Computational Ecology and Software, 2014, 4(1)

Computational Ecology and Software

Vol. 4, No. 1, 1 March 2014

International Academy of Ecology and Environmental Sciences

Computational Ecology and Software ISSN 2220-721X Volume 4, Number 1, 1 March 2014 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]

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Computational Ecology and Software, 2014, 4(1): 1-11

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Article

About a non-parametric model of hermaphrodite population dynamics

L.V. Nedorezov

University of Nova Gorica, Vipavska Cesta 13, Nova Gorica SI-5000, Slovenia

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

Received 22 November 2013; Accepted 25 December 2013; Published online 1 March 2014

Abstract

In current publication non-parametric model (model of Kolmogorov’s type) of hermaphrodite population

dynamics is analyzed. It is assumed that there are four basic variables: number of individuals, number of pairs,

and number of pregnant individuals. It is also assumed that number of pairs is fast variable: it allows

decreasing of number of differential equations. For conditions of pure qualitative type for birth and death rates

of individuals in population possible dynamic regimes are determined.

Keywords model of population dynamics; sexual structure; hermaphrodite; dynamic regimes.

1 Introduction

Sex structure plays extremely important role in population dynamics (see, for example, Maynard, 1978;

Bolshakov and Kubantsev, 1984; Geodakjan, 1965, 1981, 1991; Iannelli et al., 2005; Grechanii and Pogodaeva,

1996; Batlutskaya et al., 2010, and many others). We have to take into account existence of sex structure

analyzing epidemiological situations with sexually-transmit diseases, some methods of population size

management are based on input of sterile individuals into the system etc. Thus, constructing and testing of

mathematical models of population dynamics with sex structure are among very actual problems of modern

modeling.

In 1949 Kendall (Kendall, 1949) gave a description of model of population dynamics which contains

individuals of two types: )(tF and )(tM are the numbers of females and males respectively in population

at moment t ,

),(),(2

1MFPMFBF

dt

dF ,

),(),(2

1MFQMFBM

dt

dM . (1)

In model (1) coefficient is an intensity of death rate, 0 const , and function ),( MFB

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describes a reproduction process:

2),( RMF 0),( MFB , 0)0,(),0( FBMB ,

0F

B, 0M

B for 0, MF . (2)

In (2) }0,0:),{(2 MFMFR . Conditions (2) are rather obvious: if number of males or females

is equal to zero we have no reasons to talk about production process; increase of number of males or females

leads to increase of the respective rates.

Model (1)-(2) has the following properties. If 0)0( F or 0)0( M then for all 0t we have

0)( tF or 0)( tG respectively. At the same time other variable decreases monotonously. It means that

origin is locally stable knot. From conditions (2) we get that isocline of vertical inclines 0P is univocal

with respect to F ; isocline of horizontal inclines 0Q is univocal with respect to M . For 0)0( FF ,

0)0( MM we have

teMFtMtF )()()( 00 .

It means that within the framework of model (1)-(2) initial difference between females and males

converges to zero asymptotically. If 00 MF then for all 0t we have )()( tMtF . For the situation

when 00 MF and FMMFB ),( , we have

2

2

1FF

dt

dF .

This equation has two stationary states: stable point 01 F and unstable point 22 F . If 20 FF then

population degenerates asymptotically, 0)( tF when t . If we have the inverse inequality,

20 FF , then population size becomes equal to infinity during the finite time *t :

tCetF

1

2)( ,

0

0 2

F

FC

,

2

ln1

0

0*

F

Ft .

If we don’t want to have such dynamical effect within the framework of considering model when model

can be applied to the description of population dynamics during finite time interval, we can assume, for

example, that birth rate ),( MFB is a linear function of population size (Kendall, 1949). But it looks more

productive the following way: it is obvious that birth rate cannot increase up to plus infinity if number of males

increases unboundedly at fixed value of females; it means that the following relation is truthful:

aFMFBM

),(lim , 0 consta .

It means that limit value of birth rate depends on number of females and coefficient a which characterizes

maximum properties of females. The following relation must be truthful too: for fixed value of number of

males unlimited increasing of females gives the following result:

cMMFBF

),(lim , 0 constc .

In this relation parameter c characterizes maximum possibilities of males. In most primitive case function

),( MFB can be presented in the form:

aFcM

acFMMFB

1),( . (3)

For particular case 00 MF model (1)-(2) with function (3) has the form:

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Fg

FgF

dt

dF

2

21

1 . (4)

In (4) 02/1 constacg , 02 constcag . Equation (4) is particular case of Bazykin’

model (Bazykin, 1967, 1969, 1985) when self-regulation is absent in population ( 0 const ).

Further development of this scientific direction was connected with analysis of various modifications of

model (1)-(2) (Ginzburg and Yuzefovich, 1968; Gimelfarb et al., 1974; Nedorezov, 1979, 1986; Kiester et al.,

1981; Pertsev, 2000; Preece and Mao, 2009, and others), and in particular, with analysis of general properties

of models of (1)-(2) type within the framework of non-parametric model (model of Kolmogorov’ type;

Nedorezov, 1978). A lot of publications were devoted to very actual problem of changing of population size at

input of sterile males into the system (see, for example, Bazykin, 1967; Alexeev and Ginzburg, 1969;

Brezhnev and Ginzburg, 1974; Costello and Taylor, 1975; Brezhnev et al., 1975; Nedorezov, 1979, 1983, 1986;

Thome et al., 2010, and many others).

It is very important to point out the following problem of models of (1)-(2) type. For every fixed values of

model variables F and M we have fixed value of function B that means that we have fixed value of

pregnant females. This property of model doesn’t correspond to reality, and number of pregnant females can

vary from zero up to )(tF . Respectively, for every fixed values of model variables F and M we have to

have a certain variety of values of function B . This problem can be solved in one way only if we have one or

more additional variables which described dynamics of pregnant females or number of existing families.

Development of theory in this direction when models contain three or more variables (for families,

pregnant females, with sex-age structures etc.) was provided in a lot of publications (see, for example, Kendall,

1949; Goodman, 1953, 1967; Pollard, 1973; Yellin and Samuelson, 1974, 1977; Nedorezov, 1979, 1986;

Hadeler et al. 1988; Hadeler and Ngoma, 1990; Hadeler, 1992, 1993; Pertsev, 2000; Iannelli et al., 2005, and

others). One more well-developed sub-direction contains models with discrete time (Hadeler et al. 1988;

Hadeler and Ngoma, 1990; Hadeler, 1992, 1993; Castillo-Chaves et al., 2002; Frisman et al., 2011; Frisman, et

al., 2010 a, b).

It is possible to point out some sub-directions which are not well-developed up to current moment but their

further development look rather actual. Ginzburg (1969) analyzed model of predator-prey system dynamics in

a situation when individuals in interacting populations were divided into two sexes. In our publications

(Nedorezov, Utyupin, 2003, 2011) continuous-discrete model (system of ordinary differential equations with

impulses) of bisexual population dynamics was analyzed. These models give more adequate description for

insect population dynamics in boreal zone than models with continuous or discrete time.

In current publication we analyze non-parametric (model of Kolmogorov’ type) dynamic model of

hermaphrodite population. This sub-direction in modeling of population dynamics with sex structure is well-

developed, and it is possible to point out models of various types (see, for example, Armsworth, 2001; Stewart,

and Phillips, 2002; Cheptou, 2004; Alvarez et al., 2006; Harder et al., 2007; Kebir et al., 2010, and others)

because of very important role hermaphrodites play in ecological processes, epidemiological processes etc.

(Charnov et al. 1976; Maynard, 1978; Civeyrel and Simberloff, 1996; Barker, 2002).

2 Description of Model

Let )(tN be a number of free individuals in population at moment t , )(tS be a number of pairs, and

)(tP be a number of pregnant individuals. For every free individual N we will assume that it can die with

intensity 1k and can organize a pair S with other free individual with coefficient 2k . For coefficient 1k

we’ll assume that it depends on total population size , where SPN 2 , and the following

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conditions are truthful:

)(11 kk , 01 d

dk, * : * )(1 k . (5)

Kendall (1949) had been analyzed the model with three variables - )(tF , )(tM , and )(tS , - assuming

that speed of appearance of new pairs in system is proportional to the following function:

),min(2),( MFMFg .

is positive coefficient. Pollard (1973) had been assumed that

)(2

1),( MFMFg .

