,1r z

1 10 2

,1 2, 1

,zH q q q

r r zr

(38)

Suppose the solution for equation (38) is of the following form

0 1 20 1 2( , ; ) ( , ) ( , ) ( , )r z q q r z q r z q r z (39)

Where as

0 1 0 1

0 1 0 1

0 1 0 10 1 0 1

0 0 1 0 0 11

, ,,

, z , ,

, ,( , ) ( , )

, , ,

, , ,

q q

r rr q q

z z

q qq q

r z r z

qr z r z r z

2 0 0 2 1 12 , , , , (40)

, , ,q

r z r z r z

1

0 1 1 0 10 0 10 1 0 0

2 0 1 22 1 0

,,

,

q qz z z z

qz z z

(41)

Using (40) and (41) in (38) we get

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0 0 1 0 0 11 10 0

2 0 0 2 1 12

, , ,

, , ,

, , ,

, , ,

q qr z r z r z

qr z r z r z

0 10 0 10 1 02 2 2

2 2 2 , 0q q

z z zr r r

We first define the inverse linear operator 2 , consider equation (3), 0. Then

1,rr r zzr

10,rr r zzr

1,rr zz rr

Multiplying both sides of the above equation by r , we get

.r rr zzr r (42)

In similar way, we get 1

,zz r rrr (43)

1.rr r zzr

(44)

11

12

13

1

1

r rr zz

zz r rr

rr r zz

r r

r

r

(45)

1 1 11 2 3 1 1 1,

3 3 r rr zz zz r rr rr r zzr rr r

(46)

where 1 2 30 3

, and 1 , 2 and 3 are the solutions of homogenous equations 0r ,

0r r and 0,zz then

1 1 10

1 1 1.

3 r rr zz zz r rr rr r zzr rr r

(47)

The inverse linear operators 1r , 1

zz , and 1

rr are defined by

1 1 1, , r zz rrdr dzdz drdr (48)

Then we have,

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1 1 11 0

1 1 12 1

1 1 11

1 1 1,

3

1 1 1,

3

1 1 1

3

r rr zz zz r rr rr r zz

r rr zz zz r rr rr r zz

n r rr zz zz r rr rr r zz

r rr r

r rr r

r rr r

.n

(49)

Let us denote the quantity within the brackets by , then the following expression is obtained,

1 0

22 1 0 02

33 2 03

11 01

1,

31 1 1 1

,3 3 3 31 1

,3 3

1,

3n

n n

(50)

1 1 10

0 0

1 1 1 1.

3 3

n

nr rr zz zz r rr rr r zz nn n

n n

r rr r

Thus the inverse linear operator can be easily identified as;

2

2 1 1 1

0

1 1 1 1 (51)

3

n

rr r zz r rr zz zz r rr rr r zznn

r rr r r

Now the zeroth order problem is 0 0 , 0 0 .

(52)

The 1st order problem is:

0 01 1 01 0 02

,1 2.

,r r z zr

Operating with both sides of the above equations;

0 02 2 1 01 0 02

,1 2

,r r z r z

In order to find the initial guess of HPM, we make use of the zeroth order problem as: 20 0.

0 02 1 01 02

,1 2,

,r r z zr

Operating 2 on both sides to get,

0 01 2 01 02

,1 2,

,r r z zr

(53)

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The 2nd order problem is:

1 0 0 11 2 0 12 1 02 2

, ,2 2,

, ,r z z r z zr r

and so on. The series solution form of the problem reads as,

0 2 31lim , ;q ,r z

Where the following quantities are defined, 1 2 30 3

0 01 2 01 02

,1 2,

,r r z zr

1 0 0 11 2 0 12 1 02 2

, ,2 2,

, ,r z z r z zr r

(54)

and so on. The series form of the solution can be written as; 0

,nn

2 40 01 2 2 20 1 0 0

0 1 0 1 12 2 4

1 0 0 11 2 0 11 1 02 2

2 42 21 1 1

1 12 4

,1 22

,

, ,1 2 1 2

, ,

2

r r z zr z z

r r z z r r z zr r

z z

3 Conclusion

We have considered two cases for the available linear operators and obtained the approximation for our

problem. Of course, the selection of the linear operators mainly depends upon the given initial or boundary

conditions. We can see that for the first case, the available linear operator was split in two parts and for the

second case, we considered the full linear form of the operator without splitting it into parts. Thus, on the basis

on methodology of the Adomian decomposition and the homotopy perturbation method (Haldar, 1995), the

present analysis can be applied to a wide range of the physical and engineering problems (Shakil et al., 2013;

Wahab et al., 2013; Wahab et al., 2014; Siddiqui et al., 2014).

As compared to the Adomian decomposition method for the analysis of the problem (Haldar, 1995), we

have the great advantage of the selection of the initial guess which can be chosen on the basis of the previous

knowledge, and most importantly, the initial approximation should satisfy the given initial or boundary

conditions, which leads us to the uniformly valid approximately series solution. While, the Adomian

decomposition method does not have such advantage, because we have to select the initial guess based on the

recursive relation produced by the method. But this initial approximations sometimes, may lead to non-

uniformly valid series solution which also may contain the secular terms in the series. In homotopy

perturbation method, the initial guess satisfying the given conditions may give a uniformly valid series

solution.

On the other hand, the calculation of the Adomian polynomials is not an easy task for the nonlinear terms

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appearing in the problems. However, there are some computer programs which can calculate the Adomian

polynomials, but they are for some specific cases. In our analysis, we avoid such handy calculation because

homotopy perturbation method transforms a non-linear problem into a small number or sub-linear problems

with prescribed conditions. No matter of concern with the existence of the parameter small or large. This is

again a dominating advantage of the method over Adomian decomposition method (Shakil et al., 2013; Wahab

et al., 2013; Wahab et al., 2014; Siddiqui et al., 2014).

References

Adomian G. 1986. Application of the decomposition method to the Navier-Stokes equations. Journal of

Mathematical Analysis and Applications, 119: 340-360

Adomian G. 1989. Nonlinear Stochastic Systems: Theory and Applications to Physics. Kluwer Academic

Publishers, USA

Adomian G. 1993. Nonlinear transport in moving fluids. Applied Mathematics Letters, 6(5): 35-38

Afzal M, Wahab HA, Bhatti S., and Naeem, Qureshi MT. 2014. A Mathematical Model for the Rods with Heat

Generation and Convective Cooling. Journal of Basic and Applied Scientific, 4(6): 68-76

Haldar K. 1995. Application of adomian approximations to the Navier-Stokes equation in cylindrical

coordinates. Applied Mathematics Letters, 9(4): 109-113

He JH. 1999. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering,

178(3/4): 257-262

He JH. 2000. A coupling method of homotopy techniques and perturbation technique for nonlinear problems.

International Journal of Non-Linear Mechanics, 35(1): 37-43

He JH. 2006. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern

Physics B, 1141-1199

Shakil M, Khan T, Wahab HA, Bhatti S. 2013. A Comparison of Adomian Decomposition Method (ADM)

and Homotopy Perturbation Method (HPM) for Nonlinear Problems. International Journal of Research

in Applied, Natural and Social Sciences, 1(3): 37-48

Siddiqui AM, Wahab HA, Bhatti S, Naeem M. 2014. Comparison of HPM and PEM for the flow of non-

newtonian fluid between heater parallel plates. Research Journal of Applied Sciences, Engineering

and Technology, 7(10): 4226-4234

Squire HB. 1952. Some viscous fluid flow problems. Philosophical Magazine, 43: 942-945

Thorpe JF. 1997. Development in theoretical and applied mechanics. Shaw, WA. Pergamon Press, Oxford, UK.

Wahab HA, Shakil M, Khan T, Bhatti S, Naeem M. 2013. A comparative study of a system of Lotka-Voltera

type of PDEs through perturbation methods. Computational Ecology and Software. 3(4): 110-125

Wahab HA, Khan T, Shakil M, Bhatti S, Naeem M. 2014. Analytical approximate solutions of the systems of

non linear partial differential equations by Homotopy Perturbation Method (HPM) and Homotopy

Analysis Method (HAM). Journal of Applied Sciences and Agriculture, 9(4): 1855-1864

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Modeling the effect of pollution on biological species: A socio-

ecological problem

B. Dubey1, J. Hussain2, S. N. Raw3, Ranjit Kumar Upadhyay4 1Department of Mathematics, Birla Institute of Technology and Science, Pilani – 333031, India 2Department of Mathematics and Computer Science, Mizoram University, Aizwal – 796009, India 3Department of Mathematics, National Institute of Technology, Raipur – 492010, India 4Department of Applied Mathematics, Indian School of Mines, Dhanbad – 826004, India


Received 13 December 2014; Accepted 20 January 2015; Published 1 June 2015

Abstract

In this paper, a nonlinear spatial model is proposed and analyzed to study the effect of pollution on biological

population. It is assumed that the pollutants enter into the environment not directly by the population but by a

precursor produced by the population itself. It is further assumed that larger the population, faster the precursor

is produced, and larger the precursor, faster the pollutant is produced. Criteria for nonlinear stability and

instability for both spatial and non-spatial models are obtained. The various parameter ranges for stable

homogeneous solutions are identified. By the simulation experiments, it is observed that by applying an

appropriate effort F , the population density P can be maintained at a higher equilibrium level. It is also

shown that the equilibrium level of the concentration of precursor pollutant, concentration of pollutant in the

environment and in the population decrease due to the effort F.

Keywords precursor; pollutant; biological species; stability; conservation efforts.


1 Introduction

Our environment is getting polluted day by day due to rapid pace of urbanization, industrialization and

deforestation, and we face one of the most important present day socio-ecological problems closely related to

physiological and bio-spherical changes in the population. We do have several examples where the pollution

is responsible for increase in death rate, decrease in birth rate and migration of population (Shukla and Dubey,

1996). The effects of pollution caused by various human factors on structure and functions of ecosystems have

been studied by several researchers (Woodwell, 1970; Smith, 1981; McLaughli, 1985; Hari et al., 1986;

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Woodman and Cowling, 1987; Schulze, 1989; Ghosh, 2000; Srinivasu, 2002; Ghosh et al., 2002, 2006; Naresh

et al. 2006a, 2006b; Sundar, 2013; Sundar et al., 2014). In recent decades, some investigations have been

made to study the effect of pollution on a single or two biological species (Hallam et al., 1983a, 1983b, Hallam

and Luna, 1984; Hallam and Ma, 1986; Luna and Hallam, 1987; Freedman and Shukla, 1991; Huaping and Ma,

1991; Shukla and Dubey, 1996, 1997; Dubey, 1997; Dubey and Das, 1999; Dubey and Hussain, 2000a, 2000b,

2006; Shukla et al., 2001, 2003, 2009; Dubey et al., 2003, 2009; Naresh et al, 2006a, 2006b; Dubey and

Naranayan, 2010; Sundar et al., 2014). In particular, Hallam et al. (1983b) studied the effect of a toxicant in

the environment on a single-species population by assuming that its growth rate density decreases linearly with

the uptake concentration of toxicant. Huaping and Ma (1991) proposed and analysed a mathematical model to

study the effect of toxicant on naturally stable two-species communities. In these investigations, it has been

assumed that carrying capacity does not depend on the concentration of toxicant present in the environment.

However, in real situations the effect of toxicant is to decrease both the growth rate of species and the carrying

capacity of the environment. Taking this aspect into account, Freedman and Shukla (1991) investigated the

effect of a toxicant on a single-species and predator-prey system by considering the exogeneous introduction

of toxicant into the environment.

Shukla and Dubey (1996) studied the simultaneous effects of two toxicants on a biological species, one

being more toxic than the other. Dubey (1997) propose a mathematical model to study the depletion and

conservation of forestry resources in a polluted environment. Shukla et al. (2001) studied the effect of a

toxicant emitted into the environment from external sources on two competing biological species. They found

that the four usual outcomes of competition between two species may be altered under certain conditions

which are mainly dependent on emission rate of toxicant into the environment, uptake concentration of

toxicant by the two species and their growth rates and carrying capacities. Dubey et al. (2003) studied the

behaviour of a resource biomass in the presence of industrialization and pollution. They showed that in the

case of small periodic influx of toxicant into the environment, the resource biomass has a periodic behaviour if

the depletion rate coefficient of environmental pollution is small. However, if this coefficient increases beyond

a threshold value, then the resource biomass converges towards its equilibrium. Naresh et al. (2006a)

investigated the effect of an intermediate toxic product formed by uptake of a toxicant on a plant biomass.

Shukla et al. (2003) proposed and analysed a mathematical model and studied effects of primary and

secondary toxicants on the biomass of resources such as forestry, agricultural crops. Dubey and Hussain (2006)

investigated the survival of a biological species which is dependent on a resource in a polluted environment

and they showed that the diffusion plays a general role in stabilising the system.

In the above investigations, it is assumed that the pollutant enters into the environment by some manmade

projects which may be population (industrialization) dependent, constant, zero or periodic. In this regard,

Rescigno (1977) studied the effect of a precursor pollutant on a single species, but he did not consider the rate

of uptake concentration of the pollutant on the growth of the species. Further, in the above works the effects of

diffusion has not been considered. Keeping the above in view, in this paper we propose and analyse a

nonlinear model to study the effect of a precursor pollutant, which is formed by various human activities in the

atmosphere, on population where the effect of uptake concentration, diffusion and conservation are considered.

The paper is organized as follows. In Section 2, we discuss the model system. Under Section 3, we analyse

the model system without diffusion. In Section 4, we analyze the model system with diffusion. Section 5

describes the conservation model system. In Section 6 and 7, we analyse the conversion model system without

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and with diffusion, respectively. Section 8 depicts the numerical results. Finally, we summarize the results in

the last Section.

2 The Proposed Mathematical Model

Let us consider a polluted environment where a biological population is growing logistically in a closed region

D with smooth boundary D . We assume that the environment is polluted by various population activities. It

is further assumed that the population is affected by the pollutant formed in the atmosphere by its precursor.

Let ),,( tyxP be the population density, ),,( tyxQ the concentration of the precursor pollutant emitted by

various activities of the population, ( , , )T x y t the concentration of pollutant formed by Q in the atmosphere

and ),,( tyxU uptake concentration of pollutant by the population at coordinates Dyx ),( and time .0t

It is also assumed that the larger the population, the faster the precursor is produced. It is further assumed that

the larger the precursor, the faster the pollutant is produced. Then, system may be governed by the following

set of differential equations:

.1,0

,

,

,

,)(

)(

10

001

22110

0

21

20

PTThUt

U

TDPTUThhQt

T

QPt

Q

PDTK

PrPUr

t

P

(1)

We analyse the system (1) with the following initial and boundary conditions:

,0),()0,,( ,0),()0,,( ,0),()0,,( yxyxTyxyxQyxyxP

,0 ,),( ,0 ;),( ,0),()0,,(

tDyxn

T

n

PDyxyxyxU (2)

where n is the unit outward normal to .D We assume that the functions P, Q, T, U belong to the class

)(2 DC .