Following the idea which is on the base of Bazykin’ model (1967, 1969) we’ll assume that speed of

organizing of new pairs is proportional to 2N when number of free individuals is rather small, and it is

proportional to N when number of free individuals is rather big. Thus, function g can be presented in the

following form:

bN

aNNg

1)(

2

. (6)

In (6) 0, constba . Respectively, it allows us concluding that coefficient of appearance of new pairs

)1/()(2 bNaNk is monotonic decreasing function; in general case, we’ll assume that following

conditions are truthful:

)(22 Nkk , 0)0(2 k , 0)(2 k , 02 dN

dk, 0

dN

dg, 0

N

g

dN

d. (7)

Dynamics of free individuals can be described by the following equation:

PmkNNkNkdt

dN)1()(2)( 5

221 . (8)

In (8) coefficient 5k corresponds to time of staying of individuals in pregnant conditions, and it is naturally

to assume that 05 constk . Function m is productivity of pregnant individuals. We’ll assume that the

next conditions are truthful for this function:

)(mm , 0)0( m , 0)( m , 0d

dm. (9)

Conditions (9) are rather obvious. Increasing of total population size leads to changing of food conditions

for individuals (in a result of increasing of intensity of intra-population competition between individuals for

food), and, finally, it leads to decreasing of productivity.

Pairs S can be organized in system in a result of interaction of free individuals with coefficient 2k (7),

and can be destroyed with coefficient 3k . We’ll assume that in a result of destruction of complex S two

pregnant individuals P appear in population; coefficient 3k must be positive and constant,

03 constk . Taking it into account, dynamics of variable S can be described by the following equation:

SkNNkdt

dS3

22 )( . (10)

It is obvious that S (10) is fast variable: time of existing of complex S is much less than time of living of

free individuals and staying of individuals in pregnant condition. Thus, we can assume that 0/ dtdS ,

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0/)( 32

2 kNNkS , and 2*23 ))(0( kk .

Every pregnant individual P can die with coefficient 4k (we have no reasons to assume that coefficient

4k is equal to 1k but similar conditions to (5) are truthful for 4k ) or can transforms into 1m free

individuals with coefficient 5k . Dynamics of variable )(tP describes with following equation:

PkPkSkdt

dP)(2 453 . (11)

Taking into account that conditions (5) are truthful for coefficients 1k and 4k , we can conclude that for *)0( N and *)0( P we have for all 0t variables *)( tN and *)( tP . From (7) we

obtain that for *)0( SS we have for all 0t following inequality:

3

2*2* ))(0(

)(k

kStS

.

Thus, solutions of system of differential equations (8), (10), (11) belong to stable invariant compact

],0[],0[],0[ *** S .

Thus, we can decrease the order of system of differential equations, and determine the structure of phase space

of system (8), (10), (11) analyzing properties of system

PmkNNkNkdt

dN)1()(2)( 5

221 ,

PkPkNNkdt

dP)()(2 45

22 . (12)

Graphically all possible transitions of individuals in population are presented on Fig. 1. Note, that such

kind of interactions is observed for various species, and, in particular, for earthworm (Lumbricina), for snails

Helix pomatia and for other species. Such kind of interaction is normal for simultaneous (or synchronic)

hermaphrodites.

3 Some Properties of Model (12)

1. For non-negative and finite initial values of variables solutions of the system (12) are non-negative and

bounded.

2. Let

0)1()(2)(),( 52

211 PmkNNkNkPNF ,

0)()(2),( 452

22 PkPkNNkPNF .

(13)

From (5), (7), and (9) we obtain the following inequality:

0))((2)( 52

21

11

d

dmPkNNk

dN

d

d

dkNk

N

F.

It means that isocline of vertical inclines of system (12) 0),(1 PNF is a single-valued function with

respect to P . For isocline of horizontal inclines we have the following inequality:

0)( 445

2

d

dkPkk

P

F.

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Thus, isocline of horizontal inclines (13) is a single-valued function with respect to N . Conditions (5), (7),

(9) for coefficients jk , 5,...,1j , don’t allow determining of signs for expressions PF /1 and

NF /2 .

Fig. 1 All possible transformations of individuals in population. 1k and 4k are intensities of death rate. 2k is coefficient of forming of pair S . 3k is coefficient of destruction of pair S . 5k is a coefficient of staying of individual in pregnant state. m is number of new free individuals which are produced by one pregnant individual.

3. Previous properties of model (12) give us the following inequality:

021

P

F

N

F.

Thus, there are no limit cycles in phase space (Bendixon’ criteria; Andronov, Vitt, Khykin, 1959).

Consequently, within the framework of model (12) there are the regimes of asymptotic stabilization of

population size at any level only.

4. Origin )0,0( is stationary state of system (12). This system in sufficient small vicinity of origin can be

prersent6ed in following form:

PmkNkdt

dN)1)0(()0( 51 ,

PkPkdt

dP)0(45 .

Thus, characteristic values are negative: )0(11 k and )0(452 kk . Consequently, in all

situations origin is stable knot.

5. In a situation when we have a parametric model (model of Volterra type) we have the following main goal:

we have to present a structure of a space of model parameters and to point out dynamical regimes which

correspond to each determined part of space of parameters. When we have a non-parametric model (model of

Kolmogorov type like in current publication) we have other main goal: in a result of provided analysis we have

to present dynamical regimes which can be realized in model in principle, and their realization not in a

contradiction with considering restrictions on the types of functions in right-hand sides of equations. Below

we’ll consider some simplest dynamic regimes of model (12) – restrictions (8)-(11) and (14) don’t allow

presenting all possible dynamic regimes which can be observed within the framework of model.

If algebraic system (13) has no solutions in positive part of phase plane, origin is global stable equilibrium.

Population eliminates for all non-negative finite initial values (Fig. 2).

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Fig. 2 Regime of population elimination for all initial values of variables. 01 F and 02 F are the main isoclines of vertical and horizontal inclines of model trajectories respectively.

If algebraic system (13) has two solutions in positive part of phase plane the trigger regime is realized for

population: there are two stable attractors on phase plane (Fig. 3). Incoming separatrix y of saddle point W

divides zones of attraction of origin and stable equilibrium V . If initial sizes of variables are rather small

(within the limits of zone of elimination 1 ; Fig. 3) population eliminates asymptotically. If initial values

belong to another zone (zone of stabilization 2 ) sizes of both variables stabilize asymptotically at unique

level.

In general case within the limits of model (12) dynamic regimes with several stationary states in positive

part of phase plane can be realized (see, for example, Fig. 4). When difference between total numbers of sizes

which correspond to various stable stationary states are rather big, it can be considered as direct analog of the

regime of fixed outbreak (Isaev et al., 1978, 1980; Isaev et al., 1984, 2001). Thus, we can conclude that big

difference between pregnant individuals and free individuals can be a reason for population elimination or a

reason for transmission of system from one stable level to another one (see Fig. 4). Such kind of changing of

population size can be interpreted as unstable behavior of population within the limits of zone of population

stability (Isaev et al., 1978, 1980; Isaev et al., 1984, 2001).

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Fig. 3. Trigger regime of population dynamics. V is stable stationary state. W is saddle point. y is incoming separatrix of

saddle point W . 1 is zone of population elimination; 2 is zone of population stabilization. 01 F and 02 F

are the main isoclines of vertical and horizontal inclines of model trajectories respectively.

Fig. 4. Dynamical regime with three stable attractors: origin, 1V , and 2V . ry is incoming separatrix of saddle point 1W ,

boundary of attraction zone of origin. qy is incoming separatrix of saddle point 2W , boundary of attraction zones of 1V , and

2V . 1 is zone of population elimination (attraction zone of origin); 2 is zone of population stabilization at point 1V ;

3 is zone of population stabilization at point 2V . 01 F and 02 F are the main isoclines of vertical and horizontal

inclines of model trajectories respectively.

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4 Conclusion

Analysis of model of hermaphrodite population dynamics shows that in general case dynamic regimes with

several non-trivial stationary states can be observed for the system. It means that changing of sizes of free and

pregnant individuals (for example, under the influence of various management methods) can lead as to

transaction of system from one stable level to another one, as to extinction of population. Existence of several

stable levels in positive part of phase plane can be a reason of unstable behavior of system in zone of

population stability (Isaev et al., 1978, 1980).

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Temporal mortality-colonization dynamic can influence the

coexistence and persistence patterns of cooperators and defectors in

an evolutionary game model

YouHua Chen1, XueKe Lu2, YouFang Chen3 1Department of Renewable Resources, University of Alberta, Edmonton, T6G 2H1, Canada 2Department of Electronic Engineering and Computer Science, Queen Mary University of London, E1 4NS, UK 3School of Software, Harbin Normal University, Heilongjiang Province, China


Received 28 October 2013; Accepted 2 December 2013; Published online 1 March 2014

Abstract

In the present report, the coexistence and persistence time patterns of Prisoners’ Dilemma game players were

explored in 2D spatial grid systems by considering the impacts of the mortality-colonization temporal dynamic

specifically. Our results showed that the waiting time for triggering a colonization event could remarkably

influence and change the extinction patterns of both cooperators and defectors. Interestingly, a relatively high

frequency of stochastic colonization events could promote the persistence of defectors but not cooperators. In

contrast, a low frequency of stochastic- or constant-time colonization events could facilitate the persistence of

cooperators but not defectors. However, a long waiting time would be detrimental to the survival of both game

players and drives them to go extinction in faster rates. At last, it was found that colonization strength played a

relatively weak role on influencing the coexistence scenarios of both game players, but should be kept small if

the coexistence of game players is needed to maintain. In conclusion, our study provides evidence showing

that the temporal trade-off of mortality and colonization activities would influence the evolution of PD game

and the persistence of cooperators and defectors.