In model (1), 2

2

2

22

yx

is the Laplacian diffusion operator. 1D and 2D are the diffusion rate coefficients of ),,( tyxP and

),,( tyxT respectively in D . is the growth rate of Q due to population P , 0 the natural depletion rate

coefficient of .Q h can be interpreted as the growth rate coefficient of T due to .Q 0h can be interpreted as

the natural depletion rate coefficient of T , a fraction 0 of which goes inside the body of the population. is

the depletion rate coefficient of T due to .P 1 is the natural depletion rate coefficient of U , a fraction 1 of

which re-enters into the environment.

In model (1), the function )(Ur is the specific growth rate of the population which decreases as

U increases, i.e.

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0)0( rr and 0)( Ur for .0U (3)

The function )(TK is the carrying capacity of the population in the presence of pollutant and it decreases

as T increases. Hence we assume that

0)0( KK and 0)( TK for ,0T and a aTT such that .0)( aTK (4)

The model is analysed with and without diffusion.

3 Model Without Diffusion

In this section we take 021 DD in model (1). Then model (1) reduces to

.0)0( ,0)0( ,0)0( ,0)0(

,

,

,

,)(

)(

001

110

0

20

UTQP

PTThUdt

dU

PTUThhQdt

dT

QPdt

dQ

TK

PrPUr

dt

dP

(5)

It can be checked that there exists two non-negative equilibria (which belong to the first orthant), namely,

)0,0,0,0(0E and ),,,( UTQPE , where TQP , , and U are the positive solutions of the following

algebraic equations:

0

0

0 0 1 1

0 01

( ) ( ),

,

( ), (say),(1 ) (1 )

1{ ( ) ( )} ( ), (say).

r P r U K T

Q P

hQT f P

h P

U h f P Pf P g P

It can be verified that the equilibrium E exists if the following inequality holds at :E

.0)()())(())(()()(0 PfTKPgrPfKPgUrr (6)

By computing the variational matrix corresponding to the equilibrium 0E , it can be checked that 0E is a

saddle point with unstable manifold locally in the P direction and with stable manifold locally in the

UTQ space.

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In the following theorem, it is shown that E is locally asymptotically stable.

Theorem 3.1 Let the following inequalities hold:

),()(9

4)(

)(0

0

2

2

0 PhTK

rTTK

TK

Pr

(7)

),(3

2)}({ 012

200211 PhcPhc (8)

),(3

2001

2 Phch (9)

where

)(3

12

001

TK

rc

and .)(

2T

Urc

(10)

Then the equilibrium E is locally asymptotically stable.

Proof By taking the transformations

, , , , uUUTTqQQpPP

we first linearize model (5). Then we consider the following positive definite function in the linearized form of

model (5): 2

2 2 21 2

1( , , , } { }

2

pV p q u c q c u

P (11)

where 1c and 2c are positive constants given by (10). It can be checked that the derivative of V with respect

to t is negative definite under conditions (7)-(9), proving the theorem.

In the following theorem it is shown that the equilibrium E is globally asymptotically stable. To prove this theorem, we need the following lemma which establishes a region of attraction for system (5). The proof

of this lemma is easy and hence is omitted.

Lemma 3.1 The set }0 ,0:),,,{( 001

KUTQKPUTQP is a region of attraction for

all solutions initiating in the interior of the positive orthant, where

h0 and )}.1(),1(,min{ 11000 hh

Theorem 3.2 In addition to the assumptions (3) and (4), let )(Ur and )(TK satisfy in 1 ,

0)( ,)(0 KTKKUr m and kTK )(0 , (12)

for some positive constants mK, and .k Let the following inequalities hold:

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2

0200

KK

kKr

m

< )()(9

40

0 PhTK

r , (13)

)(3

2 01

2

0

TK

rK

, (14)

),(3

2001

2 Phch (15)

)(3

2)( 01

20011 PhPh , (16)

where 1c is same as defined in Eq. (10).

Then, E is globally asymptotically stable with respect to all solutions initiating in the interior of the positive

orthant 1 .

Proof Consider the following positive definite function around E ,

22211 )(

2

1)(

2

1)(

2ln),,,( UUTTQQ

c

P

PPPPUTQPV . (17)

Now differentiating 1V with respect to t along the solutions of (5), we get

21

20

201

201 )())(()()()(

UUTTPhQQcPPTK

r

dt

dV

))()()(())()()(())(( 01 UUPPTUTTPPTTPrQQPPc

+ ),)()(())(( 0011 UUTTPhTTQQh (18)

where

.

,)(

)(

),/()(

1

)(

1

)(

,

),(

,)()(

)(

2

TTTK

TK

TTTTTKTK

T

UUUr

UUUU

UrUrU

From (12) and the mean value theorem, we note that

)(U and 2

( ) .m

kT

K

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Now Eq. (18) can be rewritten as the sum of the quadratics:

22212

211

1 )(2

1))(()(

2

1QQaQQPPaPPa

dt

dV

23313

211 )(

2

1))(()(

2

1TTaTTPPaPPa

24414

211 )(

2

1))(()(

2

1UUaUUPPaPPa

23323

222 )(

2

1))(()(

2

1TTaTTQQaQQa

,)(2

1))(()(

2

1 24434

233 UUaUUTTaTTa

where

, ),(3

2 , ,

)(3

21440330122

011 aPhaca

TK

ra ,112 ca ),)(( 013 TTPra

,)(14 TUa ,23 ha .001134 Pha

Sufficient conditions for dt

dV1 to be negative definite are that the following conditions hold:

,2211212 aaa (19)

,3311213 aaa (20)

,4411214 aaa (21)

,3322223 aaa (22)

.4433234 aaa (23)

We note that inequality (19) is satisfied automatically for the chosen value of 1c in the theorem. We also

note that that ),20()13( ),21()14( )22()15( and ).23()16( Hence 1V is a Liapunov

function with respect to E whose domain contains the region of attraction 1 , proving the theorem.

The above theorem implies that the population living in a polluted environment attains an equilibrium

level under certain conditions. The equilibrium level of the precursor pollutant is crucial in affecting the

equilibrium level of population which decreases as the equilibrium level of precursor pollutant increases. We

also note that if and h are kept at small level, then possibility of satisfying conditions (13)-(15) increases.

This implies that the stability of the system can be maintained by lowering the rate of formations of precursor

and environmental pollutants.

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4 Model With Diffusion

In this section, we consider the complete model (1)-(2) and state the main results in the form of the following

theorem.

Theorem 4.1 (i) If the equilibrium E of model (5) is globally asymptotically stable, then the corresponding uniform steady state of the initial-boundary value problems (1)-(2) is also globally asymptotically stable.

(ii) If the equilibrium E of model (5) is unstable even then the uniform steady state of the initial-boundary value problems (1)-(2) can be made stable by increasing diffusion coefficients to sufficiently large values.

Proof Let us consider the following positive definite function

D

dAUTQPVtUtTtQtPV ),,,())(),(),(),(( 12

where 1V is given by Eq. (17).

We assume that 1V is differentiable and the functions P, Q, T, U belong to the class )(2 DC .

Then we have,

D

dAt

U

U

V

t

T

T

V

t

Q

Q

V

t

P

P

V

dt

dV 11112

1 2 ,I I (24)

where

D

dAdt

dVI 1

1 and .212

2112 dAT

T

VDP

P

VDI

D

(25)

We note the following properties of ,1V namely,

011

DD T

V

P

V,

and for all points of ,D

,012

12

12

12

12

12

UT

V

UQ

V

TQ

V

UP

V

TP

V

QP

V

0 ,0 ,021

2

21

2

21

2

T

V

Q

V

P

Vand .0

21

2

U

V

Now we consider 2I and determine the sign of each term. We utilize the following formula known as

Green’s first identity in the plane:

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D D D

dAGFdsn

GFdAGF ,).( 2

where n

G

is the directional derivative in the direction of the unit outward normal to D and s is the arc

length.

Then with P

VF

1 and PG we obtain

D DD

dAPP

Vds

n

P

P

VdAP

P

V.1121

,.1

D

dAPP

V since .0

n

P

Now .ˆˆ21

2

21

21 j

y

P

P

Vi

x

P

P

V

P

V

Hence

dAPP

V

D

21 ,022

21

2

dAy

P

x

P

P

VD

Similarly .0 21

D dATT

V

i.e., .02 I (26)

Thus we note that if ,01 I then .0211 II

dt

dV This shows that if E is globally asymptotically

stable in the absence of diffusion, then the uniform steady state of the initial-boundary value problems (1)-(2)

also must be globally asymptotically stable. This proves the first part of Theorem 4.1.

We further note that if ,01 dt

dVi.e., if ,01 I then E may become unstable in the absence of diffusion.

But, Eqs. (24) and (26) show that by increasing diffusion coefficients 1D and 2D to sufficiently large values,

dt

dV2 can be made negative even if .01 I This proves the second part of Theorem 4.1.

The above theorem implies that diffusion with reservoir boundary conditions may stabilize a system which

is otherwise unstable.

We shall explain the above theorem for a rectangular habitat D defined by

}0 ,0:),{( byaxyxD (27)

in the form of the following theorem.

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Theorem 4.2 In addition to assumptions (3) and (4), let )(),( TKUr satisfy the inequalities in (12). Let the

following inequalities hold:

2 2 2 2 2 2 20 0 0 0 1 2

02 2 2 2 2 20

( ) ( )4

9 ( )m

r K k K r D P a b D a bh P

K K a b a bK T

(28)

,)(

)(3

2222

0

22210

1

2

0

baK

baPD

TK

rK

(29)

22

2222

0032 )(

3

2

ba

baDPhch

, (30)

,)(

3

2}{

22

2222

012

1100

ba

baDPhPh

(31)

where

.)(

)(3 2220

22210

20

3

baK

baPD

TK

rc

(32)

Then the uniform steady state of the initial-boundary value problems (1)-(2) is globally asymptotically stable

with respect to all solutions initiating in the interior of the positive orthant.

Proof Let us consider the rectangular region D given by equation (27). In this case 2I , which is defined in

Theorem (4.1), can be written as

}.)()){((})(){()( 2221

2

222

21

2

12 y

T

x

T

T

VDdA

y

P

x

P

P

VDI

DD

(33)

From Eq. (17), we obtain

221

2

P

P

P

V

and .121

2

T

V

Hence

dAy

T

x

TDdA

y

P

x

P

K

PDI

DD

22

2

22

20

12 .

Now

dAx

PD

2

DdA

x

PP2

)(

b a

dxdyx

PP

0 0

2)(

.

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Let ,a

xz then

b

Ddzdy

z

PP

adA

x

P

0

1

0

22)(1

.

Now using the inequality (Denn, 1972),

,1

0

21

0

22

dxPdxx

P

we obtain

b

DdzdyPP

adA

x

P

0

1

0

222

)(

b a

dxdyPPa 0 0

22

)(

.)( 22

2

D

dAPPa

Similarly, .)()( 22

22

DDdAPP

bdA

y

P

Thus, .)()(

)()( 2

22

22222

2220

2221

2

DD

dATTba

baDdAPP

baK

baPDI

Now from (18) and (24), we obtain

D

QQcPPbaK

baPD

TK

r

dt

dV 201

2

2220

222102 )()}(

)(

)({[

21

222

2222

0 )()}()(

{ UUTTba

baDPh

))(}()({))(( 03 TTPPTTPrQQPPc

))(}()({ UUPPTU

,)])(}({ 1100 dAUUTTPh (34)

where )(T and )(U are defined in Eq. (18).

Now Eq. (34) can be written as the sum of the quadratics

D

QQbQQPPbPPbdt

dV 22212

211

2 )(2

1))(()(

2

1{

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23313

211 )(

2

1))(()(

2

1TTbTTPPbPPb

24414

211 )(

2

1))(()(

2

1UUbUUPPbPPb

23323

222 )(

2

1))(()(

2

1TTbTTQQbQQb

,})(2

1))(()(

2

1 24434

233 dAUUbUUTTbTTb

where

,b },)(

{3

2 , },

)(

)({

3

214422

2222

03301222220

22210

11

ba

baDPhbcb

baK

baPD

TK

rb

,312 cb ),)(( 013 TTPrb ,)(14 TUb ,23 hb .001134 Phb

Sufficient conditions for dt

dV2 to be negative definite are that the following conditions hold:

,22112

12 bbb (35)

,33112

13 bbb (36)

,44112

14 bbb (37)

,3322223 bbb (38)

.4433234 bbb (39)

We note that inequality (35) is automatically satisfied for the value of 3c given in (32). We further note

(28) (36), (29) (37), (30) (38) and (31) (39). Hence 2V is a Liapunov function with respect to E

whose domain contains the region of attraction ,1 proving the theorem.

From the above theorem we note that inequalities (28)-(31) may be satisfied by increasing 1D and 2D to

sufficiently large values. This implies that in the case of diffusion stability is more plausible than the case of

no diffusion. Thus, in the case of diffusion the population converges towards its carrying capacity faster than

the case of no diffusion, and hence the survival of the population may be ensured.

5 Conservation Model

In the previous section, it has been noted that uncontrolled human activities that are polluting the environment

may harm itself considerably. Therefore, some kind of efforts must be adopted to stop further deterioration of

the environment. In this section a mathematical model is proposed and analysed to control the undesired level

of precursor pollutant by some mechanisms. It is assumed that the effort applied to control the precursor

pollutant is proportional to the undesired level of the precursor pollutant. Then the dynamics of the system is

assumed to be governed by the system of differential equations given below:

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2

201

0 1

20 1 1 2

1 0 0

1 1

0 1

( ) ,( )

,

,

,

( ) ( ) ,

0 , 1.

c c

r PPr U P D P

t K T

QP Q r F

tT

hQ h T U PT D TtU

U h T PTt

FQ Q H Q Q F

t

(40)

The above model (40) is to be analysed with following initial and boundary conditions:

,0),()0,,( ,0),()0,,( ,0),()0,,( yxyxTyxyxQyxyxP

;),( ,0),()0,,( ,0),()0,,( 1 DyxyxyxFyxyxU

,0 ,),( ,0

tDyxn

T

n

P (41)

where n is the unit outward normal to .D Again we assume that the functions P, Q, T, U, F belong to the

class )(2 DC .

In model (40), ),,( tyxF is the density of effort applied to control the undesired level of precursor

pollutant formed by the population. 01 r is depletion rate coefficient of ),,( tyxQ due to the effort .F 1

is the growth rate coefficient of F and 1 its natural depreciation rate coefficient. cQ is the critical level of

precursor pollutant which is assumed to be harmless to the population. In the last equation of system (40),

)(tH denotes the unit step function which takes into account the case for which .cQQ

6 Conservation Model Without Diffusion

In this section we take 021 DD in model (40). Then model (40) has only one interior equilibrium,

namely, ),,,,,( FUTQPE where UTQP ,,, and F are the positive solutions of the system

of algebraic equations given below:

(say) ),()1()1(

)(

(say) ),(

),()(

21100

1

11101

111

0

PfPh

PhfT

Pfr

QrPQ

TKUrPr

c

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. ),(

,0

(say) ),()()(1

1

1

32001

cc

c

QQQQ

QQ

F

PfPfPhU

As earlier, it is easy to check that E exists if the following inequality holds at E :

.0))(()()())(()()( 32230 PfrPfTKPfKPfUrr (42)

In the following theorem, it is shown that E is locally asymptotically stable. The proof is similar to

Theorem 3.1 and hence is omitted.

Theorem 6.1 Let the following inequalities hold:

2

0 002

4( ) ( ),

( ) 9 ( )

r P rK T T h P

K T K T

(43)

),(9

4001

2 Phch (44)

),(3

2)}({ 012

200211

PhcPhc (45)

where

)(3 200

1 TK

rc

and .)(

2

T

Urc

Then E is locally asymptotically stable.

In the following lemma a region of attraction for system (40) without diffusion is established. The proof of

this lemma is easy and hence is omitted.

Lemma 6.1 The set }0,0 ,0:),,,,{(1

01002

K

FK

UTQKPFUTQP is a

region of attraction for all solutions initiating in the interior of the positive orthant, where

h0 and )}.1(),1(),min{( 11000 hh

The following theorem gives criteria for global stability of E , whose proof is similar to Theorem 3.2 and

hence is omitted.

Theorem 6.2 In addition to the assumptions (3) and (4), let )(Ur and )(TK satisfy in 2 ,

0)( ,)(0 KTKKUr m and 0 ( )K T k ,

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for some positive constants mK, and .k Let the following inequalities hold:

),()(9

40

0

2

200

PhTK

rK

K

kKr

m

(46)

,)(3

2 01

2

0

TK

rK

(47)

),(9

4001

2 Phch (48)

),(3

2)( 01

20011

PhPh (49)

where .)(3 2

001

TK

rc

Then E is globally asymptotically stable with respect to all solutions initiating in the positive orthant.

Theorems (6.1) and (6.2) show that if suitable efforts are made to control the undesired level of precursor

pollutant formed by the activities of populations in the environment, the population density may be maintained

at a desired level under certain conditions.

7 Conservation Model With Diffusion

We now consider the case when )2,1( 0 iDi in model (40). Under an analysis similar to Section 4, it can

be established that if the interior equilibrium E of model (40) with no diffusion is globally asymptotically

stable, then the corresponding uniform steady state of system (40)-(41) is also globally asymptotically stable

with respect to solutions such that

,0),( ,0),( ,0),( ,0),( ,0),( 1 yxyxyxyxyx .),( Dyx

Further, it should be noted that if system (40) with no diffusion is unstable even then the corresponding

uniform steady state of system (40)-(41) can be made stable by increasing diffusion coefficients to sufficiently

large values.

Thus, we conclude that diffusion in our model plays the general role of stabilizing the system.

8 Numerical Simulations

In this section, numerical simulation results are presented to illustrate the results of previous sections. Matlab

7.5 is used for numerical simulation to study the dynamical behaviour of the model system (5). Model (5) is

integrated numerically using the fourth order Runge-Kutta method. We consider the following particular form

of the function in model (5):

0 10

0 1

( ) ,

( ) .

r U r r U

K T K K T

(50)

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The model system (5) displays stable focus for the following set of parameter values given in Eq. (51):

).5,5,5,5(),,,( conditions initialwith

,06.0,02.0,01.0,0.7,20.0,30.0

,04.0,05.0,0.50,08.0,0.60,07.0,0.20

0000

1010

010100

UTQP

hh

KKKrr m

(51)

With above values of parameters, it is found that condition (6) for the existence of interior equilibrium E

is satisfied and it is given by

( , , , ) (58.8821,73.5844, 6.0276, 3.0440).P Q T U (52)

We also note that for the values of parameters given above, all conditions of Theorem (3.1) and (3.2) are

satisfied. This shows that E is locally as well as globally asymptotically stable. The time series analysis of

model system (5) is presented in Fig.1 which shows that the positive equilibrium E is a stable focus.

Fig. 1 Time-series corresponding to the individuals of the model system (5) with parameter values given in Eq. (51).

To study the dynamical behaviour of the model system (5), the temporal evolution of T and U are

observed for different values of control parameters. We observe the temporal dynamics of the concentration of

pollutant T in the atmosphere formed by Q for different control parameters and found that it increases for the

increasing value of the growth rate parameter of Q due to P (i.e., ) but it is of decline nature as we increase

the value of the parameter 0 , the natural depletion rate coefficient of Q. We have presented the increasing and

decreasing nature of T in Fig. 2.

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(a) (b) Fig. 2 Temporal evolution (t vs. T) for the model system (5) with variation of parameters (a) , (b) 0 , and other parameters are given in Eq. (51).

We have also studied the temporal dynamics of uptake concentration of pollutant by the population due to

the variation of parameters , 0 and 1 . It is found that the uptake concentration of pollutant by population

increases if we increase the growth rate coefficient of Q due to P (i.e. ), and it decreases if we increase the

values of natural depletion rate coefficient of Q and U respectively (see Fig. 3).

(a) (b)

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(c) Fig. 3 Temporal evolution (t vs. U) for the model system (5) with variation of parameters (a) , (b) 0 , (c) 1 , and other parameters are given in Eq. (51).

Now if we compare the nature of Fig. 2 with Figs.3(a)-(b), it is found that the growth rate and the natural

depletion rate of Q have almost same impact on the dynamics of the pollutant formed by Q in the atmosphere

and on the uptake concentration of pollutant by the population.

To study the dynamical behaviour of model system (40) without diffusion, we select the same particular

form of the function as given in Eq. (50) and values of parameters are given below in Eq. (53):

0 10 0 1 0

0 1 0 1

1 1 1

20.0, 0.07, 60.0, 0.08, 0.05, 0.04,

0.30, 0.20, 7.0, 0.01, 0.02, 0.06,

0.09, 12.0, 0.9, 0.14,c

r r K K

h h

r Q

(53)

0 0 0 0 0with initial conditions ( , , , , ) (5.0, 5.0, 5.0, 5.0, 5.0).P Q T U F

With above values of parameters, it is found that condition (42) for the existence of interior equilibrium *E is satisfied and is given by

)9893.31,1229.0,2390.0,9680.2,9511.59(),,,,( FUTQP . (54)

By choosing 50.0mK , we note that all conditions of Theorem (6.1) and (6.2) are satisfied. This shows

that equilibrium *E is locally as well as globally asymptotically stable. The time series of model (40) without

diffusion is presented in Fig.4 which shows that the positive equilibrium *E is a stable focus.

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Fig. 4 Time-series corresponding to the individuals of the model system (40) without diffusion with parameter values given in eq. (53).

By comparing Figs.1 and 4, we note that due to effort F, the equilibrium level of the population has

increased whereas equilibrium level of the concentration of precursor pollutant, concentration of pollutant in

the environment and population have decreased.

Fig. 5 Graph of F versus Q for the different initial starts for the set of parameter value given in Eq. (53).

The phase plane analysis of the model system (40) without diffusion in the (Q, F) plane is shown in Fig. 5

which also shows that the positive equilibrium is a stable focus.

The time series analysis of F, the effort applied to control the undesired level of precursor pollutant

formed by the population is shown in Fig.6. It shows the positive and negative impact as we increase the value

of growth rate of Q due to P (i.e., ) and the depletion rate of Q due to F (i.e., 1r ) respectively.

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(a) (b) Fig. 6 Temporal evolution (t vs. F) for the model system (40) without diffusion with variation of parameters (a) , (b) 1r , and other parameters are given in Eq. (53).

The effect of different control parameters on the dynamical behaviour of the conservation model is

presented in Table 1. After varying one of the control parameter in its range, while keeping all others constant,

we monitor the changes in the dynamical behaviour of the model system, thereby fixing the regimes in which

the system exhibit either stable focus or stable limit cycle solution. We have varied the control parameters in

the following ranges:

0 10 0 115 53 26 49 57 149 0 01 0 34 3 6 12 5r , r , K , . h . , . μ . .

From Table 1, it is found that for the parameters 0r in the ranges [22.0, 53.0], 10r in the range [26.0, 27.0],

0K in the range [57.0, 92.0], h in the range [0.01, 0.28] and 1 in the range [3.6, 8.5], the system dynamics

converging to the stable equilibrium and for other ranges it exhibits limit cycle solution. For the lower values

of all the control parameters except for 0r , the intrinsic growth rate of population, the dynamics settled on

equilibrium position and for higher values it shows the periodic nature.

9 Discussions and Conclusions

The proposed nonlinear model is analysed to study the effect of pollution on a population, which is living in an

environment polluted by its own activities. The model has been studied with and without diffusion. In the case

of no diffusion, it has been shown that population density settles down to its equilibrium level, the magnitude

of which depends upon the equilibrium levels of emission and washout rates of pollutant as well as on the rate

of precursor formation and its depletion. It has been noted that the rate of precursor formation is critical in

effecting the population. It has further been noted that if the concentration of pollutant increases unabatedly,

the survival of the population would be threatened.

In case of a model with diffusion, it has been shown that the uniform steady state of the system is

globally asymptotically stable if the corresponding steady state is globally asymptotically stable in case of

without diffusion. It has further been noted that if the positive equilibrium of the system with no diffusion is

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unstable, then the corresponding uniform steady state of the system with diffusion can be made stable by

increasing diffusion coefficients appropriately. Thus, it has been concluded that the global stability is more

plausible in the case of diffusion than the case of no diffusion. It is found that the uptake concentration of

pollutant by population increases if we increase the growth rate coefficient of Q due to P, and it decreases, if

we increase the values of natural depletion rate coefficient of Q, T and U respectively.

Table 1 Simulation experiment of model (40) without diffusion with parameter values 1 40,K 1,

0 0.01, 1 0.3,r 0 0.2,h 1 0.02, 0 0.01, 1 7.0, 0.06, 1 0.4, and 0.14cQ with

initial condition 0 0 0 0 0( , , , , ) (5.0,5.0,5.0,5.0,5.0)P Q T U F and SF: Stable Focus; SLC: Stable Limit Cycle.

Parameter varied

Range in which parameter varied

Dynamical outcome

(P,Q)

(P,T)

(P,U)

(P,F)

(Q,T)

(Q,U)

(Q,F)

(T,U)

(T,F)

(U,F)

0

015 53

r

r

15-21 22-53

SLC SF

SLC SF

SLC SF

SLC SF

SLC SF

SLC SF

SLC SF

SLC SF

SLC SF

SLC SF

10

1026 49

r

r

26-27 28-49

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

0

057 149

K

K

57-92 93-149

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

0.01 0.34

h

h

0.01-0.28 0.29-0.34

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

1

13.6 12.5

3.6-8.5 9.0-12.5

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

SF SLC

In case of conservation model, it has been shown that if the rate of formation of the precursor pollutant is

controlled by some external means, its effect on the population can be minimised. All the above results in the

absence of diffusion are well supported by computer simulations as explained in Section 8. It is also found that

the system dynamics converging to the stable equilibrium for lower values of all the control parameters except

for the intrinsic growth rate parameter 0r of the population and for the higher values it exhibit the limit cycle

solution.

From this study, it can be concluded that the uncontrolled human activities that polluting the environment

may be harmful to itself. Therefore some kind of efforts must be adopted to control the further deterioration of

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the environment. This study also gives some idea about how to prevent the biological species living in an

environment polluted by its own activity and to develop the model related to socio-ecological problems and

about its solution.

Acknowledgements

The first author (BD) gratefully acknowledges the support received by UGC, New Delhi, Grant No.

F.510/2/DRS/2010 (SAP-I).

References

Denn MM. 1972. Stability of Reaction and Transport Processes. Prentice-Hall, Englewood, Cliffs, NJ, USA

Dubey B. 1997. Modelling the effect of toxicant on forestry resources. Indian Journal of Pure and Applied

Mathematics, 28: 1-12

Dubey B, Das B. 1999. Models for the survival of species dependent on resource in industrial environments.

Journal of Mathematical Analysis and Applications, 231: 374-396

Dubey B, Hussain J. 2000a. A model for the allelopathic effect on two competing species. Ecological

Modelling, 129: 195-207

Dubey B, Hussain J. 2000b. Modelling the interaction of two biological species in a polluted environment.

Journal of Mathematical Analysis and Applications, 246: 58-79

Dubey B, Hussain J. 2006. Modelling the survival of species dependent on a resource in a polluted

environment. Nonlinear Analysis: Real World Applications, 7: 187-210

Dubey B, Narayanan AS. 2010. Modelling effects of industrialization, population and pollution on a renewable

resource. Nonlinear Analysis: Real World Applications, 11: 2853-2848

Dubey B, Sharma S, Sinha P, Shukla JB. 2009. Modelling the depletion of forestry resources by population

and population pressure augmented industrialization. Applied Mathematical Modelling, 33: 3002-3014

Dubey B, Upadhyay RK, Hussain J. 2003. Effects of industrialization and pollution on resource biomass: a

mathematical model. Ecological Modelling, 167: 83-95

Freedman HI, Shukla JB. 1991. Models for the effect of toxicant in single-species and predator-prey systems.

Journal of Mathematical Biology, 30: 15-30

Ghosh M. 2000. Industrial pollution and asthma: a mathematical model. Journal of Biological Systems, 8(4):

347-371

Ghosh M, Chandra P, Sinha P. 2002. A mathematical model to study the effect of toxic chemicals on a prey-

predator type fishery. Journal of Biological Systems, 10(2): 97-105

Ghosh M, Chandra P, Sinha P, Shukla, JB. 2006. Modelling the spread of bacterial infectious disease with

environmental effect in a logistically growing human population. Nonlinear Analysis: Real World

Applications, 7(3): 341 – 363

Hallam TG, Clark CE, Jordan GS. 1983a. Effects of toxicants on populations: a qualitative approach II. First

order kinetics. Journal of Mathematical Biology, 18: 25-37

Hallam TG, Clark CE, Lassiter RR. 1983b. Effects of toxicants on populations: a qualitative approach I.

Equilibrium environmental exposure. Ecological Modelling, 18: 291-304

173


IAEES www.iaees.org

Hallam TG, Luna De JT. 1984. Effects of toxicants on populations: a qualitative approach III. Environmental

and food chain pathways. Journal of Theoretical Biology, 109: 411-429

Hallam TG, Ma Z. 1986. Persistence in population models with demographic fluctuations. Journal of

Mathematical Biology, 24: 327-339

Hari P, Raunemaa T, Hautojarvi A. 1986. The effect on forest growth of air pollution from energy production.