Keywords species coexistence; game theory; colonization-extinction dynamics; individual-based modeling.

1 Introduction

The classical Prisoner’s Dilemma (PD) game has been broadly studied in evolutionary biology (Hui and

McGeoch, 2007; Zhang and Hui, 2011; Zhang et al., 2005; Nowak and May, 1993, 1992; Zhang, 2012).

Spatial version of Prisoner’s Dilemma could allow the emergence of complex defense-cooperation dynamic

patterns and make the cooperation become more possible (Langer et al., 2008; Zhang et al., 2005).



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In a previous study, the evolution of cooperation under habitat destruction has been well quantified (Zhang

et al., 2005). One important part of the model used by the previous work (Zhang et al., 2005) is to model the

dynamic between mortality and colonization. However, the trade-off between the occurrence frequency of

colonization and mortality events and the relevant impacts on the coexistence and survival of both game

players have not been extensively evaluated yet. Because in the previous study (Zhang et al., 2005),

colonization and mortality events are allowed to happen at each simulation time step. A detailed study on the

trade-off between colonization and mortality occurrence frequency would allow ones to better understant the

real-time habitat degeneration and isolation processes on influencing the coexistence patterns of cooperators

and defectors.

In the present report, We quantify the condition of coexistence of both defectors and cooperators by

varying the occurrence frequency of colonization and mortality events. In specific, We fix the mortality

frequency during the simulation (allowed to occur one time per one time step), and evaluate the influence of

waiting time of triggering a colonization event on the persistence of cooperators and defectors.

As a summary, the central objective of the present study is to reveal the impacts of the temporal trade-off

of the occurrence frequency of colonization and mortality events on the persistence time of PD game players.

2 Materials and Methods

The payoff matrix of a typical evolutionary PD game is defined as (Zhang et al., 2005),

C D

C

D

(1)

where >0 and >0. C represents the cooperator, while D represents the defector.

Assuming that each patch is only allowed to inhabit one individual, the ip score for the individual in the

patch i, taking into account of the rewards during the evolutionary game interaction, is defined as follows

(Zhang et al., 2005),

( 1) ( 1)( ) ( )

2 2i i i i

i i i ii C D C D

x x x xp f f f f (2)

Here we adopt the same notation used in the previous study (Hui et al., 2005). Where ix =1 if patch We is

occupied by a cooperator; ix =-1 if the patch is occupied by a defector; and ix =0 if it is empty. iCf is the

fraction of cooperators in the two neighboring patches of the patch We and iDf is the fraction of defectors.

Clearly,iCf +

iDf 1.

Degeneration of habitat quality is thought to be related to mortality rate, while patch isolation is related to

colonization rate of species (Zhang et al., 2005). As such, as mentioned above, the trade-off between mortality

and colonization frequency actually reflects the dynamic of habitat degeneration and isolation on the

persistence of both game players.

The mortality rate of individuals for taking into account of the degeneration of habitat quality is defined as

(Zhang et al., 2005),

exp( )( )

1 exp( )i

ii

pM p m

p

(3)

and the colonization rate of individuals is (Zhang et al., 2005),

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1( )

1 exp( )ii

C p cup

(4)

Here, m and c are regarded to be related to habitat degeneration and isolation respectively, being in the range

of [0, 1]. Higher values of m and/or c indicates higher degrees of degeneration and/or isolation of the habitat.

Hereafter, We call m and c as mortality and colonization coefficients (or strengths) respectively.

For modeling the temporal impact of trade-off between mortality and colonization events, We define and

use the waiting time of triggering a colonization event. In detail, a mortality event is assumed to happen for

each time step, while a colonization event could happen only when the next time for triggering it satisfying the

waiting time (WT) setting.

Two strategies are used to configure the waiting time setting for triggering a colonization event during the

simulation. The first one is to assume the waiting time of a colonization (WT) event is deterministic and

constant, which is fixed to be an integral. As such, the colonization or mortality events could happen at the

time steps when they are the integral multiples of the waiting time value. For example, if a waiting time for a

colonization event is set to WT=12, then the colonization events could happen in the time steps 12, 24, 36 and

so on. As such, WT measures how many time steps are required to trigger a colonization event. When the

simulation has a total time step of 100, the overall colonization event number should 100/12 8.

The second strategy is to assume the waiting time of a colonization event being stochastic. The stochastic

waiting time is modeled by an acceptance rate (still use WT to indicate the acceptance rate, being less than 1

and larger than 0) and a variate randomly drawn from the uniform distribution [0,1]. Different from the

constant WT cases, for stochastic WT, for each time step, a colonization event could be allowed to happen

only when the randomly drawn variate is smaller than the acceptance rate WT. Consequently, for stochastic

WT cases, an acceptance rate WT indicates how many colonization events could happen during the simulation.

For example, if WT=0.5, and the simulation time is 100 as a total, then the overall colonization event number

for the simulation is 1000.5=50.

Finally, it is worth noting that, whether stochastic or constant strategies are applied, WT=1 always implies

a perfect synchrony between colonization and mortality events because both are allowed to happen at each

time step.

Based on the above definitions, for each time step, a mortality event has to happen, for which an individual

has the probability of ( )iM p to die and the patch becomes vacant again. In contrast for each time step, the

colonization of the vacant sites could be allowed only when the WT setting for a colonization event is satisfied.

When a colonization event can be allowed to happen, the vacant patch will be colonized by an offspring of

another individual from the neighboring patches (four neighboring cells are used in the present study: up, low,

left and right). Whether the offspring is a cooperator or defector is determined by following probabilities,

( 1)1( )

2 2i

j ji j

j S

x xPC C p

(5)

And

( 1)1( )

2 2i

j ji j

j S

x xPD C p

(6)

where iPC and iPD represent the probability of an offspring of the cooperators and defectors from the

neighboring patches of patch We to colonize the vacant patch i. If iPC > iPD , then the patch is colonized by

a cooperator offspring; if iPC < iPD , the patch is colonized by a defector offspring.

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During the simulation, we use the occurrence frequency of species (fraction of grid cells occupied) as the

index to quantify the influence of varying colonization waiting time on the coexistence and survival of both

game players. In our simulation, a 2D square grid system with periodic boundary conditions is employed with

a size of 5050. For each simulation, 1000 time step is used.

3 Results

3.1 Coexistence of game players by varying mortality and colonization strengths

When evaluating the coexistence of both players as the function of mortality and colonization coefficients,

apparently, as indicated by the 3D surface plot (Fig. 1), lower mortality coefficient could allow the coexistence

and survival of both game players. Colonization coefficient c has little effect on the coexistence scenarios of

both game players. The coexistence of players is principally determined by mortality strength m.

3.2 Coexistence of game players by varying cooperation and defense rewards

When evaluating the coexistence of both players as the function of cooperation and defense rewards, as

indicated by the 3D surface plot (Fig. 2), the linear combination between and reward could allow the

coexistence of both game players. Interestingly, for all the area with < and part of the region with >

could maintain the coexistence of both species. The latter could be applicable only when their difference is not

too large (Fig. 2). Otherwise, cooperators would dominate the community.

3.3 Coexistence of game players by varying defense reward and colonization strength

As showed in Fig. 3, high colonization coefficient c will lead to the dominance of defectors in the community,

while cooperators would die out. In contrast, when the colonization coefficient c is low, coexistence of both

game players are possible, regardless of the values of defense reward .

3.4 Coexistence of game players by varying waiting time and colonization strength

As showed in Fig. 4, varying either waiting time or colonization strength could not change the coexistence

pattern of cooperators and defectors. Both game players could coexist throughout the simulation, but

cooperators have higher population densities.

3.5 Coexistence of game players by varying waiting time and defense reward

As showed in Fig. 5, only when waiting time is small and stochastic, the coexistence of both players is

possible. Otherwise, cooperators would dominate the community and defectors go extinct. Interestingly, the

population density of cooperators would become highest for the cases of defense reward >10 when WT is

around 13 (Fig. 5). Increasing or decreasing WT from the optimum would reduce the population of cooperators

in the community, regardless of the existence of defectors.

Another interesting thing is that the cooperators could not occupy all the vacant sites even when defectors

have been removed out of the community for most of parameter space (Fig. 6). An exception is found at the

bottom-left area which has the parameter space with small and long stochastic waiting time of triggering

colonization events (small WT approaches zero) (Fig. 6).