Atmospheric Environment, 20(1): 129-137

Huaping L, Ma Z. 1991. The threshold of survival for system of two species in a polluted environment. Journal

of Mathematical Biology, 30: 49-61

Luna De JT, Hallam TG. 1987. Effects of toxicants on populations: a qualitative approach IV. Resource-

Consumer-Toxicant models. Ecological Modelling, 35: 249-273

McLaughlin SB. 1985. Effects of air pollution on forests. Journal of Air Pollution and Control Association, 35:

512-534

Naresh R, Sundar S, Shukla JB. 2006a. Modelling the effect of an intermediate toxic product formed by uptake

of a toxicant on plant biomass. Applied Mathematics and Computation, 182: 151-160

Naresh R, Sundar S, Upadhyay RK. 2006b. Modelling the removal of primary and secondary air pollutants by

precipitation. International Journal of Nonlinear Sciences and Numerical Simulation, 7(3): 285-294

Rescigno A. 1977. The struggle for life-V. One species living in a limited environment. Bulletin of


Schulze ED. 1989. Air pollution and forest decline in a Spruce (Picea abies) forest. Science, 224: 776-783

Shukla JB, Agrawal AK, Dubey B, Sinha P. 2001. Existence and survival of two competing species in a

polluted environment: a mathematical model. Journal of Biological Systems, 2: 89-103

Shukla JB, Agrawal AK, Sinha P, Dubey B. 2003. Modelling effects of primary and secondary toxicants on

renewable resources. Natural Resource Modelling, 16: 99-120

Shukla JB, Dubey B. 1996. Simultaneous effect of two toxicants on biological species: a mathematical model.

Journal of Biological Systems, 4(1): 109-130

Shukla JB, Dubey B. 1997. Modelling the depletion and conservation of forestry resources: effects of

population and pollution. Journal of Mathematical Biology, 36: 71-94

Shukla JB, Sharma S, Dubey B, Sinha P. 2009. Modeling the survival of a resource dependent population:

effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors.

Nonlinear Analysis: Real World Applications, 10: 54-70

Smith WH. 1981. Air Pollution and Forests. Springer-Verlag, New York, USA

Srinivasu PDN. 2002. Control of environmental pollution to conserve a population. Nonlinear Analysis: Real

World Applications, 3: 397-411

Sundar S. 2013. An ecological type nonlinear model for the removal of carbon dioxide from the atmosphere by

introducing liquid species. Computational Ecology and Software, 3(2): 33-43

Sundar S, Naresh R, Mishra, AK, Tripathi A. 2014. Modelling the dynamics of carbon dioxide removal in the

atmosphere. Computational Ecology and Software, 4(4): 248-268

Woodman JN, Cowling EB. 1987. Airborne chemical and forest health. Environmental Science and

Technology 21: 120-126

Woodwell GM. 1970. Effects of pollution on the structure and physiology of ecosystems. Science, 168: 429-

431

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IAEES www.iaees.org

Article

Hopf bifurcation and stability analysis for a delayed logistic equation

with additive Allee effect

E.M. Elabbasy, Waleed A.I. Elmorsi University of Mansoura, Mansoura, Egypt


Received 19 November 2014; Accepted 25 December 2014; Published online 1 June 2015

Abstract

In this paper the linear stability of the delayed logistic equation with additive Allee effect is investigated. We

also analyze the associated characteristic transcendental equation, to show the occurrence of Hopf bifurcation

at the positive equilibrium. To determine the direction of Hopf bifurcation and the stability of bifurcating

periodic solution, we use the normal form approach and a center manifold theorem. Finally, a numerical

example is given to demonstrate the effectiveness of the theoretical analysis.

Keywords time delay; logistic equation; stability; Hopf bifurcation; additive Allee effect.

1 Introduction

Considering the fact that the environment has limited resources, the Belgium mathematician Pierre-Francois

Verhulst (Verhulst, 1838) proposed one of the most famous equation that used to model a lot of applications in

ecology and biology. The logistic equation - also known as Verhulst model- is a model of population growth

first proposed by Verhulst (1845, 1847). Verhulst (Agarwal et al., 2014) argued that the unlimited growth in

the exponential growth model must be restricted by the Malthusian “struggle for existence”

and he proposed the model

(1)

Equation (1) is called logistic growth in a population where is the intrinsic growth rate and

is the carrying capacity (the maximum number of individuals that the environment can support). We can see

that is globally stable steady state for equation (1) with any initial condition. If the initial condition

is more than (less than) then the population decreases (increase) approaching as tends to ∞ .

The logistic equation has a lot of applications in many fields like economy (Shone, 2002), ecology (Pastor,

2008), biology (Murray, 2002), medicine (Forys and Marciniak-Czochra, 2003) and neurosciences

(Gershenfeld, 1999). To know more about the history of the logistic equation see Kingsland (1982).



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In the last few years the importance of embedding the time delay into dynamical systems was increased,

especially in ecological and biological systems because in these systems the reproduction is not instantaneous.

Incorporating the time delay into system allow the system rate of change to depend on his own past history.

Also by using time delay in equations that model eco-systems or bio-systems, phenomena as feeding time,

reaction time, maturation periods, etc., can be represented.

Time delays have been incorporated into biological and ecological models to fix the deficiencies of

ordinary differential equation that ignored important phenomena. Furthermore, so many of the processes, both

natural and man-made, in medicine, diseases, physics, chemistry, bio-systems, eco-systems, economics, etc.,

involve time delays. In general, delay differential equations exhibit much more complicated dynamics than

ordinary differential equations. For these reasons, the researchers in many fields pay great attention for

studying delayed systems (Agarwal et al., 2014; Ding et al., 2013; Engelborghs et al., 2002; Ruan, 2006;

Kuang, 1993; Braddock, 1983; Bi and Xiao, 2014; Hu and Li, 2012).

To make the logistic equation more realistic, Hutchinson [Hutchinson, 1948] proposed incorporating the

effect of delay and he introduced the delayed logistic equation

(2)

where is time delay. For other formula of delayed logistic equation see (Arino et al., 2006).

Hutchinson suggested that the equation (2) can be used to model the dynamics of a single species

population growing towards a saturation level with a constant reproduction rate (Kuang, 1993;

Gobalsamy, 1992; Cuching, 1977).

More interesting topological changes in the population size as limit cycles, chaos and damped oscillations

are produced in the existence of delay (Storgaz, 1994).

Noticing the behavior of species one can see that some species often help each other in their search for

food or habitat and to escape from their predators. For example, some social species such as ants, bees, etc.,

have developed complex cooperative behavior involving division of labor, altruism, etc. Such cooperative

processes have a positive feedback influence since individuals have been provided a greater chance to survive

and reproduce as density increase.

The ecologist Warder Clyde Allee (Allee, 1931) paid a lot of attention to aggregation and associated

cooperative and social characteristics among members of a species in animal populations, and his work has

been among the most influential for animal behavioral research.

In numerous writings (Allee, 1931; Allee, 1941; Allee et al., 1949) Allee shows that for a variety of

biological reasons positive (negative) feedback effects can happen at low (high) population density. The

positive feedback is called Allee effects (Dennis, 1989; Stephens et al., 1999). In population dynamics, the

Allee effect refers to a process that reduces the growth rate for small population densities.

The so-called Allee effect (Elabbasy et al., 2007) refers to a population that has a maximal per capita

growth rate at intermediate density. This occurs when the per capita growth rate increases as density increases

and decreases after the density passes a certain value.

Modelling Allee effects in population dynamics and fields that related to it as a multi-species interactions

in eco-systems, disease dynamics and the spread of epidemics, etc., has great interest in mathematical literature.

(Dennis, 1989; Elaydi and Sacker, 2010; Courchamp et al., 2008; Schreiber, 2003; Cushing and Hudsona,

2012; Lewis and Kareiva, 1993)

The equation

(3)

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is the prototypical model for a multiplicative Allee effect where is the intrinsic growth rate and is the

carrying capacity. If it shows the weak Allee effect, while if , it shows the

strong Allee effect.

The strong Allee effect introduces a population threshold (the minimal size of the population required to

survive), and the population must surpass this threshold to grow. In contrast, a population with a weak Allee

effect does not have a threshold [Wang et al, 2011; Wang and Kot, 2001].

Dennis (Dennis, 1989) who first introduced the equation that modeled the additive Allee effect in the

form

, (4)

and then it used in [Aguirre et al, 2009].

The term is called the additive Allee effect where and are called Allee

constants with . If then the equation (4) exhibits a weak Allee effect and if then it

exhibits a strong Allee effect (Wang and Kot, 2001).

In our paper, we study the delayed logistic equation with additive Allee effect in the form

(5)

2 Local Stability and Existence of Hopf Bifurcation

The model – at - has a trivial equilibrium , and positive equilibrium

. And at the model (5) has a trivial equilibrium and two positive

equilibrium

and .

For convenience, we indicate to the next lemma which consolidates our stability analysis.

Lemma 1 (Hale and Lunel, 1993)

All roots of the characteristic equation , where are real, have negative real

parts if and only if

,

and

where is the root of , if and if .

Theorem 1

(I) At

1. The equilibrium of equation (5) is unstable.

2. The equilibrium of Eq. (5) is stable if and is unstable if .

(II) At 0

1. The equilibrium 0 of equation (5) is stable.

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2. The two equilibrium and of equation

5 are stable if and are unstable if .

Proof.

I Linearizing equation about equilibrium using , it becomes

6

It is easy to show that equation has the characteristic equation in the form

(7)

Since 0 then 0

Then the model is unstable.

Again, by linearizing about equation 5 will be

1 (8)

Or

(9)

Where 1 and

Equation (9) has the characteristic equation

0 (10)

Let be the root of equation 10 ; then:

cos sin 0

cos sin 0 (11)

Then, by separating and equating real parts and imaginary parts

cos 0 (12.a)

sin 0 (12.b)

Then (13)

From 13 and using 12. , 12.

tan (14)

By the same way, we can prove part (II).

Theorem 2

If τ α τ i τ denote a root of Eq. (10) near τ τ , such that τ and

α τ 0 then

dα τ

dτ0

Proof.

By differentiating the characteristic equation (10) with respect to we get

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1

This gives

Then

Sign sign sign

sign sign1

0

Theorem 2 stated the last condition for the occurrence of Hopf bifurcations and the results can be introduced as

follows.

Theorem 3.

(I) in the case if 0 , When the parameter τ passes through the critical value τ , there

are Hopf bifurcations at the equilibrium

to a periodic orbit.

(II) In the case if 0 , when the parameter τ passes through the critical

value τ , there are Hopf bifurcations at the equilibriums

and to a periodic orbit.

3 Stability and Direction of the Hopf Bifurcation

Let , then the equation 5 written as

1

1 (15)

In 1, 0 , equation 15 written as

, (16)

Where

0 1

, 0 1

0 (17)

Using the Riesz representation theorem, there exists a function , of bounded variation for 1,0

such that

, 0 for (18)

We can choose

, 1 (19)

where is the Dirac delta function.

For 1,0 , ; define

1,0

, θ 0 (20)

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and

0 1,0

, 0 21

Then the system 16 can be written as operation equation

22

where

1,0

The equation 22 is more mathematically pleased because this equation involves a single unknown variable

.

For ψ 0, 1 , the adjoint operator of is defined as

ψ

ψ 1,0

, ψ s 0 23

For 1,0 and ψ 0, 1 , the bilinear inner product defined as

ψ , ψ 0 0 ψ 24

To determine the Poincare normal form of the operator we need to calculate the eigenvector and

of and that corresponding to the eigenvalues and respectively.

It is easy to be verified that and .

In order to assure that , 1, we need to determine the value. From 24

, 0 0

, 1 1

1

1 25

Hassard et al. (1981) introduced a method to compute the co-ordinates that describe the center manifold

at 0 .

Tracking Hassard method, for , a solution of 22 at 0 , we define:

, and , 2 26

On the center manifold we have:

, , , , Where

, , 27

where are local co-ordinates for center manifold in in the direction of and . Note that ,

is real if is real. We shall deal with real solution only.

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Now, for solution of equation 22

, , ,

Then since 0

, ,

0 , 28

Equation (28) can be written in abbreviated form as

, 29

Where

, 0 ,

,2 2

30

Since from 26

Then

2 0 , 1,0

2 0 , , 0 31

This can be written as

, , 32

where

, , 2 2

33

Since

, 0 ,

0 0,

0 1 0 (34)

Using equations 26 in 34 and comparing coefficient with 30 , we find

2 35.

(35.b

2 35.

0 0 1 1 2 0

0 35.d

Since , 36

Using 32 and substitution by the expansions of previous functions and comparing coefficients we find that

2 – 37

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38

2 – 39

Since from 31 , 32 and 33 we find

40

41

42

From 37 and 40

2

2 43

This equation has the solution

3

44

By the same way

45

Where and are constants and they are evaluated from the formulas

2

2 46

2

/ 47

We can also compute

0 2

2| || |3 2

0

2 0

48

Theorem (Hassard et al., 1981): In (48), the directions of Hopf bifurcation are determined by the sign of

and the stability of bifurcating periodic solutions by the sign of . In this case, if 0 0 , then the

Hopf bifurcation is supercritical (subcritical) and if 0 0 the bifurcating periodic solutions are

orbitally stable (unstable).

4 Numerical Example

In this section, we give some numerical simulations supporting our theoretical analysis. In the first case

( 0), by choosing r 3 , 4 , 0.5 ; 0.75 and 0.5 for , 0 , fig. 1

at 0.6 shows the existence of Hopf bifurcation and limit cycle behavior for model (5).

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(a) ( b)

Fig. 1 Wave form plot (a) and phase plot (b) at 0.6 for the case 0 for model (5).

The wave form plot and the phase plot in Fig. 2 show the periodicity of the solution and existence of an

attractor for model (5) at 1.

(a) (b)

Fig. 2 Wave form plot (a) and phase plot (b) at 1 for the case ( 0 ) for model (5).

In the second case ( 0 ) - we will choose 1 , 4, 0.9 ; 0.6 and

0.5 for , , 0 -Fig. 3 shows that the equilibrium point for the model (5) is asymptotically stable at

=1.5.

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(a) (b)

Fig. 3 Wave form plot (a) and phase plot (b) at 1.5for the case ( 0 ) for model (5).

In Fig. 4, wave form plot and phase plot at =2 show the existence of Hopf bifurcation and limit cycle

behavior for model (5).

(a) (b)

Fig. 4 Wave form plot (a) and phase plot (b) at =2 for the case ( 0 ) for model (5).

5 Conclusions

In this paper, we have investigated the stability and Hopf bifurcation of a delayed logistic equation with

additive Allee effect. Also we have obtained stability conditions and we showed that a Hopf Bifurcation will

occur when the time delay parameter pass through critical values; that is, a family of periodic orbits bifurcates

from the equilibrium. The direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are

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discussed by applying the normal form approach and the center manifold theorem. Numerical simulations have

shown that the analytical results are correct.