3.6 Persistence time of both cooperators and defectors for different waiting time situations of triggering

colonization events

For the persistence time of defectors, a unimodal pattern was identified (Figs. 5 and 7). The longest persistence

time for the players could be found at WT=0.7 (indicating that for each time step, a 0.7 probability of

triggering a colonization event). The overall number of colonization events during a simulation with 800 time

step would be around 560. However, when WT becomes larger and the probability of triggering colonization

events becomes higher, the persistence time of the defectors in the community is decreased and they could not

survive until the end of the simulation.

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Fig. 1 Coexistence of cooperators and defectors under different combinations of mortality and colonization coefficients m and c. The settings for other parameters: =1, =1.5, WT=0.5, = =0.9. The initial populations of both players are set to 1/3 of the number of total grids (=833).

Fig. 2 Coexistence of cooperators and defectors under different combinations of rewards and for cooperators and defectors. The settings for other parameters: m=0.1, c=0.6, WT=0.5, = =0.9. The initial populations of both players are set to 1/3 of the number of total grids (=833).

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Fig. 3 Coexistence of cooperators and defectors under different combinations of defense reward and colonization coefficient c. The settings for other parameters: =1, m=0.1, WT=0.5, = =0.9. The initial populations of both players are set to 1/3 of the number of total grids (=833).

Fig. 4 Coexistence of cooperators and defectors under different combinations of waiting time WT and colonization coefficient c. The settings for other parameters: =1, =1.5, m=0.1, = =0.9. The initial populations of both players are set to 1/3 of the number of total grids (=833).

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Fig. 5 Coexistence of cooperators and defectors under different combinations of waiting time WT (0.1~40) and defense reward (1~20). The settings for other parameters: =5, m=0.1, c=0.6, = =0.9. The initial populations of both players are set to 1/3 of the number of total grids (=833).

Fig. 6 Contour plot of the frequency of defectors under different combinations of waiting time WT and defense reward based on Figure 5. The values marked on each contour line indicated the frequency of the defectors in the community.

waiting time

be

ta

0.1

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.8

0.9

1

0 10 20 30 40

51

01

52

0

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Fig. 7 The influence of different random and constant waiting time of triggering colonization events on the persistence time of defectors. Each curve represented the locally weighted scatterplot smoothing of the mean values of persistence time from 500 replicates of simulation with 800 time step. Values showed in the x-axis indicated the waiting time for a colonization event (WT). When WT<1, random waiting time of colonization during the simulation was used. For each time step, the acceptance probability of triggering a colonization event was the corresponding value. When WT>1, constant waiting time of colonization was used. Parameter settings: =5, m=0.1, c=0.6, = =0.9.

Fig. 8 The influence of different random and constant waiting time of triggering colonization events on the persistence time of cooperators. Each curve represented the locally weighted scatterplot smoothing of the mean values of persistence time from 500 replicates of simulation with 800 time step. Values showed in the x-axis indicated the waiting time for a colonization event (WT). When WT<1, random waiting time of colonization during the simulation was used. For each time step, the acceptance probability of triggering a colonization event was the corresponding value. When WT>1, constant waiting time of colonization was used. Parameter settings: =5, m=0.1, c=0.6, = =0.9.

For the persistence time of cooperators, the situation is remarkably different (Figs. 5 and 8). For the cases

of large defense award , persistence time of cooperators would have two peaks at low WT<0.3 and

WT=13~17 respectively (Fig. 8). For both peaks, the frequency of triggering colonization events is low. When

WT<0.3, the overall number of colonization events during a simulation with 800 time step would be less than

0

100200

300

400

500600

700

800900

1000

0.1 0.3 0.5 0.7 0.9 5 13 21 29 37

Waiting time

Per

sist

ence

tim

e

beta=1

beta=4

beta=7

beta=10

beta=13

beta=16

beta=19

beta=22

beta=25

beta=28

150

250

350

450

550

650

750

850

0.1 0.3 0.5 0.7 0.9 5 13 21 29 37

Waiting time

Per

sist

ence

tim

e

beta=1

beta=4

beta=7

beta=10

beta=13

beta=16

beta=19

beta=22

beta=25

beta=28

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240. When WT=13~17, the overall number of colonization events during a simulation with 800 time step

would be around 47~61.

For both game players, too long constant waiting time of triggering colonization events actually will be

detrimental to both species (Figs. 5, 7-8). The persistence time of both players would decline drastically when

WT>20. Further, too long stochastic waiting time of triggering colonization events is disadvantageous to the

survival of defectors as well (WT<0.3; Figs. 5 and 7), but not cooperators (Figs. 5 and 8).

4 Discussion

The principal finding of the present study is that waiting time of colonization for the game players during the

simulation could considerably affect the survival patterns of cooperators and defectors in the spatial PD game.

When comparing the persistence time patterns of both game players by varying different waiting time settings,

it is found that stochastic waiting time setting (0<WT<1) could allow a longer persistence time of defectors but

not cooperators during the evolutionary game. Higher probability of triggering a colonization event (WT 1)

indicated a lower fluctuation of waiting time, which in turn indicated a high synchrony between mortality and

colonization events. Based on the persistence time curve patterns (Fig. 7), a relatively higher synchrony

(WT~0.7) of the two quantities would allow a longer persistence time of defectors. In contrast, a remarkably

temporal asynchrony between colonization and mortality is beneficial to the persistence of cooperators (Fig. 8).

That is, a low frequency of triggering colonization events could make cooperators to survive better. As a

consequence, a temporal synchrony between colonization and mortality would have opposite influences on the

survival of defectors and cooperators. As such, the present study is different from a previous study, which

suggested that a strong synchrony of within-population reproduction activity could promote species

coexistence (Chen and Hsu, 2011).

A longer waiting time may hinder the survival of both players in the simulation. It should be true because

species are very vulnerable for extinction when the morality events take place too frequent during the

simulation in comparison to the colonization events. In the present study, such an assertion could be evidenced

by the shorter persistence time of both game players when WT>40 (Figs 5, 7-8). However, a stochastic longer

waiting time might have a different scenario. As showed in Fig. 8, when WT<0.3, the persistence time of

cooperators could be facilitated actually (or no worse than the other higher WT cases).

Our present study found that both cooperators and defectors could coexist in the community as long as the

mortality coefficient is low enough (m<0.4; Fig. 1). Such an observation is contradictory to the previous study

(Zhang et al., 2005), which suggested that cooperators would dominate the patches only when the ratio

between morality and colonization strengths is moderate. In our study, the coexistence of both game players

could be maintained in the small m situations, being irrelevant to the colonization coefficient c (Fig. 1).

Interestingly, the weak effect of colonization strength c on the coexistence of game players could be further

evidenced by evaluating the influence on players’ coexistence for the pair of c and WT. For the combination

between c and WT, the role of c could change the frequencies of both game players, but never driving them to

go extinct (Fig. 4). Moreover, the colonization strength c plays some important and interesting roles, as

evidenced by the combination between c and : coexistence of cooperators and defectors is possible only

when c is controlled to be <0.6 (Fig. 3). Too strong colonization strength actually would lead to the extinction

of cooperators.

Acknowledgements

The study was supported by China Scholarship Council.

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References

Chen Y, Hsu S. 2011. Synchronized reproduction promotes species coexistence through reproductive

facilitation. Journal of Theoretical Biology, 274: 136-144

Hui C, McGeoch M. 2007. Spatial patterns of prisoner’s dilemma game in metapopulations. Bulletin of

Mathematical Biology, 69: 659-676

Hui C, Zhang F, Han X, Li Z. 2005. Cooperation evolution and self-regulation dynamics in metapopulation:

stage-equilibrium hypothesis. Ecological Modelling, 184: 397-412

Langer P, Nowak M a, Hauert C. 2008. Spatial invasion of cooperation. Journal of Theoretical Biology, 250:

634-641

Nowak M, May R. 1992. Evolutionary games and spatial chaos. Nature, 359: 826-829

Nowak M, May R. 1993. The spatial dimemmas of evolution. International Journal of Bifurcation and Chaos,

3: 35-78

Zhang F, Hui C. 2011. Eco-evolutionary feedback and the invasion of cooperation in prisoner’s dilemma

games. PLoS One, 6: e27523

Zhang F, Hui C, Han X, Li Z. 2005. Evolution of cooperation in patchy habitat under patch decay and isolation

Ecological Research, 20: 461-469

Zhang WJ. 2012. Computational Ecology: Graphs, Networks and Agent-based Modeling. World Scientific,

Singapore

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Article

An online calculator for spatial data and its applications

Kalle Remm, Tiiu Kelviste Institute of Ecology and Earth Sciences, University of Tartu. 46 Vanemuise St., 51014 Tartu, Estonia


Received 6 December 2013; Accepted 10 January 2014; Published online 1 March 2014

Abstract

An online calculator (http://digiarhiiv.ut.ee/kalkulaator/) for statistical analysis of spatial data is introduced.

The calculator is applicable in a wide range of spatial research and for courses involving spatial data analysis.

The present version of the calculator contains 35 web pages for statistical functions with several options and

settings. The input data for most functions are pure Cartesian coordinates and variable values, which should be

copied to the input cell on the page of a particular spatial operation. The source code for the computational part

of all functions is freely available in C# programming language. Examples are given for thinning spatially

dense observation points to a predefined minimum distance, for calculating spatial autocorrelations, for

creating habitat suitability maps and for generalising movement data into spatio-temporal clusters.

Keywords spatial statistics; online tool; habitat suitability; autocorrelation; spatio-temporal clustering.

1 Introduction

Statistical problems in ecology, earth and environmental sciences, human geography, and other fields of

science and technology are often related to location or distance between observations. Spatial statistics deals

with functions that involve location in one way or another. For example, calculating mean precipitation level at

weather stations is not spatial statistics, as the location of stations is not involved in the summarising. Finding

out how the difference in precipitation values is related to the distance between stations is spatial analysis since

location is directly involved.

Scientific studies can be divided into descriptive (exploratory) and inferential (confirmatory) approach.

Exploratory spatial statistics deals with finding generalised values, relationships, spatial patterns, spatial

clustering and segmentation (Zhang, 2010). The first goal of inferential spatial analysis is usually to prove the

non-randomness of the pattern, followed by attempts to find possible reasons and to model the processes which

created the non-randomness.

Spatial statistical functions are included into major commercial GIS (Geographical Information Science)

software packages. Tools for spatial statistical analysis in freeware packages like QGIS are much more limited.



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There are ample online calculators for mathematical functions and ordinary statistical tests (see, e.g.

http://www.martindalecenter.com/Calculators2.html), but an online calculator for spatial analysis is hard to

find. Existing online calculators for spatial data mostly transform geodetic coordinates only.

The goal of this publication is to introduce the spatial data calculator at http://digiarhiiv.ut.ee/kalkulaator/

that is applicable in a wide range of spatial research and supports several courses on data management and

programming at the University of Tartu. The idea was formed, the original user interface designed and the

coding and testing project mainly developed by the first author. From the technological side, the calculator was

developed as Microsoft ASP.NET project, functions written in the C# programming language. The calculator

was created primarily for teaching and learning purposes, although has already been used in research (Kotta et

al., 2013). Computations for the following examples are made using the online calculator.

2 Main Characteristics of the Calculator

The calculator currently involves a home page and 35 web pages for statistical functions; 28 of these are

directly for spatial analysis. Most pages have several options to set the initial parameters for calculations. The

calculator does not demand any client-side installation other than a web browser. Input data and the parameters

set by the user are transferred to the server, which responds with computation results to the client's browser.

The calculator has no extra demands on the memory or processing unit of the client's device except for data

transfer and browsing ability. The amount of input data and the choice of functions are restricted, as online

applications are expected corresponding within a reasonable period of time. The restrictions depend on

function. Generally, for larger data sets and more complicated tasks, special GIS and/or statistical software

should be used instead.

Data input to most functions must be pure Cartesian coordinates and variable values. The input cell of a

web page in the calculator can be filled with example data by clicking the "Example data" button.

The measurement units of output distances and input parameters are the same as for coordinates, e.g. if the

coordinates are given in metres, then distance intervals and search radii must also be in metres. The decimal

separator must be a point, and the column separator a space, tab or semicolon. Empty cells are not allowed –

these should be filled or removed during data mounting. It is easy to prepare the input matrix as a spreadsheet

(Excel, Access) and to copy the values without column headers to the input cell of the online calculator. The

introductory text at every statistical function, as well as example data, indicate the necessary format of input

data. The point data can be in one data set or alternatively as separate samples of source locations and

destination points. In the last case, distances are measured only between two types of points. The web pages of

the calculator functions also include buttons for viewing a scatter plot of input data in a pop-up window, a

button that initiates calculation, and a button that opens the source code window (Fig. 1).

The meaning of X and Y coordinates as input columns can be switched with regard to which is the west-

east direction and which is south-north (Fig. 2). The X axis of example data in the calculator is always directed

to the east. For the European cartographic system, where the first coordinate axis is directed to the north, the

north direction must be set to the first position (select option "N direction first").

Suitability surface as an Idrisi unpacked headerless rst format raster can be added in several functions. It

enables the user, for example, to delimit the location of generated random points, and to define an irregularly

shaped, uneven and/or patchy study surface.

The location of input data points can be visualised in a pop-up window as an XY scatterplot or a bubble

chart, if the data points have values. Area borders set by the user determine the extent of the scatterplot, the

borders of the suitability surface (if used), and delimit random locations for the null model. A map background

from the Web Map Service (WMS) server of the Estonian Land Board can be added to scatterplots of source

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data and to output maps, given the coordinates are acceptable for this WMS service.

Fig. 1 Input cell of the calculator and the main buttons around it.

Fig. 2 Alternative meaning of X and Y coordinates in input data.

Pressing the button labelled "Calculate" initiates calculation on the server. Every click on the "Calculate"

button is counted to see the usage intensity and dynamics of every function. The calculation result is output as

a tab-delimited text that can be easily imported to other software solutions. The results include the time spent

on calculations on the server. The time spent on transmission between the server and the client’s computer is to

be added to the pure calculation time.

The source code for the computational part of the calculator web pages is public – advanced users can see

in detail how the result is calculated. The authors do not charge a fee for the use of this online calculator, and

do not take responsibility for any incorrect or unexpected results obtained by using this calculator. The

calculator is still in development, the code is continuously improved according to the developers’ skill, and

additional functions will be added if useful in teaching or for research. Users can send comments and proposals

for advancing this application directly from the home page.

3 Application Examples

3.1 Habitat suitability of Potentilla fruticosa and the patchiness of found sites

Potentilla fruticosa is as a flowering shrub mainly known as an ornamental cultivar. P. fruticosa subsp.

fruticosa grows naturally in a few locations in Sweden, Great Britain, Estonia and Latvia (Elkington, 1969;

Leht and Reier, 1999).

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For the following example, we used points on GPS-recorded field observation tracks and found locations

of this species from a study area of 10 km × 10 km square (sheet 6382 of the Estonian 1: 20 000 base map) in

the middle of the natural distribution area of P. fruticosa in north-west Estonia. Coordinates of observation

tracks and P. fruticosa observation locations were recorded using a Garmin Vista HC+ GPS recorder during

field trips on foot by the first author in summers 2008–2013. The initial data contained 12,299 track points and

1469 observation locations. Firstly, all automatically recorded track points closer than 50 m from active

observation points were removed using the online calculator’s thinning function. Then, both recorded track

points and observation locations were thinned to have at least 50 m between each accepted site. The thinned

data contained 499 find sites and 1122 absent sites (332 actively and 790 automatically recorded) (Fig. 3).

Fig. 3 Potentilla fruticosa observation sites within map sheet 6382 of the Estonian 1: 20 000 base map. Black dots – absent sites, red dots – find sites, background green intensity – habitat suitability representing the probability to find P. fruticosa according to soil and land use category.

The find sites of P. fruticosa are spatially clustered, lying on the tracked observation routes and

aggregating at more suitable habitats. P. fruticosa occurrence may also be spatially clustered because of the

limited dispersal ability of the plant. To assess the effect of habitat on the apparent spatial clustering of the P.

fruticosa find sites, we need a habitat suitability map covering the study area.

The online calculator offers several options for suitability mapping according to landscape characteristics

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by calculating: 1) combinations of presence and absence frequency; 2) probability of occurrence, and 3)

expected presence or absence according to similarity of sites using either the k nearest neighbours (kNN), d

nearest neighbours (dNN) or sumsim algorithm. For the kNN algorithm, the number of the most similar

exemplars included has to be fixed; dNN algorithm sets a limit on the acceptable similarity level; the sumsim

algorithm, which is applied in the Constud software system (Remm and Kelviste, 2011; Remm and Remm,

2008), needs the sum of similarity of the most similar accepted exemplars to be given.

For this study, habitat suitability of P. fruticosa was calculated using probability mapping within a map

sheet. The resulting map indicates a higher probability to find the plant in alvar grasslands, where the land

cover category is either bush land or so-called other (unmanaged) open land, and the favourite soil categories

are Gleisoil, Endogleyic Luvisol, and Rendzic Leptosol. The bushes in these alvar grasslands are mainly

junipers.

The second part of this example is analysing the spatial pattern of P. fruticosa find sites. A spatial pattern

of points: its clustering, regularity or randomness can be described in a generalised form using several formal

statistics presented in the online calculator: the nearest neighbour distances, distribution of all distances

between points, the mean squared distance, K(t), L(t)−t, O(r), G(r), F(r), and J(r) functions. Clustering of P.

fruticosa find sites relative to total area, to estimated habitat suitability, to observation sites, and to suitability

weighted observation sites according to K(t), L(t)−t and O(r) statistics with 95% confidence envelope,

calculated using the online calculator, is presented here as an example.