References

Agarwal RP, O’Regan D, Saker S. 2014. Oscillation and Stability of Delay Models in Biology. Springer,

Netherlands

Aguirre P, Gonzalez-Olivares E, Saez E. 2009. Two limit cycles in a Leslie–Gower predator–prey model with

additive Allee effect. Nonlinear Analysis: Real World Applications, 10: 1401-1416

Aguirre P, Gonzalez-Olivares E, Saez E. 2009. Three limit cycles in a Leslie–Gower predator–prey model with

additive Allee effect. SIAM Journal of Applied Mathematics, 69: 1244-1269

Allee WC. 1931. Animal Aggregation: A study In General Sociology. The University of Chicago Press, USA

Allee WC. 1941. The Social Life of Animals (Third edition). William Heineman, London and Toronto, UK

Allee WC, Park O, Park T, et al. 1949. Principles of Animal Ecology. WB Saunders, Philadelphia, USA

Arino J, Wang L, Wolkowicz GSK. 2006. An alternative formulation for a delayed logistic equation. Journal

of Theoretical Biology, 241: 109-119

Bi P, Xiao H. 2014. Bifurcations of tumor-immune competition systems with delay. Abstract and Applied

Analysis, (10.1155/2014/723159)

Braddock RD, Driessche PV. 1983. On a two lag differential equation. Journal of the Australian Mathematical

Society Series B, 24: 292-317

Courchamp F, Berec L, Gascoigne J.2008. Allee Effects in Ecology and Conservation. Oxford University

Press, UK

Cushing JM. 1977. Integro-differential Equations and Delay Models in Population Dynamics. Lecture Notes in

Biomathematics, Springer, Berlin, Germany

Cushing JM, Hudsona JT. 2012. Evolutionary dynamics and strong Allee effects. Journal of Biological

Dynamics, 6: 941-958

Dennis B. 1989. Allee effects: Population growth, critical density, and the chance of extinction. Natural

Resource Model, 3: 481-538

Ding Y, Jiang W, Yu P. 2013. Bifurcation analysis in a recurrent neural network model with delays.

Communications in Nonlinear Science and Numerical Simulation, 18: 351-372

Elabbasy EM, Saker SH, EL-Metwally H. 2007. Oscillation and stability of nonlinear discrete models

exhibiting the Allee effect, Mathematica Versita Slovaca, 57: 243-258

Elaydi SN, Sacker RJ. 2010. Population models with Allee effect: A new model. Journal of Biological

Dynamics, 4: 397-408

Engelborghs K, Luzyanina T, Roose D. 2002. Numerical bifurcation analysis of delay differential equations

using dde-biftool. ACM Transactions on Mathematical Software, 28: 1-21

Forys U, Marciniak-Czochra A. 2003. Logistic equation in tumor growth modelling. International Journal of

Applied Mathematics and Computer Science, 13: 317-325

Gershenfeld NA.1999. The Nature of Mathematical Modeling. Cambridge University Press, Cambridge, USA

Gopalsamy K. 1992. Stability and Oscillations in Delay Differential Equations of Population Dynamics,

Kluwer Academic Press, London, UK

Hale J, Lunel MV.1993. Introduction to Functional Differential Equations. Springer, New York, USA

Hassard BD, Kazarinoff ND, Wan YH. 1981. Theory and Applications of Hopf Bifurcation. Cambridge

University Press, Cambridge, USA

185


IAEES www.iaees.org

Hu GP, Li XL. 2012. Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey,

Chaos, Solitons and Fractals, 45: 229-237

Hutchinson GE. 1948. Circular casual systems in ecology. Annals of New York Academy of Sciences, 50:

221-246

Kingsland S. 1982.The refractory model: The logistic curve and the history of population of ecology. The

Quarterly Review of Biology.57: 29-52

Kuang Y. 1993. Delay Differential Equations with Applications in Population Dynamics. Academic Press

Incorporation, USA

Lewis MA, Kareiva P. 1993. Allee dynamics and the spread of invading organisms. Theoretical Population

Biology, 43: 141-158

Murray JD. 2002. Mathematical Biology I. An Introduction (Third edition). Springer, Netherlands

Pastor J. 2008. Mathematical Ecology of Populations and Ecosystems. Wiley-Blackwell, USA

Ruan S. 2006. Delay differential equations in single species dynamics. In: Delay Differential Equations and

Applications (Arino O, Hbid ML, eds). 477-517, Springer, Berlin, Germany

Schreiber SJ. 2003. Allee effects, extinctions, and chaotic transients in simple population models. Theoretical

Population Biology, 64: 201-209

Shone R. 2002. Economic Dynamics: Phase Diagrams and Their Economic Application. Second edition.

Cambridge University Press, USA

Stephens PA, Sutherland WJ, Freckleton RP. 1999. What is the Allee effect? Oikos, 87: 185-190

Storgaz SH. 1994. Nonlinear dynamics and Chaos. Perseus Books Publishing, USA

Verhulst PF. 1838. Notice sur la loique la population suit dans son accroissement. Corresp. Math. et Phys., 10:

113-121

Wang J, Shi J, Wei J. 2011. Predator–prey system with strong Allee effect in prey. Journal of Mathematical

Biology, 62: 291-331

Wang MH, Kot M. 2001. Speeds of invasion in a model with strong or weak Allee effects. Mathematical

Bioscience, 171: 83-97

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Article

Bifurcation and complex dynamics of a discrete-time predator–prey

system

S. M. Sohel Rana Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh


Received 22 December 2014; Accepted 10 February 2015; Published online 1 June 2015

Abstract

In this paper, we investigate the dynamics of a discrete-time predator-prey system of Holling-I type in the

closed first quadrant 2R . The existence and local stability of positive fixed point of the discrete dynamical

system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-

Sacker bifurcation in the interior of 2R by using bifurcation theory. It has been found that the dynamical

behavior of the model is very sensitive to the parameter values and the initial conditions. Numerical simulation

results not only show the consistence with the theoretical analysis but also display the new and interesting

dynamic behaviors, including phase portraits, period-9, 10, 20-orbits, attracting invariant circle, cascade of

period-doubling bifurcation from period-20 leading to chaos, quasi-periodic orbits, and sudden disappearance

of the chaotic dynamics and attracting chaotic set. In particular, we observe that when the prey is in chaotic

dynamic, the predator can tend to extinction or to a stable equilibrium. The Lyapunov exponents are

numerically computed to characterize the complexity of the dynamical behaviors. The analysis and results in

this paper are interesting in mathematics and biology.

Keywords discrete-time predator-prey system; chaos; flip and Neimark-Sacker bifurcations; Lyapunov

exponents.

1 Introduction

The dynamics of predator-prey interaction is the starting point for many variations that yield more realistic

biological and mathematical problems in population ecology. Predation is a direct interaction which occurs

when individuals from one population derive their nourishment by capturing and ingesting individuals from

another population. There are many articles devoted to the study of predator-prey interaction both from the

experimental and the modeling point of view. It is well known the Lotka-Voltera predator-prey model is one of

the fundamental population models; a predator-prey interaction has been described firstly by two pioneers

Lotka (1924) and Voltera (1926) in two independent works. After them, more realistic prey-predator model



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were introduced by Holling suggesting three types of functional responses for different species to model the

phenomena of predation (Holling, 1965).

Qualitative analyses of prey-predator models describe by set of differential equations were studied by

many authors (Brauer and Castillo, 2001; Hastings and Powell, 1991; Klebanoff and Hastings, 1994; May,

1974; Murray, 1998; Zhu et al., 2002). Another possible way to understand a prey-predator interaction is by

using discrete-time models. These models are more reasonable than the continuous time models when

populations have non-overlapping generations (Brauer and Castillo, 2001; Murray, 1998) and lead to

unpredictable dynamic behaviors from a biological point of view. This suggests the possibility that the

governing laws of ecological systems may be relatively simple and therefore discoverable. The author (May,

1975, 1976) had clearly documented the rich array of dynamic behavior possible in simple discrete-time

models. Recently, there is a growing evidence showing that the dynamics of the discrete-time prey-predator

models can present a much richer set of patterns than those observed in continuous-time models (Agiza et al.,

2009; Danca et al., 1997; Elsadany et al., 2012; Hasan et al., 2012; He and Lai, 2011; Jing and Yang, 2006; Li,

1975; Liu, 2007; Hu et al., 2011; He and Li, 2014). However, there are few articles discussing the dynamical

behaviors of predator-prey models, which include bifurcations and chaos phenomena for the discrete-time

models. The authors (He and Lai, 2011; Jing, 2006; Liu, 2007; Hu et al., 2011) obtained the flip bifurcation by

using the center manifold theorem and bifurcation theory. But in (Agiza et al., 2009; Danca et al., 1997;

Elsadany et al., 2012), the authors only showed the flip bifurcation and Hopf bifurcation by using numerical

simulations. In this work, we confine our interest to present, by using both analytic and numerical methods, the

domains of the values of the parameters under which the system predicts that the populations will be able to

persist at a steady state, the conditions for flip and/or Neimark-Sacker bifurcations by using the normal form

theory of the discrete system (see section 4, Kuznetsov, 1998) and the domain for the presence of chaos in the

system by measuring the maximum Lyapunov exponents.

In ecology, many species have no overlap between successive generations, and thus their population

evolves in discrete-time steps (Murray, 1998). Such a population dynamics is described by difference equation.

Let nx denotes the number of prey population and ny the number of predator population in the n th

generation. Our model is described by the following system of nonlinear difference equations in non-

dimensional form:

nnnn

nnnnn

dyybxy

yaxxrxxH

1

1 )1(: (1)

In the system (1), the prey grows logistically with intrinsic growth rate r and carrying capacity one in

the absence of predation. The predator consumes the prey with functional response Holling type I. All

parameters dbar ,,, have positive values that stand for prey intrinsic growth rate, per capita searching

efficiency of the predator, conversion rate, and the death rate of the predator, respectively. From mathematical

and biological point of view, we will pay attention on the dynamical behaviors of (1) in the closed first

quadrant 2R . Starting with initial population size 00 , yx , the iteration of system (1) is uniquely

determined a trajectory of the states of population output in the following form

00 ,, yxHyx nnn , where ,2,1,0n .

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Our results in this paper are extension to those in (Danca et al., 1997; Elsadany et al., 2012). This paper is

organized as follows. In Section 2, we discuss the existence and local stability of positive fixed point for

system (1) in 2R . In Section 3, we show that there exist some values of the parameters such that (1)

undergoes the flip bifurcation and the Neimark-Sacker bifurcation in the interior of 2R . In section 4, we

present the numerical simulations which not only illustrate our results with theoretical analysis but also exhibit

complex dynamical behaviors such as the cascade periodic-doubling bifurcation in periods 2, 4, 8, 9, 10, 20-

orbits, quasi-periodic orbits and chaotic sets. Finally a short discussion is given in Section 5.

2 Existence and Local Stability of Fixed Points

In this section, we shall first discuss the existence of fixed points for (1), then study the stability of the fixed

point by the eigenvalues for the Jacobian matrix of (1) at the fixed point.It is clear that the system (1) has the

following fixed points in the ),( yx -plane:

0,00E ,

0,1

1 r

rE and **

2 , yxE , whereb

dx

1* and ab

d

a

ry

111*

.

To discuss the existence of fixed points, we say that fixed points will not exist if any one of its

components is negative. The fixed point 0E always exists. The existence condition for 1E is 1r . Finally,

the feasibility condition for the positive fixed point 2E is

db

br

1 (or 1,

1

)1(

rr

drb ).

Now we study the stability of the positive fixed point (we left the others) only. Note that the local stability

of the fixed point ),( yx is determined by the modules of eigenvalues of the characteristic equation at the

fixed point.

The Jacobian matrix due to the linearization of (1) evaluated at 2E is given by

1

)1()1(

)1()1(1

, **

a

rdrbb

da

b

rd

yxJ

and the characteristic equation of the Jacobian matrix J can be written as

0212 (2)

whereb

rdtrJ

)1(21

and d

b

drdJ

21)1(det2 .

Therefore, the eigenvalues of J are

b

rd

2

)1(1

21

2,1

(3)

where db

drd

b

rd

1)1(

2

)1(1

2

2

2

2

1 .

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Using Jury’s criterion (Elaydi, 1996), we have necessary and sufficient condition for local stability of the

fixed point 2E which are given in the following proposition.

Proposition 1. Whendb

br

1, then system (1) has a positive fixed point 2E and

(i) it is a sink if one of the following conditions holds:

(i.1) 0 and 1

)2(

)1(3

)3)(1(

r

rdb

rdd

rdd;

(i.2) 0 and 1

)2(

r

rdb .

(ii) it is a source if one of the following conditions holds:

(ii.1) 0 and

1

)2(,

)1(3

)3)(1(max

r

rd

rdd

rddb ;

(ii.2) 0 and 1

)2(

r

rdb .

(iii) it is non-hyperbolic if one of the following conditions holds:

(iii.1) 0 and rdd

rddb

)1(3

)3)(1(

;

(iii.2) 0 and 1

)2(

r

rdb .

(iv) it is a saddle for the other values of parameters except those values in (i)–(iii).

Following Jury’s criterion, we can see that one of the eigenvalues of 2EJ is 1 and the others are

neither 1 nor 1 if the term (iii.1) of Proposition 1 holds. Therefore, there may be flip bifurcation of the

fixed point 2E if r varies in the small neighborhood of 2EFB where

0,,,1,0,

)3)(1(

)3(:,,,

2dbar

dbd

dbrdbarFBE .

Also when the term (iii.2) of Proposition 1 holds, we can obtain that the eigenvalues of 2EJ are a pair

of conjugate complex numbers with module one. The conditions in the term (iii.2) of Proposition 1 can be

written as the following set:

0,,,1,0,

2:,,,

2dbar

db

brdbarNSE

and if the parameter r varies in the small neighborhood of 2ENS ; then the Neimark-Sacker bifurcation will

appear.

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3 Flip Bifurcation and Neimark-Sacker Bifurcation

In this section, we choose the parameter r as a bifurcation parameter to study the flip bifurcation and the

Neimark-Sacker bifurcation of 2E by using bifurcation theory in (see Section 4 in Kuznetsov, 1998; see also

Guckenheimer and Holmes, 1983; Robinson, 1999; Wiggins, 2003).

We first discuss the flip bifurcation of (1) at 2E . Suppose that 0 , i.e.,

01)1(2

)1(1

2

d

b

drd

b

rd. (4)

Let )3)(1(

)3(1 dbd

dbr

, then the eigenvalues of J are

1)( 11 r , and db

dbr

3

436)( 12 .

The condition 1)( 12 r leads to

13

436

db

db. (5)

Let *~ xxx , *~ yyy and **, yxJrA , we transform the fixed point **, yx of system

(1) into the origin, then system (1) becomes

ryxF

ryxF

y

xrA

y

x

,~,~,~,~

~

~

~

~

2

1 (6)

where

,~~,~,~

,~~~,~,~4

2

421

XyxbryxF

XyxaxrryxF

(7)

and TyxX ~,~ . It follows that

,2),(

, 122111

2

1,0

12

1 yaxyaxyrxyxrF

yxBkj

kjkj

,),(

, 1221

2

1,0

22

2 ybxybxyxrF

yxBkj

kjkj

,0),(

,,2

1,,0

13

1

lkjlkj

lkj

uyxrF

uyxC

,0),(

,,2

1,,0

23

2

lkjlkj

lkj

uyxrF

uyxC

and 1rr .

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Therefore,

yxB

yxByxB

,

,,

2

1 and

uyxC

uyxCuyxC

,,

,,,,

2

1 are symmetric multilinear vector

functions of uyx ,, R2.