The K(t) statistic was introduced by B.D. Ripley (Ripley, 1976, 1977, 1981) as the mean number of

neighbouring objects within radius t from a source point divided by the mean density of objects. Letters d and r

often stand for radius in the K(t) function instead of Ripley's original sign, t. In the following, r is preferred to

make the notation of radius uniform. The expected number of randomly located neighbours within radius r is

λK(r), where λ is the mean density of objects. The K(r) statistic is widely accepted in science, since it does not

depend on the density of points (He and Duncan, 2000).

Ripley's K(r) is often transformed to L(r)–r statistic (1), for which the expected value in case of spatial

randomness is zero, not depending on the radius.

(1)

The K(r) and L(r)–r statistics are cumulative functions of radius, contrary to differential statistics

measuring the frequency of neighbours in distance intervals. The cumulative function is more stable, but

pattern properties close to the source affect values of the statistic at larger distances. Differential functions are

calculated separately for every distance zone and the result becomes unstable as the number of objects per

interval diminishes.

The function characterising neighbour density according to distance is called radial distribution in physics.

In ecology it has different names: pair correlation (Law et al., 2009), O-ring statistic (Wiegand et al., 1999;

Wiegand and Moloney, 2004), relative neighbourhood density (Condit et al., 2000), and neighbour density

distribution (Remm and Luud, 2003). O-ring statistic and neighbour density is not normalised by the mean

density; pair correlation and radial distribution function are normalised to have the expected density equal to

one. All point pattern statistics that use distances between objects can be used to characterise a pattern of

uniform point objects or for describing relationship (spatial association, segregation) between different

category point objects. In the last case, source and target objects are different.

In case of ecological data, habitat properties can seldom be considered equal across the study area. More

rrK

rrL

)()(

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IAEES www.iaees.org

suitable regions have more resources per surface unit and support a higher density of living objects. In order to

estimate relative location preferences of objects not caused by unequal distribution of resources, the density of

objects should be counted not relative to geometrical area, but to the suitability field (Malanson, 1985; Remm

et al., 2006).

A common issue affecting both observed and predicted values of neighbourhood statistics is the edge effect

– no target objects are counted outside the study area, although distance zones around source objects closer to

the boundary than the counting radius are partly outside the study area. Therefore, the local density of

neighbours may be underestimated near boundaries of the study area. The algorithm used in the online

calculator includes an edge effect correction option for K(r), L(r)–r and O(r) functions. Instead of using area

within a radius, the area of arbitrary grid units (pixels) within the study area is summed. The grid units have

suitability values if the suitability field is included; otherwise, all pixels have the same value. When computing

boundary and suitability corrected L(r)–r statistic in the calculator, the explicit radius r in the right side of the

formula (1) is replaced by a variable derived from the suitability corrected area (A) within the boundaries of

the study plot and within the given radius r (2).

(2)

To describe statistical properties of the spatial pattern of P. fruticosa find locations, K(r), L(r)–r and O(r)

functions were calculated within 200-m-wide intervals up to 5 km distance using the online calculator. These

functions were calculated relative to untransformed area, relative to the mapped habitat suitability, relative to

observation sites and relative to suitability weighted observation sites to reveal the effect of habitat suitability

and uneven location of observations. Suitability field was calculated based on proportion of finds in

combinations of land use and soil categories as described above (Fig. 3). A raster layer where pixels of

observation sites were assigned a value of one and all other study area of zero was used to calculate the

functions relative to observed sites. To count suitability at observed sites, the observed pixels received

suitability value from the suitability coverage; all other pixels remained equal to zero and were excluded from

the calculation of zone area. The 95% confidence envelope for the null model of random location of

neighbours was obtained from 1000 iterations.

The density of neighbours calculated per suitability of observation tracks should remove the effect of

habitat patchiness, clustering of observations retaining the effect of the species’ limited spreading ability, and

probably some uncontrolled effects as well. Relating the O(r) statistic only to observed sites is justified, since

the data on presence/absence of the species at unobserved sites is not known and therefore should not affect the

results. The density of neighbouring finds should be much higher if the density of observations was higher.

The area divisor available in the calculator for correcting spatial units and densities was used to set the total

study area to 100 units, regardless of the sum of suitability values and the area of observed sites.

As expected, the O(r) indicates how clustering depends on distance in more detail than the cumulative

L(r)–r function (Fig. 4, 5). P. fruticosa find sites are remarkably spatially clustered, with the density of

neighbours about 20 times higher within 200 m from a find site than the average density of finds in the study

area. This clustering is partly caused by the general patchiness of the habitat suitability in the study area and by

clustering of observation sites on moving tracks. The effect of habitat suitability in the neighbourhood is

evident only at close distances up to about 700 m, where per-suitability calculated density of neighbours (Fig.

4, green line) is less than per-equal-properties-area calculated density of neighbours (Fig. 4, black line).

The clustering of observations has a more evident effect on clustering of find locations. The density of

)()(

)(rArK

rrL

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neighbours calculated relative to observation sites (Fig. 4, blue line) and per suitability of observation sites is

higher than expected in case of randomness up to 4 km of distance (Fig. 4, red line). The O(r) statistic

calculated per suitability weighted observation sites is relatively stable up to this distance, indicating the

existence of larger patches of this species in the study area, although most of the visible clustering is caused by

patchiness of the observation sites and habitat properties.

Fig. 4 Neighbour density function [O(r)] of Potentilla fruticosa find sites relative to the edge-corrected area within the given radius, relative to habitat suitability corrected area, relative to observation sites, and relative to suitability-weighted observation sites. Dashed lines indicate 95% confidence limits for the null model of random location for neighbouring sites within a suitable area. The vertical axis has a logarithmic scale.

Fig. 5 The L(r)–r function of Potentilla fruticosa find sites relative to the edge-corrected area within the given radius, relative to habitat suitability corrected area, relative to observation sites, and relative to suitability-weighted observation sites. Dashed lines indicate 95% confidence limits for the null model of random location for neighbouring sites within a suitable area.

3.2 Spatial autocorrelation of precipitation in the Baltic states

Spatial relationships depend on the location of data points and can be related to the distance and/or direction

between observations but also to region and scale. Autocorrelation indicates the relationship between values of

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the same variable. It can be both positive, meaning a higher distance-related similarity than expected from

random placement of values, and negative, meaning dissimilarity compared to spatial randomness. As an

analogue to spatial relationships, temporal autocorrelation and correlation depend on difference and direction

in time.

As a rule, close observations tend to be similar because they are affected by the same spatially continuous

factors. If spatial phenomena are recorded, interrelated close data points add less new knowledge than an

independent observation. Measuring autocorrelation is essential to estimate the effect of spatial continuity and

to estimate at which distance the observations can be considered to be independent, as classical statistical

methods presume independence of observations. Ordinary statistical tests overestimate the degrees of freedom

in the spatially autocorrelated data, yielding a higher probability of type one statistical errors, i.e. false

meaningful conclusions (Legendre, 1993; Malanson, 1985). Neglecting the effect of autocorrelation when

comparing the explanatory variables leads to overestimation of positively autocorrelated factors (Lennon,

2000). An overview of species and habitat distribution models including spatial relationships is given in

(Miller et al., 2007).

The online spatial data calculator offers calculation of classical correlation coefficients between numerical

variables, both Pearson linear correlation and Spearman rank correlation. For autocorrelation, the present

version of the calculator includes two options for the Moran's I statistic. The Moran’s I is a quotient expressing

the ratio between autocovariation and variance of data values with a predefined relative location (distance

and/or direction) (Moran, 1950). The two options are: 1) a general Moran's I where the global variance is used

for standardising; and 2) I(d) that is calculated separately for distance zones and standardised by the variance

of data values belonging to the same distance interval only. The formula for Moran's I (3) differs from the

algorithm for Pearson R by including relative location (usually distance) dependent weights.

(3)

where N is the number of observations, i and j are site indices of the members of observation pairs (i ≠ j), wij is

the weight depending on distance between sites i and j (wij = 1 if the distance between sites i and j is within a

given value, otherwise wij = 0); z is the mean of z values.

The expected value E(I) in case of spatial randomness of data values approaches zero if the number of

observations is large:

(4)

The Moran's I assumes spatial stability of the variance (homoscedasticity), otherwise the global variance does

not represent the variance at a particular distance interval and the range of I values exceeds –1…+1. In case of

calculating I(d) separately for each distance interval, the variance among values not belonging to the interval

does not affect the result, and therefore the range of Moran's I(d) (5) is within –1…+1.