We know that A has simple eigenvalue 1)( 11 r , and the corresponding eigenspace cE is one-

dimensional and spanned by an eigenvector q R2 such that qqrA )( 1 . Let p R2 be the adjoint

eigenvector, that is, pprAT )( 1 . By direct calculation we obtain

Tbybxdq ** ,1~ ,

Taybxdp ** ,1~ .

In order to normalize p with respect to q , we denote

Taybxdp **1 ,1

where

**2*11

1

yabxbxd .

It is easy to see 1, qp , where , means the standard scalar product in R2:

2211, qpqpqp .

Following the algorithms given in (Kuznetsov, 1998), the sign of the critical normal form coefficient

11 r , which determines the direction of the flip bifurcation, is given by the following formula:

qqBIAqBpqqqCpr ,)(,,2

1,,,

6

1 111

(8)

From the above analysis and the theorem in (Kuznetsov, 1998; Guckenheimer and Holmes, 1983; Robinson,

1999; Wiggins, 2003), we have the following result.

Theorem 1. Suppose that **, yx is the positive fixed point. If the conditions (4), (5) hold and 011 r ,

then system (1) undergoes a flip bifurcation at the fixed point **, yx when the parameter r varies in a

small neighborhood of 1r . Moreover, if 011 r (respectively, 011 r ), then the period-2 orbits that

bifurcate from **, yx are stable (respectively, unstable).

In Section 4, we will give some values of the parameters such that 011 r , thus the flip bifurcation

occurs as r varies (see Figure 1).

We next discuss the existence of a Neimark-Sacker bifurcation by using the Neimark-Sacker theorem in

(Kuznetsov, 1998; Guckenheimer and Holmes, 1983; Robinson, 1999; Wiggins, 2003).

It is clear that the eigenvalues 2,1 given by (3) are complex for 0det42 JtrJ , which leads to

0 , i.e.,

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01)1(2

)1(1

2

d

b

drd

b

rd (9)

Let db

br

22 ,

then we have 1)(det 2 rJ .

For 2rr , the eigenvalues of the matrix associated with the linearization of the map (6) at

0,0, ** yx are conjugate with modulus 1, and they are written as

)()(422

)(2

2122

212,1 rr

ir (10)

and 1)( 2 ri , 02

)1)(2()(

2

b

ddb

dr

rd

rr

i, 2,1i . Note that 12 .

In addition, if 1,0)( 2 rtrJ , which leads to

3,2)1( 2

b

rd,

then we have 1)( 2 rki for .4,3,2,1k

Let q C2 be an eigenvector of )( 2rA corresponding to the eigenvalue )( 21 r such that

qrqrA )()( 212 , qrqrA )()( 212 .

Also let p C2 be an eigenvector of the transposed matrix )( 2rAT corresponding to its eigenvalue,

that is, )()( 2221 rr ,

prprAT )()( 222 , prprAT )()( 222 .

By direct calculation we obtain

Tbybxdq **1 ,~ ,

Taybxdp **2 ,~ .

In order to normalize p with respect to q , we denote

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Taybxdp **22 ,

where

**2*2

2

1

yabxbxd

.

It is easy to see 1, qp , where , means the standard scalar product in C2:

2211, qpqpqp .

Any vector X R2 can be represented for r near 2r as qzqzX ,for some complex z . Indeed,

the explicit formula to determine z is Xpz , . Thus, system (6) can be transformed for sufficiently

small r (near 2r ) into the following form:

),,()(1 rzzgzrz ,

where )(1 r can be written as )(1 )(1)( rierr (where )(r is a smooth function with

0)( 2 r ) and g is a complex-valued smooth function of rzz and,, , whose Taylor expression with

respect to ),( zz contains quadratic and higher-order terms:

2

)(!!

1),,(

lk

lkkl zzrg

lkrzzg , with klg , ,1,0, lk .

By symmetric multilinear vector functions, the Taylor coefficients klg can be expressed by the formulas

qqBprg ,,220 , qqBprg ,,211 ,

qqBprg ,,202 , qqqCprg ,,,221 ,

and the coefficient 22 r , which determines the direction of the appearance of the invariant curve in a

generic system exhibiting the Neimark-Sacker bifurcation, can be computed via

2

02

2

111120)(

)(2)(21

)(

22 4

1

2

1

12

21Re

2Re

2

222

gggge

eeger

ri

ririri

,

where )( 21)( 2 re ri .

For the above argument and the theorem in (Kuznetsov, 1998; Guckenheimer and Holmes, 1983;

Robinson, 1999; Wiggins, 2003), we have the following result.

Theorem 2. Suppose that **, yx is the positive fixed point. If 022 r (respectively, 0 ) the

Neimark-Sacker bifurcation of system (1) at 2rr is supercritical (respectively, subcritical) and there exists

a unique closed invariant curve bifurcation from **, yx for 2rr , which is asymptotically stable

(respectively, unstable).

In Section 4 we will choose some values of the parameters so as to show the process of a Neimark-Sacker

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bifurcation for system (1) in Figure 2 by numerical simulation.


In this section, our aim is to present numerical simulations to explain the above theoretical analysis, especially

the bifurcation diagrams, phase portraits and Maximum Lyapunov exponents for system (1) around the

positive fixed point 2E and show the new interesting complex dynamical behaviors. It is known that

Maximum Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories

and frequently employ to identify a chaotic behaviour. We choose the growth rate of prey, r as the real

bifurcation parameter (varied parameter) and other model parameters are as fixed parameters, otherwise stated.

For showing the dynamics of the system (1) change, the bifurcation parameters are considered in the following

cases:

Case (i): varying r in range 43 r , and 25.0,95.1,3 dba fixing.

Case (ii): varying r in range 97.21 r , and 25.0,5.4,5.3 dba fixing.

Fig. 1 Bifurcation diagrams and maximum Lyapunov exponent for system (1) around 2E . (a) Flip bifurcation diagram of system

(1) in ( yxr ) space, the initial value is 061.0,641.0, 00 yx (b) Flip bifurcation diagram in ( xr ) plane (c)

Maximum Lyapunov exponents corresponding to (b) and (d) Maximum Lyapunov exponents are superimposed on Flip

bifurcation diagram.

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For case (i). The bifurcation diagrams of system (1) in ( yxr ) space and in ( xr ) pane are given

in Fig. 1(a-b). After calculation for the fixed point 2E of map (1), the flip bifurcation emerges from the fixed

point 062.0,641.0 at 3.31 rr and 2

,, EFBdba . It shows the correctness of proposition 1. At

1rr , we have 167.1411 r , which determines the direction of the flip bifurcation and shows the

correctness of Theorem1.From Fig. 1(b), we see that the fixed point 2E is stable for 3.3r and loses its

stability at the flip bifurcation parameter value 3.3r , we also observe that there is a cascade of period

doubling bifurcations for 3.3r . The maximum Lyapunov exponents corresponding to Fig. 1(b) are

computed and plotted in Fig. 1(c), confirming the existence of the chaotic regions and period orbits in the

parametric space.

For case (ii). The bifurcation diagrams of system (1) in the ( yxr ) space, the ( xr ) plane and the

( yr ) plane are given in Fig. 2(a-b-c). After calculation for the fixed point 2E of map (1), the Neimark-

Sacker bifurcation emerges from the fixed point 127.0,2778.0 at 22 rr and 2

,, ENSdba . It

shows the correctness of proposition 1. For 2rr , we have ,691661.0722222.02,1 i 12,1 ,

,03125.0)(

2

rr

i

dr

rd ,65106.188889.020 ig ,57389.461111.102 ig

,30523.125.111 ig ,021 g and 625.522 r . Therefore, the Neimark-Sacker bifurcation is

supercritical and it shows the correctness of Theorem 2.

From Fig. 2(b-c), we observe that the fixed point 2E of map (1) is stable for 2r and loses its

stability at 2r and an invariant circle appears when the parameter r exceeds 2 , we also observe that

there are period-doubling phenomenons. The maximum Lyapunov exponents corresponding to Fig. 2(b-c) are

computed and plotted in Fig. 2(d), confirming the existence of the chaotic regions and period orbits in the

parametric space. From Fig. 2(d), we observe that some Lyapunov exponents are bigger than 0, some are

smaller than 0, so there exist stable fixed points or stable period windows in the chaotic region. In general the

positive Lyapunov exponent is considered to be one of the characteristics implying the existence of chaos. The

bifurcation diagrams for x and y together with maximum Lyapunov exponents is presented in Fig. 2(e).

Fig. 2(f) is the local amplification corresponding to Fig. 2(b) for ]948.2,7.2[r .

The phase portraits which are associated with Fig. 2(a) are disposed in Fig. 3, which clearly depicts the

process of how a smooth invariant circle bifurcates from the stable fixed point 127.0,2778.0 . When r

exceeds 2 there appears a circular curve enclosing the fixed point 2E , and its radius becomes larger with

respect to the growth of r . When r increases at certain values, for example, at 745.2r , the circle

disappears and a period-9 orbits appears, and some cascades of period doubling bifurcations lead to chaos.

From Fig. 3, we observe that as r increases there are period-9, 10, 20-orbits, quasi-periodic orbits and

attracting chaotic sets. See that for 97.2&95.2r , where the system is chaotic, is the value of maximal

Lyapunov exponent positive that confirm the existence of the chaotic sets.

196

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Fig. 2 Bi

diagram o

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M, Codreanu

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EM, EL-Me

II. Nonlinear

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k, USA

S, Bako B.

s, 23: 11-20

Introduction

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xponent for sy

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10: 116-129

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of parameters

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system (1) of

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ndergo a flip

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r

l

-

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199


IAEES www.iaees.org

prey-predator system. Computational Ecology and Software, 2(3): 169-180

Guckenheimer J, Holmes P. 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector

Fields. Springer, New York, USA

Hasan KA, Hama, M. F. 2012. Complex Dynamics Behaviors of a Discrete Prey-Predator Model with

Beddington-DeAngelis Functional Response. International Journal of Contemporary Mathematical

Sciences, 7(45): 2179-2195

Hastings A, Powell T. 1991.Chaos in three-species food chain. Ecology, 72: 896-903

He ZM, Lai X. 2011.Bifurcations and chaotic behavior of a discrete-time predator-prey system. Nonlinear

Analysis: Real World Applications, 12: 403-417

He ZM, Li Bo. 2014.Complex dynamic behavior of a discrete-time predator-prey system of Holling-III type.

Advances in Difference Equations, 180

Hu ZY, Teng Z, Zhang L. 2011. Stability and bifurcation analysis of a discrete predator-prey model with

nonmonotonic functional response. Nonlinear Analysis, 12: 2356-2377

Holling CS. 1965. The functional response of predator to prey density and its role in mimicry and population

regulation. Memoirs of the Entomological Society of Canada, 45: 1-60

Jing ZJ, Yang J. 2006. Bifurcation and chaos discrete-time predator-prey system. Chaos, Solutions and

Fractals, 27: 259-277

Klebanoff A, Hastings A. 1994. Chaos in three species food-chain. Journal of Mathematical Biology, 32: 427-

245

Kuznetsov YK. 1998. Elements of Applied Bifurcation Theory (3rd edn). Springer, New York, USA

Li TY, Yorke JA. 1975. Period three implies chaos. American Mathematical Monthly, 82: 985-992

Liu X, Xiao DM. 2007. Complex dynamic behaviors of a discrete-time predator prey system. Chaos, Solutions

and Fractals, 32: 80-94

Lotka AJ. 1925. Elements of Mathematical Biology, Williams and Wilkins, Baltimore, USA

May RM. 1974.Stability and Complexity in Model Ecosystems. Princeton University Press, NJ, USA

May RM. 1975. Biological populations obeying difference equations: stable points, stable cycles and chaos.

Journal of Theoretical Biology, 51(2): 511-524

May RM. 1976. Simple mathematical models with very complicated dynamics. Nature; 261:459-467

Murray JD. 1998. Mathematical Biology. Springer-Verlag, Berlin, Germany

Robinson C. 1999. Dynamical Systems, Stability, Symbolic Dynamics and Chaos (2nd edn). CRC Press, Boca

Raton, USA

Volterra V. 1926.Variazioni e fluttuazionidelnumerod’individui in specie animaliconviventi, Mem. R. Accad.

Naz. Dei Lincei, Ser. VI, vol. 2

Wiggins S. 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos (2nd edn). Springer, New

York, USA

Zhu H, Campbell SA, Wolkowicz G. 2002. Bifurcation analysis of a predator-prey system with nonmonotonic

functional response. SIAM Journal on Applied Mathematics, 63: 636-682

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Article

Modeling and analysis of the survival of a biological species in a

polluted environment: Effect of environmental tax

Shyam Sundar1, Ram Naresh2

1Department of Mathematics, P. S. Institute of Technology, Kanpur-208020, India 2Department of Mathematics, H. B. Technological Institute, Kanpur-208002, India


Received 30 December 2014; Accepted 10 February 2015; Published online 1 June 2015

Abstract

In this paper, a nonlinear dynamical model is proposed and analyzed to study the survival of biological species

in a polluted environment considering the effect of environmental tax which can be used further to improve

environmental quality. The environmental tax is imposed to control the emission of pollutants/toxicants only

when the equilibrium concentration of pollutants go beyond its threshold level causing harm to the biological

species and its ecosystem under consideration. Local and nonlinear stability conditions are obtained by

considering suitable Liapunov function. Numerical simulation of the dynamical system is performed in order

to illustrate the analytical findings. It is shown that the density of biological species decreases as the

concentration of pollutants increases and may even become extinct if the concentration is very high. It has also

been shown that the environmental tax plays an important role to control the concentration of pollutants in the

atmosphere and maintaining the density of biological species at a desired level.

Keywords modeling; biological species; polluted environment; environmental tax; stability.

1 Introduction

It has been observed during last several years that various kinds of toxicants (pollutants) such as toxic gases,

smoke, particulate matters, cement dust, chemicals, etc. discharged from various industries and other sources

have made considerable change in the both terrestrial and aquatic environment in the form of deforestation, air

pollution, water pollution, etc. The survival of biological species is threatened instantly due to polluted air,

water, soil, land and vegetation, etc. caused by toxicants. Therefore, it is crucial to investigate the effect of

toxicants on biological species and the reduction in concentration in the atmosphere by imposing

environmental tax on emitters which may in turn reduce environmental damage and minimizing harm to

economic growth.

Some investigations have been made to study the effect of toxicants released to the water bodies, gaseous

pollutants and particulate matters on the environment and ecology as well as on biological species (Lovett and

Kinsman, 1990; Hopke, 2009; Woo, 2009; Cambra-Lopez, 2010; Pertsev and Tsaregorodtseva, 2011). For

example, Cambra-Lopez (2010) have reviewed the effect of airborne particulate matters from the livestock

production systems and have shown that high concentration of particulate matters can deteriorate the

environment as well as the health of human and animals.


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In recent years, several studies have been conducted to comprehend the effect of toxicants on biological

species living in a polluted environment (Hallam and Clark, 1981; Freedman and Shukla, 1991; Liu and Ma,

1991; Chattopadhyaya, 1996; Shukla and Dubey, 1996, 1997; Shukla et al., 2001, 2009, 2013; Mukherjee,

2002; Liu et al., 2003; Samanta and Matti, 2004; Dubey and Hussain, 2006; Naresh et al., 2006; Dubey, 2010;

Samanta, 2010; Misra and Kalra, 2012; Naresh et al., 2014). In this regard, Samanta and Matti (2004) have

proposed and analyzed a nonlinear mathematical model to study the effect of toxicant on a single species

living in a polluted environment by considering three cases; instantaneous input of toxicant, constant input of

toxicant and fluctuating emission rate of toxicant into the environment. In the analysis, it has been shown that

instantaneous emission of toxicants has no significant effect on population but the population settles down to a

steady state if the toxicants are emitted incessantly. Dubey and Hussain (2006) proposed a mathematical model

for the survival of species dependent on a resource in polluted environment considering the effect of diffusion

on the system. They have shown that the equilibrium level of the density of population decreases as the

environmental concentration of the pollutant increases. Naresh et al. (2006) have studied the dynamics of the

plant biomass in a polluted environment by considering the effect of intermediate toxic product formed by

uptake of a toxicant on plant biomass. It has been shown that intermediate toxic product is mainly responsible

for the decrease in the intrinsic growth of plant biomass and the equilibrium label of the density of plant

biomass depends upon the rate of emission of toxicant into the atmosphere. Shukla et al. (2009) have studied

the effect of toxicants on population emitted from extraneous sources as well as formed by its precursors by

developing a nonlinear mathematical model and have shown that the densities of population and its resource

decrease due to increase in the concentration of toxicants in the environment.

At present, deterioration in environmental quality due to discharge of toxicants in the atmosphere is a

burning issue in India and elsewhere. Rapid growth of industries, fast population growth, increasing demand of

resources and deforestation are all exacerbating problems that need to be comprehended. In this regard, some

mitigation options are to be required to improve environmental quality. Environmental tax policy may be one

of the most potent mitigation strategies that must be imposed to the emitters keeping in mind that it is imposed

neither to enhance net additional tax revenue nor to reduce the overall energy consumptions but is

implemented to get clean environment. During previous years, effect of implementation of environmental tax

has been discussed by policy makers to reduce environmental damages and to get a clean environment

(Symons et al., 1994; Bovenberg et al., 1996; Stern, 2006; Sterner, 2007; Braathen and Greene, 2011; Liu,

2012). Environmental tax, to be imposed to the emitters, is generally based on the following factors,

1. The quantity of the pollutants/toxicants discharged into the environment

2. The use of resources

3. The products responsible for environmental degradation

4. The vehicle excise duty

In India, about 64% policy makers have considered environmental tax as a most important significant

factor making a clean environment (Kanabar, 2011). It is mentioned here that the tax is levied to the emitters

only when the concentration of pollutants in the atmosphere crosses a threshold limit. Threshold means the

concentration of pollutants below which there is no harm to the population and its environment. The tax is

imposed on the basis of per unit emission of pollutants (beyond its threshold limit) in the environment. The

study of implementation of environmental tax to reduce the concentration of pollutants in the atmosphere has

less understood and received little attention using nonlinear mathematical models. In this regard, Agarwal and

Devi (2010) have studied the effect of environmental tax on the survival of biological species in a polluted

environment using a mathematical model but they have not considered the formation of intermediate toxic

product inside the biomass.

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In view of the above, in this paper, we have proposed and analyzed a nonlinear mathematical model to

study the effect of environmental tax to reduce the concentration of toxicants in the atmosphere forming

intermediate toxic product inside the biomass due to interaction of toxicants with sap (biomass fluid).

2 Mathematical Model

Consider a biological species living in a polluted environment affected by toxicants emitted into the

atmosphere from different sources. To model the phenomenon, we have made the following assumptions,

1. The cumulative rate of emission of toxicants is constant (say Q ) though it may be a function of time.

2. The growth rate of biological species is affected by an intermediate toxic product (formed inside the

biomass due to the interaction of uptaken toxicants and the liquid present in the biomass).

3. The carrying capacity of biological species is affected by the concentration of toxicants emitted into the

environment and it decreases with increase in the concentration of toxicants emitted into the environment.

4. Environmental tax is assumed to be imposed only when the toxicant concentration crosses a threshold and as

such, it is taken to be directly proportional to the difference of toxicants concentration and its threshold level.

Threshold concentration implies the level up to which there is no harmful effect on biological species.

Let )(tN be the density of biological species, )(),( tUtT and )(1 tU be the concentrations of

toxicants emitted into the environment, the toxicants uptaken by biological species and the intermediate toxic

product formed, respectively. Let )(tI be the environmental tax imposed on the emitters. It is assumed that the

depletion of toxicants is directly proportional to the concentration of toxicants as well as the density of

biological species i.e. )()( tTtN , being the interaction rate coefficient of toxicants with biological

species, is the uptake rate coefficient of toxicants due to biological species and )1( is the rate by

which biological species are directly affected by toxicants. The uptake concentration of toxicants is assumed to

be depleted naturally by a rate 0 and 1 is the interaction rate coefficient of toxicants uptaken by the

biological species. When toxicants uptaken interact with the fluid (sap) inside biological species, intermediate

toxic product is formed which is mainly responsible for deterioration of biological species. Let be the rate

of formation of intermediate toxic product and the constants 0 and 1 are the depletion rate coefficients of

intermediate toxic product due to excretion and depuration of toxicants. To control the emission of toxicants

into the atmosphere, environmental tax is assumed to be imposed on the emitters when toxicant concentration

crosses a threshold level and it is assumed to be proportional to the difference of toxicants concentration and

its threshold value i.e. )( 0TT , 0T being the threshold concentration of toxicants and is the tax rate

coefficient. If 0TT , no tax will be imposed on the emitters. Since it is difficult to implement and maintain a

foolproof tax system due to some practical problems like pilferages, natural and administrative problems, it is,

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therefore, obvious to consider a term E0 as a depletion of environmental tax due to these factors. The

constant μ is the tax repulsion rate coefficient.

)()()1()())((

)())((

)( 01 tTtNtN

tTK

tNrtUr

dt

tdN

(1)

)()()()()()(

0 tEtTtNtTtQdt

tdT (2)

)()()()()()(

10 tNtUtUtTtNdt

tdU (3)

)()()()()(

11101 tNtUtUtU

dt

tdU (4)

)())(()(

00 tETtTdt

tdE (5)

0)0(,0)0(,0)0(,0)0(,0)0( 10 EUUTTN

In the model, the function )( 1Ur denotes the intrinsic growth rate of biological species in the presence of

intermediate toxic product formed inside it, as discussed above, and 0r is the maximum intrinsic growth rate

of biological species in the absence of intrinsic toxic product. The intrinsic growth rate )( 1Ur decreases as the

concentration of intermediate toxic product 1U increases and hence, we assume that,

0)0( 0 rr , 0)( 1 Ur for 01 U

The function )(TK denotes the carrying capacity of species in presence of toxicants in the environment

and 0K is the maximum carrying capacity in the absence of toxicants. The carrying capacity )(TK decreases

as the concentration of toxicants T increases and hence,

0)0( 0 KK , 0)( TK for 0T

Remarks

1. As discussed above, the rate of discharge of toxicants (Q ) into the atmosphere is assumed to be constant

which is controlled by introducing a term E (environmental tax), given in equation (2). From (2), we note

that as (the tax repulsion coefficient) increases, the concentration of toxicant into the atmosphere decreases.

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2. It is further remarked here that the cumulative concentration of toxicants/pollutants must be greater than its

threshold concentration (i.e. 0TT ) for the physical significance of the model system and in this situation the

tax will be imposed on the industrialists and would continue till 0TT . If 0TT , then dt

dE will be negative

and no tax will be imposed to the concerned industrialists. Further, if 0 i.e. no tax is imposed to the

industrialists, the toxicants concentration would cross its harmful limit (threshold concentration) and the

survival of biological species will be threatened and it might become extinct.

It is, therefore, desirable that environmental tax must be levied to keep the toxicants emission under control.

Lemma If 00

TQ

, then the set

mmm EEUUUUQ

TKNEUUTN 0,0,0,0,0:),,,,( 110

01

is the region of attraction for all solutions of the model system (1) – (5) initiating in the interior of positive

octant, where 00

0

QK

U m , mm UU0

1

,

000

TQ

Em

.

Since 0

Q is the maximum concentration of toxicants and therefore it is remarked here that the condition

00

QT for the existence of region of attraction implies that the environmental tax can be imposed to the

industrialists only when the concentration of toxicants crosses its threshold value.

3 Equilibrium Analysis

The model under consideration has following two nonnegative equilibria,

(i) ),0,0,,0(0 ETE

where

00

00

TQ

T and

00

00

0

TQE

(ii) ),,,,( **1

**** EUUTNE

The positive solution of *E is given by the following system of algebraic equations,

0)1()(

)( 01 T

TK

NrUr (6)

00 ETNTQ (7)

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010 UNUTN (8)

01110 NUUU (9)

0)( 00 ETT (10)

From equations (7) and (10), we have

0)( 00

0 TTTNTQ (11)

Equation (11), (8) and (9) can also be respectively written as,

)(

00

00 Nf

N

TQ

T

(12)

)()(

10

NgN

NfNU

(13)

)()(

101 Nh

N

NgU

(14)

Now, from (6), we have

0)1()(

))(( 0 TTK

NrNhr (15)

To show the existence of nontrivial equilibrium *E , we plot the isoclines given by equations (11) and (15) in TN plane as follows,

From equation (11) we note the following,

(i) 0N 0

00

00

TQ

T

(ii) 000

T

N

dT

dN

in first quadrant.

(iii)

00

N and 0T are the asymptotes.

From equation (15), we also note the following,

(i) 0T 0KN

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(ii) 0N 0)1(

0

rT

(iii) 0

)()())((

)1()()(

0

20

TK

rNgNgr

TKTK

Nr

dT

dN

in first quadrant, provided 11

002

N

In view of the above, it is shown in figure 1 that the isoclines given by (11) and (15) intersect at a unique point

),( ** TN in the interior of first quadrant in TN plane showing that the steady-state values of *N and

*T are within the invariant region.

Knowing the values of *N and *T , we can find the values of *

1* ,UU and *E from the equations (13), (14)

and (10) respectively.

It is noted that,

N

TNU

10

0

TN

mUQK

00

0

This shows that the steady-state value of *U is within the invariant region.

We also note that,

N

UU

101

0

U

0

mU mU1

This shows that the steady-state value of *

1U is within the invariant region.

Further,

0

0 )(

TT

E

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0

00

TQ

mE

which shows that the steady-state value of *E is within the invariant region.

Fig. 1 Existence of *E in TN plane.

4 Stability Analysis

4.1 Local stability of equilibria

The local stability analysis of an equilibrium point determines the behaviour of the dynamical system. It

characterizes whether or not the system settles down to the equilibrium point if it initiates very close to

equilibrium point. The local stability of an equilibrium point can be determined by computing the eigenvalues

of variational matrix corresponding to that equilibrium point.

To establish the local stability behaviour of equilibria, we compute the following Jacobian matrix M

for model system (1) – (5),

0

1011

101

0

12

200

1

000

0)(0

00)()(

00)(

0)(0)1()()(

)1()(

2)(

NU

NNUT

NT

NUrNTKTK

NrT

TK

NrUr

M

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It can easily be checked that the equilibrium ),0,0,,0(0 ETE is stable or unstable according as

)1(0

rT or

)1(0

rT (necessary condition for the existence of *E ) respectively. This implies that

0E is saddle point if *E exist otherwise it is locally asymptotically stable.

To study the local stability behaviour of the model system about ),,,,( **1

**** EUUTNE , we define

following positive definite function *E ,

25

214

23

22

21 2

1

2

1

2

1

2

1

2

1ekukukknkV (16)

where 1,,, uun and e are small perturbations about *E and )5...,,2,1( iki are positive constants to be

chosen appropriately.

Differentiating (16) with respect to ''t and using linearized system of (1) – (5), we get,

2*103

2*02

2*

*0

1 )()()(

uNkNknTK

Nrk

dt

dV

205

21

*104 )( ekuNk

nuUTknTkNTKTK

Nrk )()1()(

)(*

1*

3*

2**

*2

2*0

1

uNknuUkNUrk )())(( *31

*114

**11

1452 )()( uukekk

Now, dt

dV will be negative definite under the following conditions,

)()(9

4)1()(

)(*

0*

*0

21

2

*2

***2

2*0

1 NTK

NrkkTkNTK

TK

Nrk

)()(9

4)( *

10*

*0

12*

1*

3 NTK

NrkUTk

)()(3

2))(( *

10*

*0

412*

114**

11 NTK

NrkkUkNUrk

))((9

4)( *

10*

022*

3 NNkNk

0*

0522

52 )(3

4)( Nkkkk

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))((3

2 *10

*103

24 NNkk

Choosing 121 kk ,

2*

*0

2*1

**

*0*

103 )(

)(,

)(

1

)(min)(

9

4

N

N

UTTK

NrNk

,

32

*10

*10

4

))((

3

2k

NNk

and

5k , dt

dV will be negative definite provided the conditions

(17) and (18) are satisfied implying that ),,,,( **1

**** EUUTNE is locally asymptotically stable.

Theorem 4.1 The interior equilibrium ),,,,( **1

**** EUUTNE is locally asymptotically stable if the

following conditions hold,

)()(9

4)1()(

)(*

0*

*0

2

****2

2*0 N

TK

NrTNTK

TK

Nr

(17)

)()(3

2))(( *

10*

*0

42*

114**

1 NTK

NrkUkNUr (18)

This theorem implies that if the interaction rate coefficient of toxicants with biological species (i.e. ) and

)( *1Ur are large then the conditions (17) and (18) may not be satisfied. This implies that these parameters

destabilize the system.

4.2 Nonlinear stability of equilibrium ),,,,( **1

**** EUUTNE

In this section we discuss the nonlinear stability character of an interior equilibrium

),,,,( **1

**** EUUTNE inside the region of attraction by using Liapunov second method.

To establish the nonlinear stability behaviour of ),,,,( **1

**** EUUTNE , we consider the following

positive definite function,

2*52*11

42*32*2*

**1 )(

2)(

2)(

2)(

2log EE

mUU

mUU

mTT

m

N

NNNNmW

where )5...,2,1( imi are positive constants to be chosen appropriately.