,

1

2

1 1

N

ii

jiij

N

i

N

jjiij

)z (zw

)z )(zz (zwN

I

.1

1

N

IE

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(5)

where N(d) is the number of observation points satisfying the distance condition d and W(d) is the number of

observation pairs at distance d.

The online calculator has options to calculate both cumulative and discrete spatial (auto)correlation. In the

first case, d means a radius, in the second case a distance interval. An analysis in distance intervals gives a

more detailed picture of the relationship, but may yield unstable and noisy results if the number of

observations per interval is not sufficient. The dependence of (auto)correlation on distance is graphically

depicted as an (auto)correlogram (Fig. 6).

Long-term mean annual precipitation values at 245 meteorological stations in the Baltic states (Estonia,

Latvia and Lithuania) are used in the following autocorrelation examples. More details on these data can be

found in (Jaagus et al., 2010; Remm et al., 2011).

The spatial autocorrelation of the annual amount of precipitation is significantly positive up to a distance of

90 km between stations (Fig. 6). This can be considered the typical extent of the maritime rainy coastal belt,

especially along the coast of Lithuania and western Latvia, and the more continental part of the study area.

Spatial autocorrelation of the mean annual precipitation is negative at distances of 120–190 km.

Autocorrelation is statistically insignificant (p > 0.05) at a distance of 10 or fewer kilometres, since the number

of station pairs that are this close is only 16 in these data.

Fig. 6 Distance-dependent autocorrelation of the long-term mean annual precipitation in the Baltic countries, along with the 95% probability envelopes for the null model (red dashed lines). Data from Remm et al. (2011).

The strength of spatial relationships depends on location and can be mapped if calculated in multiple

locations. The online calculator enables users to calculate local spatial autocorrelation and correlation at nodes

,)(

)(

)(

1

2

1 1

N

ii

N

i

N

jji

)z (zdW

)z )(zz (zdN

dI

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of a regular grid covering the entire study area. In addition to the interval between grid nodes, local search

radius around grid nodes and the distance between included observations must be set in the calculator. The

local spatial autocorrelation map of the long-term mean annual precipitation highlights regions in western

Lithuania and Latvia where similar values are spatially aggregated, that is are located closer than expected

from randomness (Fig. 7). Here, the contrast between maritime and continental climate is most expressed.

Negative autocorrelation values in eastern Latvia are related to a small number of stations attended by

relatively variable measured mean annual precipitation amount in neighbouring stations.

Fig. 7 Statistically significant local spatial autocorrelation values of the long-term mean annual precipitation in the Baltic countries. Interval between grid points of 10 km, search radius around grid points of 100 km, and maximum distance between included stations of 50 km. Red marks positive local Moran's I values, and blue – negative; n. s–p > 0.05, n. c– not calculated. Data from Remm et al. (2011).

3.3 Thinning and spatial clusters of movement data

Currently, movement data are of major interest in many domains, e.g. GPS tracking of wildlife, pets and

vehicles, studies of human migration and tourist movement patterns. The raw data for a movement study are

usually tracking data consisting of millions of location coordinates, which are senseless without some sort of

generalisation and summarisation. To distinguish different aspects of movement (coverage, anchor points,

temporal pattern, etc.) several mathematical operations are needed. The choice of specific method largely

depends on the research subject and aim.

The present version of the online calculator includes k-means clustering, QT (Quality Threshold) and

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DBSCAN (Density-Based Spatial Clustering for Applications with Noise) clustering. The k-means algorithm

demands a prefixed number of clusters, which is not desirable while identifying recurring places from

movement data, as the number of favourite places is a priory not known. The QT algorithm needs only a user-

decided search radius to delimit the extent of clusters. The input parameters for DBSCAN clustering are the

search radius and the minimum number of objects in a cluster, but not the given number of clusters. The level

of generalisation of the resulting clusters can easily and flexibly be controlled by the two parameters. A special

advantage of the DBSCAN algorithm is the acceptance of noise in input – clusters highlight regions where the

density of points is higher, leaving some points and some parts of the study area un-clustered.

To illustrate the online calculator functions, the active mobile phone positioning data of a volunteer living

in Tartu, Estonia, was used for one year (January 2, 2010 to December 31, 2010). Active positioning data in

this case means that the location of the user’s mobile phone was registered continuously after every 15 minutes.

The average positioning error is expected to be about 3 km. The number of raw tracking points was 32,169

(Fig. 8A).

Fig. 8 A – Location of track points in Estonia, B – Main overnight locations of the study object, C – On-stay DBSCAN clusters depicted by a convex hull polygon and a dot at the cluster centre, D – DBSCAN clusters of on-move track points (transport connections between Tartu and Tallinn: the railway is at the north-east, motorway in the centre).

Traditionally, the movement studies involve only two spatial dimensions (Zhou et al., 2007), although

movement is related to time. Therefore, several functions in the online calculator include a spatio-temporal

option where the input includes X and Y coordinates, plus Z dimension, which can be time in the general Date

Time format. For example, thinning a set of raw spatio-temporal data starting at 3 a.m. to a one-day clearance

interval enables places to be extracted where the object was most frequently located at 3 a.m.; that is, where

he/she most frequently overnights. Thinning of the example data resulted in 339 retained separate night

locations. Spatial clustering of these points by the DBSCAN module, using a search radius of 6000 m and

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cluster size of 30 resulted in three frequent overnight places for this person: at home in Tartu, somewhere in

the capital, Tallinn, and one about 20 km west of Tallinn (Fig. 8B). The other locations can be considered

occasional.

A second exercise was separating two kinds of tracking points and generalising these into two types of

clusters: on-move clusters that reflect the trajectories of movement, and on-stay clusters that indicate locations

where the study object has spent more time. To get on-stay clusters, the search radius along XY plane should

be delimited to a shorter value (6000 m in this case) and the time difference should be at least as long as the

study period. The minimum number of objects in a cluster was set to 30. Most of the track points were

classified as on-stay category at this stage. The rest (515) were considered on-move points, which were

generalised into clusters using a 12-hour maximum time difference and not limiting search radius. The

minimum number of objects in an on-move cluster was set to 5, since in some cases only 5 track points were

recorded during Tallinn–Tartu train travel.

As a result, five on-stay clusters and two on-way clusters were generalised for this person. The largest on-

stay cluster is around the Tartu residence; the on-way clusters represent one-day travel routes (49 times

between the hometown and capital, Tallinn), one tour near Tartu, and three longer trips to western Estonia (Fig.

8C, D).

4 Conclusions

The online calculator introduced above supports a wide range of spatial analysis operations, although, the

application is still merely a calculator primarily developed for teaching purposes. For larger datasets and more

comprehensive studies, special software is needed or should be developed. The code for all functions used in

the online calculator is freely open for assessment, criticism, and development of other applications.

The presented examples of spatial analysis were all calculated using the online calculator. The example

cases yielded the following results.

(1) Spatial analysis of Potentilla fruticosa find locations reveals that the main reasons for the obvious

clustering of finds are the patchiness of suitable habitats and clustering of observation sites. Although the seeds

of this plant are not spreading far, the inherent dispersal-related patchiness has a weaker effect than soil

properties and land use type combined.

(2) Spatial autocorrelation of the annual amount of precipitation at the Baltic meteorological stations is

significantly positive at a distance of 10–90 km between stations.

(3) Spatio-temporal clustering of movement data supports insights into large sets of tracking data,

generalisation and visualisation of movement.

Acknowledgments

The authors thank Markus Unt for participating in code development, colleagues Jaanus Remm for developing

an earlier solution for directional analysis and Ain Kull for advice supporting the present module of directional

analysis in the calculator; and also the proof reader Dirk Lloyd. The development of this calculator was

financially supported by the Estonian Ministry of Education and Research (Project IUT 2-16).

References

Condit R, Ashton PS, Baker P, et al. 2000. Spatial patterns in the distribution of tropical tree species. Science,

288(5470): 1414-1418

Elkington TT. 1969. Cytotaxonomic variation in Potentilla fruticosa L. New Phytologist, 68(1): 151-160

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He F, Duncan RP. 2000. Density-dependent effects on tree survival in an old-growth Douglas fir forest.

Journal of Ecology, 88(4): 676-688

Jaagus J, Briede A, Rimkus E, Remm K. 2010. Precipitation pattern in the Baltic countries under the influence

of large-scale atmospheric circulation and local landscape factors. International Journal of Climatology,

30(5): 705-720

Kotta J, Remm K, Vahtmäe E, et al. 2013. In-air spectral signatures of the Baltic Sea macrophytes and their

statistical separability. Journal of Applied Remote Sensing (submitted manuscript).

Law R, Illian J, Burslem DFRP, et al. 2009. Ecological information from spatial patterns of plants: insights

from point process theory. Journal of Ecology, 97(4): 616–628

Legendre P. 1993. Spatial autocorrelation: trouble or new paradigm? Ecology, 74(6): 1659-1673

Leht M, Reier Ü. 1999. Origin, chromosome number and reproduction biology of Potentilla fruticosa

(Rosaceae) in Estonia and Latvia. In: Chorological Problems in the European Flora. (Uotila P, ed).