Differentiating it, we get

dt

dUUUm

dt

dTTTm

dt

dN

NNNm

dt

dW)()(

1)( *

3*

2*

1

dt

dEEEm

dt

dUUUm )()( *

51*

114

2*13

2*2 )()( UUNmTTNm

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2*03

2*02

2**

01 )()()(

)(UUmTTmNN

TK

rm

2*05

2*11104 )())(( EEmUUNm

))(](})()1[{( ***210 TTNNTmmTNr

))(}()({))()(( *11

**11411

***1

*3 UUNNUmUmUUNNUTm

))(())(}({))(( *11

*4

**52

**3 UUUUmEETTmmUUTTNm

where

*

11*

1

*11*

11

*11

1

)(

,)()(

)(

UUUr

UUUU

UrUrU and

*

*2

**

*

)(

)(

,)(

1

)(

1

)(

TTTK

TK

TTTT

TKTKT

Now dt

dW will be negative definite provided the following conditions are satisfied,

0*0

212*

210 )(9

4]})()1[{(

TK

rmmTmmTNr

0*0

12*

1*

3 )(9

4)(

TK

rmUTm

0*0

12*

11411 )(3

2})({

TK

rmUmUm

0022

3 9

4)( mNm

00522

52 3

4)( mmmm

0032

4 3

2 mm

Let )( 1Ur and )(TK satisfying in such that 0)( KTKK m , pUr )(0 1 , qTK )(0 ,

where qpKm ,, are some positive constants.

Using mean value theorem, we get pU )( 1 and qT )( .

Now maximizing LHS and choosing 121 mm ,

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20

02*

1**

003 )(

,)(

1

)(min

9

4

KUTTK

rm

,

5m , 3200

4 3

2mm

, dt

dV will be

negative definite inside the region of attraction , provided the conditions (19) and (20) are satisfied

implying that ),,,,( **1

**** EUUTNE is nonlinearly asymptotically stable.

Theorem 4.2 Let )( 1Ur and )(TK satisfying in such that 0)( KTKK m , pUr )(0 1 ,

qTK )(0 , where qpKm ,, are some positive constants then equilibrium

),,,,( **1

**** EUUTNE will be nonlinearly asymptotically stable provided the inequalities are satisfied,

)(9

4)1(

*00

2

*200 TK

rT

K

qKr

m

(19)

)(3

2)(

*00

12*

114 TK

rmpUm

(20)

where

20

02*

1**

003 )(

,)(

1

)(min

9

4

KUTTK

rm

and 32

004 3

2mm

This theorem implies that if the interaction rate coefficient of toxicants with biological species (i.e. )

and p are large then the conditions (19) and (20) may not be satisfied. This implies that these parameters have

destabilizing effect on the model system.

5 Permanence of Solution

From a biological point of view, permanence (persistence) is defined as the long-term survival of all

interacting populations in an ecosystem. It also deals with the growth of biological species as well as other

components of the system. It is noted that the steady state level of all species settles asymptotically above a

certain threshold. Mathematically, persistence is defined as,

Let )(tN be the population density at any time ‘ t ’ then it is said to be persistent (Freedman and Waltman,

1984), if

0)(inflim

tNt

provided 0)0( N . If there exists 0 such that

)(inflim tNt

then the population is said to be uniformly persistent in an ecological system. Thus, the population is said to be

permanent, if it is uniformly persistent and if the bound of population size does not depend on initial

conditions as t .

Theorem 5

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If 00

)1(

QQr

then the system (1)-(5) is uniformly persistent.

Proof

From equation (1), we have

0

01 )1(

))((

)())(()(

)(

Q

tTK

tNrtUrtN

dt

tdN

00

0

0

)1()(

)(

Q

K

tNrQrtN

)()1(

)(

000

0

0

0 tNQQ

rr

K

K

tNr

This implies that

inf000

0 )1()(inflim NQQ

rr

KtN

t

(let)


0

0000 )()()(

)(T

QtTKtTtQ

dt

tdT

)()( 00000

tTKTQ

Q

This implies that

inf00

000

)()(inflim T

K

TQ

Q

tTt

(let)


)()()(

01011 tUKTNdt

tdU

This implies that

inf010

infinf

)()(inflim U

K

TNtU

t

(let)


)()()()(

1010inf1 tUKtU

dt

tdU

This implies that

)()(inflim

010

inf1 K

UtU

t

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Similarly, we can find from equation (5), that

0

0inf )()(inflim

TT

tEt

Thus, we have,

0000

0 )(suplim)(inflim)1( KtNtNQQ

rr

Ktt

000

000 )(suplim)(inflim

)(

Q

tTtTK

TQ

Q

tt

0010

infinf )(suplim)(inflim)(

QtUtU

K

TNtt

011

010

inf )(suplim)(inflim)(

QtUtU

K

Utt

0

000

0inf )(suplim)(inflim)(

TQ

tEtETT

tt

Hence the theorem.


In this section, we have performed some numerical simulations using software Maple7 in the presence and

absence of environmental tax considering the effect of intermediate toxic product on biological species. For

this, we have assumed the following set of parameters,

3.0,2.0,1.0,3.0,5.0,9,4,01.0,01.0,5 010 KrbaQ

6.0,2.0,4.0,4.0,03.0,02.0,2.0 0010 T

The equilibrium values corresponding to ),,,,( **1

**** EUUTNE are given by,

695243.1,049509.1,781606.0,447621.1,285064.8 **1

*** EUUTN

Eigenvalues corresponding to ),,,,( **1

**** EUUTNE are given by,

0.0073i0.2656 0.0073i,0.2656 1.0300, 2.3388,4.2706, . Since all the eigenvalues are

either negative or have negative real parts and therefore equilibrium ),,,,( **1

**** EUUTNE is locally

asymptotically stable.

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The existence of ),,,,( **1

**** EUUTNE in TN plane is shown in figure 1. The nonlinear

stability behavior of ),,,,( **1

**** EUUTNE is shown in figure 2 where the trajectories with different

initial starts have been plotted. It is noted that all the trajectories with different initial starts approach to the

equilibrium point *E . The variation of density of biological species, concentration of toxicants, uptaken concentration, concentration of intermediate toxic product and the amount of environmental tax with time ''t

for different values of rate of emission of toxicants is shown in figures 3 – 7 respectively. From these figures, it

can easily be observed that the density of biological species decreases while the concentrations of toxicants,

uptaken toxicants and intermediate toxic product increase as the rate of emission of toxicants increases. Further,

it has also been shown in figure 7 that the environmental tax increases as the rate of emission of toxicants in

the environment increases beyond its threshold level.

Fig. 2 Nonlinear stability in 1UU plane.

Fig. 3 Variation of N with time ''t for different values of Q

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Fig. 4 Variation of T with time ''t for different values of Q

Fig. 5 Variation of U with time ''t for different values of Q

The effect of intermediate toxic product on biological species with time ''t has been shown in figure 8. It

can be seen from figure 8 that as the rate of formation of intermediate toxic product inside the biological

species increases the density of biomass decreases. The variation of density of biological species )(N and the

concentration of toxicants )(T with time ''t in the presence and absence of environmental tax has been shown

in figures 9 and 10 respectively. From these figures, it is noted that in the absence of environmental tax,

density of biological species decreases as a result of increase in the concentration of toxicants while in the

presence of environmental tax, density of biological species increases due to decrease in the concentration of

toxicants. The variation of density of biological species )(N and the concentration of toxicants )(T with time

''t for different values of tax rate coefficient is shown in figures 11 and 12 respectively. It is shown that as the

tax rate coefficient increases, the equilibrium level of density of biological species increases and the

concentration of toxicants decreases.

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Fig. 6 Variation of 1U with time ''t for different values of Q

Fig. 7 Variation of E with time ''t for different values of Q

Fig. 8 Variation of N with time ''t for different values of

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Fig. 9 Variation of N with time ''t in the presence )4.0( and absence )0( of environmental tax.

Fig. 10 Variation of T with time ''t in the presence )4.0( and absence )0( of environmental tax

Fig. 11 Variation of N with time ''t for the different values of tax rate coefficient

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Fig. 12 Variation of T with time ''t for the different values of tax rate coefficient

7 Conclusions

In this paper, our main aim is to study the survival of biological species living in a polluted environment and to

reduce the concentration of toxicants into the atmosphere by imposing environmental tax on emitters. The

concentration of toxicants into the atmosphere can only be reduced by reducing the cumulative emission rate

of toxicants. It is assumed that the environmental tax is levied on emitters only when the concentration of

pollutants goes beyond a threshold level, as discussed earlier. Existence of equilibria and their stability

behavior has been obtained. It is shown that the first equilibrium (i.e. E ) corresponding to the extinction of

biological species is unstable. The nontrivial equilibrium (i.e. *E ) is locally and globally stable under certain conditions within the region of attraction. It is shown that the equilibrium density of biological population

decreases while the concentration of toxicants, uptaken concentration and the concentration of intermediate

toxic product increase as the emission rate of toxicants increases. It is further shown that as the rate of

emission of toxicants increases, the environmental tax to be imposed on emitters, also increases. This implies

that it is advantageous to levy tax to reduce the discharge of toxicants and to improve the environmental

quality. It is also noted here that the tax revenue, thus generated, can be used for environmental protection so

that we have a clean environment and the biological species may survive potentially.

References

Agarwal M, Devi S. 2010. The effect of environmental tax on the survival of biological species in a polluted

environment: a mathematical model. Nonlinear Analysis: Modelling and Control, 15(3): 271-286

Bovenberg AL, Lawrence HG. 1996. Optimal environmental taxation in the presence of other taxes: General

equilibrium analyses. The American Economic Review, 86(4): 985-1000

Braathen NA, Greene J. Taxation, Innovation and the Environment. OECD, 2011

www.oecd.org/env/taxes/innovation

Cambra-Lopez M, Aarnink AJ, Zhao Y, Calvet S, Torres AG. 2010. Airborne particulate matter from livestock

production systems: A review of an air pollution problem. Environmental Pollution, 158: 1-17

Chattopadhyaya J. 1996. Effect of toxic substance on a two species competitive system. Ecological Modelling,

84(1-3): 287-289

Dubey B. 2010. A model for the effect of pollutant on human population dependent on a resource with

environmental and health policy. Journal of Biological Systems, 18(3): 571-592

219


IAEES www.iaees.org

Dubey B, Hussain J. 2006. Modelling the survival of species dependent on a resource in a polluted

environment. Nonlinear Analysis: Real World Applications, 7: 187-210

Freedman HI, Shukla JB. 1991. Models for the effect of toxicant in single-species and predator–prey systems.

Journal of Mathematical Biology, 30: 15-30

Freedman HI, Waltman P. 1984. Persistence in models of three interacting predator prey populations.

Mathematical Bioscences, 68: 213-231

Hallam TG, Clark CE. 1981. Non-autonomous logistic equations as models of populations in a deteriorating

environment. Journal of Theoretical Biology, 93: 303-311

Hopke PK. 2009. Contemporary threats and air pollution. Atmospheric Environment, 43: 87-93

Kanabar D. 2011. Tax trends in emerging India: A survey. KPMG, kpmg.com/in

Liu H, Ma Z. 1991. The threshold of survival for system of two species in a polluted environment. Journal of


Liu B, Chen L, Zhang Y. 2003. The effects of impulsive toxicant input on a population in a polluted

environment. Journal of Biological Systems, 11(3): 265-274

Liu AA. 2012. Tax Evasion and Optimal Environmental Taxes, Resources for the Future. Washington DC,

USA

Lovett GM, Kinsman JD. 1990. Atmospheric pollutant deposition to high-elevation ecosystem. Atmospheric

Environment, 11: 2767-2786

Misra OP, Kalra P. 2012. Modelling effect of toxic metal on the individual plant growth: A two

compartment model. American Journal of Computational and Applied Mathematics, 2(6): 276-289

Mukherjee D. 2002. Persistence and global stability of a population in a polluted environment with delay.

Journal of Biological Systems, 10: 225-232

Naresh R, Sundar S, Shukla JB. 2006. Modelling the effect of an intermediate toxic product formedby uptake

of a toxicant on plant biomass. Applied Mathematics and Computation, 182(1): 51-160

Naresh R, Sharma D, Shyam Sundar S. 2014. Modeling the effect of toxicant on plant biomass with time

delay. International Journal of Nonlinear Science, 17(3): 254-267

Pertsev NV, Tsaregorodtseva GE. 2011. A mathematical model of the dynamics of a population affected by

harmful substances. Journal of Applied and Industrial Mathematics, 5(1): 94-103

Samanta GP, Matti A.2004. Dynamical model of a single species system in a polluted environment. Journal of

Applied Mathematics and Computing, 16(1-2): 231-242

Samanta GP. 2010. A two-species competitive system under the influence of toxic substances. Applied

Mathematics and Computation, 216(1): 291-299

Shukla JB, Dubey B. 1996. Simultaneous effect of two toxicants on biological species: a mathematical model.

Journal of Biological Systems, 4(1): 109-130

Shukla JB, Dubey B. 1997. Modelling the depletion and conservation of forestry resources: effects of

population and pollution. Journal of Mathematical Biology, 36: 71-94

Shukla JB, Agarwal A, Dubey B, Sinha P. 2001. Existence and survival of two competing species in a polluted

environment: A mathematical model. Journal of Biological Systems, 9(2): 89-103

Shukla JB, Sharma S, Dubey B, Sinha P. 2009. Modeling the survival of resource-dependent population:

Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors.

Nonlinear Analysis: Real World Applications, 10: 54-70

Shukla JB, Sundar S, Shivangi, Naresh R. 2013. Modeling and analysis of the acid rain formation due to

precipitation and its effect on plant species. Natural Resource Modeling, 26(1): 53-65

Stern N. 2006. Stern’s Review on Economics of Climate Change. Cambridge University Press, USA

220


IAEES www.iaees.org

Sterner T. 2007. Gasoline Taxes: A useful instrument for climate policy. Energy Policy, 35: 3194-3202

Symons E, Proops J, Gay P. 1994. Carbon Taxes, consumer demand and carbon dioxide emissions: A

simulation analysis for the UK. Fiscal Studies, 15(2): 19-43

Woo SU. 2009. Forest decline of the world: A linkage with air pollution and global warming. African Journal

of Biotechnology, 8(25): 7409-7414

221

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Computational Ecology and Software ISSN 2220-721X

Volume 5, Number 2, 1 June 2015 Articles Fluctuating asymmetry and developmental instability in Protoreaster nodosus (Chocolate Chip Sea Star) as a biomarker for environmental stress D.J. V. Trono, R. Dacar, L. Quinones, S. R. M. Tabugo 119-129 Distinguishing niche and neutral processes: Issues in variation partitioning statistical methods and further perspectives YouHua Chen 130-138 Application of homotopy perturbation method to the Navier-Stokes equations in cylindrical coordinates H. A. Wahab, Anwar Jamal, Saira Bhatti, et al. 139-151 Modeling the effect of pollution on biological species: A socio-ecological problem B. Dubey, J. Hussain, S. N. Raw, Ranjit Kumar Upadhyay 152-174 Hopf bifurcation and stability analysis for a delayed logistic equation with additive Allee effect E.M. Elabbasy, Waleed A.I. Elmorsi 175-186 Bifurcation and complex dynamics of a discrete-time predator–prey system S. M. Sohel Rana 187-200 Modeling and analysis of the survival of a biological species in a polluted environment: Effect of environmental tax Shyam Sundar, Ram Naresh 201-221

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Computational Ecology and Software, 2015, Vol. 5, Iss. 2

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Transcript of Computational Ecology and Software, 2015, Vol. 5, Iss. 2