Proceedings of the VIII meeting of the Committee for Mapping the Flora of Europe. 191-196, Helsinki,

Finland

Lennon JJ. 2000. Red-shifts and red herrings in geographical ecology. Ecography, 23(1): 101-113

Malanson GP. 1985. Spatial autocorrelation and distributions of plant species on environmental gradients.

Oikos, 45(2): 278-280

Miller J, Franklin J, Aspinall R. 2007. Incorporating spatial dependence in predictive vegetation models.

Ecological Modelling, 202(3-4): 225-242

Moran PAP. 1950. Notes on continous stochastic phenomena. Biometrika, 37(1-2): 17-23

Remm J, Lõhmus A, Remm K. 2006. Tree cavities in riverine forests: what determines their occurrence and

use by hole-nesting passerines? Forest Ecology and Management, 221(1-3): 267-277

Remm K, Jaagus J, Briede A, et al. 2011. Interpolative mapping of mean precipitation in the Baltic countries

by using landscape characteristics. Estonian Journal of Earth Sciences, 60(3): 172-190

Remm K, Kelviste T. 2011. Constud Tutorial. University of Tartu, Chair of Geoinformatics and Cartography,

Tartu, Estonia

Remm K, Luud A. 2003. Regression and point pattern models of moose distribution in relation to habitat

distribution and human influence in Ida-Viru county, Estonia. Journal for Nature Conservation, 11(3): 197-

211

Remm M, Remm K. 2008. Case-based estimation of the risk of enterobiasis. Artificial Intelligence in Medicine,

43(3): 167-177

Ripley BD. 1976. The second-order analysis of stationary point processes. Journal of Applied Probability,

13(2): 225-266

Ripley BD. 1977. Modelling spatial patterns. Journal of the Royal Statistical Society, B 39(2): 172-212

Ripley BD. 1981. Spatial Statistics. Wiley, New York, USA

Wiegand T, Moloney KA. 2004. Rings, circles and null-models for point pattern analysis in ecology. Oikos,

104(2): 209-229

Wiegand T, Moloney KA, Naves J, Knauer F. 1999. Finding the missing link between landscape structure and

population dynamics: a spatially explicit perspective. The American Naturalist, 154(6): 605-627

Zhang WJ. 2010. Computational Ecology: Artificial Neural Networks and Their Applications. World

Scientific, Singapore

Zhou C, Bhatnagar N, Shekhar S, et al. 2007. Mining Personally Important Places from GPS Tracks. Data

Engineering Workshop, IEEE 23rd International Conference, 517-526

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Article

Global behavior of an anti-competitive system of fourth-order rational

difference equations

A. Q. Khan1, Q. Din1, M. N. Qureshi1, T. F. Ibrahim2,3 1Department of Mathematics, University of Azad Jammu & Kashmir, Muzaffarabad, Pakistan 2Department of Mathematics, Faculty of Sciences and Arts King Khalid University, Abha, Saudi Arabia 3Permanent address: Department of Mathematics, Faculty of Sciences Mansoura University, Mansoura 35516, Egypt

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

Received 22 October 2013; Accepted 28 November 2013; Published online 1 March 2014

Abstract

In the present work, we study the qualitative behavior of an anti-competitive system of fourth-order rational

difference equations. More precisely, we study the local asymptotic stability, global character of the unique

equilibrium point, and the rate of convergence of the positive solutions of the given system. Some numerical

examples are given to verify our theoretical results.

Keywords system of rational difference equations; stability; global character; rate of convergence.

1 Introduction

Difference equations or discrete dynamical systems are diverse field which impact almost every branch of pure

and applied mathematics. Every dynamical system determines a difference equation and vice

versa. Recently, there has been great interest in studying difference equations systems. One of the reasons for

this is a necessity for some techniques which can be used in investigating equations arising in mathematical

models describing real life situations in many applied sciences. The theory of discrete dynamical systems and

difference equations developed greatly during the last twenty-five years of the twentieth century. Applications

of discrete dynamical systems and difference equations have appeared recently in many areas. The theory of

difference equations occupies a central position in applicable analysis. There is no doubt that the theory of

difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference

equations of order greater than one are of paramount importance in applications. Such equations also appear

naturally as discrete analogues and as numerical solutions of differential and delay differential equations which

model various diverse phenomena in biology, ecology, physiology, physics, engineering, economics,

probability theory, genetics, psychology and resource management. It is very interesting to investigate the



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behavior of solutions of a system of higher-order rational difference equations and to discuss the local

asymptotic stability of their equilibrium points. Systems of rational difference equations have been studied by

several authors. Especially there has been a great interest in the study of the attractivity of the solutions of such

systems. For more results for the systems of difference equations, we refer the interested reader to Cinar

(2004), Stevic (2012a, b), Bajo and Liz (2011), Kalabusic et al. (2009, 2011), Kurbanli (2011), Kurbanli et al.

(2011), Touafek and Elsayed (2012a, b), Elsayed and Ibrahim (in press), Din (a, b; in press).

Zhang et al. (2012) studied the dynamics of a system of rational third-order difference equation:

, , 0,1, .

Din et al. (2012) investigated the dynamics of a system of fourth-order rational difference equations

∏ ,

∏, 0,1, ,

Our aim in this paper is to investigate the qualitative behavior of an anti-competitive system of fourth order

rational difference equations

∏ ,

∏, 0,1, , (1)

where the parameters , , , , , and initial conditions , , , , , , , are

positive real numbers. This paper is natural extension of (Shojaei et al., 2009; Din et al., 2012; Zhang et al.,

2012).

Let us consider eight-dimensional discrete dynamical system of the form:

, , , , , , , , , , , , , , , , 0,1, , (2)

where : and : are continuously differentiable functions and , are some

intervals of real numbers. Furthermore, a solution , of system (2) is uniquely determined by

initial conditions , for 3, 2, 1,0 . Along with the system (2) we consider the

corresponding vector map

, , , , , , , , , .

An equilibrium point of system (2) is a point , that satisfies

, , , , , , ,

, , , , , , ,

The point , is also called a fixed point of the vector map .

Definition 1. Let , be an equilibrium point of the system (2).

(i) An equilibrium point , is said to be stable if for every 0 there exists 0 such

thatfor every initial conditions , , 3, 2, 1,0 if ∑ , ,

implies , , for all 0, where · is usual Euclidian norm in .

(ii) An equilibrium point , is said to be unstable if it is not stable.

(iii) An equilibrium point , is said to be asymptotically stable if there exists 0 such that

∑ , , and , , as ∞.

(iv) An equilibrium point , is called global attractor if , , as ∞. (v) An equilibrium point , is called asymptotic global attractor if it is a global attractor andstable.

Definition 2. Let , be an equilibrium point of a map

, , , , , , , , ,

where and are continuously differentiable functions at , . The linearized system of (2) about the

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equilibrium point , is given by

,

where and is Jacobean matrix of the system (2) about the equilibrium point , .

To construct corresponding linearized form of the system (1) we consider the following transformation:

, , , , , , , , , , , , , , , (3)

where ∏

, , , , ∏, ,

, The Jacobian matrix about the fixed point , under the transformation (3) is given by

,

0 0 01 0 0 0 0 0 0 0000000

100000

0 0 0 0 0 01 0 0 0 0 000 0 1 0 0 00 0 0 1 0 00 0 0 0 1 0

,

where , , and .

Theorem 1. (Sedaghat, 2003) For the system , 0,1, , of difference equations such

that be a fixed point of . If all eigenvalues of the Jacobian matrix about lie inside the open

unitdisk | | 1, then is locally asymptotically stable. If one of them has a modulus greater than one, then

is unstable.

2 Main Results

Let , be an equilibrium point of the system (1), then system (1) has only one equilibrium point namely

0,0 .

Theorem 2.Let , be a positive solution of the system (1), then for every 0 the following result

hold:

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0

, 8 1,

, 8 2,

, 8 3,

, 8 4,

, 8 5,

, 8 6,

, 8 7,

, 8 8.

0

, 8 1,

, 8 2,

, 8 3,

, 8 4,

, 8 5,

, 8 6,

, 8 7,

, 8 8.

Proof. It follows from induction.

Lemma 1.Let0 1, then every solution , of the system (1) is bounded.

Proof. Assume that

max , , , , , , , ,

and

max , , , , , , , .

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Then from Theorem 2 one can see that 0 and 0 for all 0,1, .

Theorem 3.If 0 1 then equilibrium point (0, 0) of the system (1) is locally asymptotically stable.

Proof. The linearized system of (1) about the equilibrium point (0, 0) is given by:

0, 0 ,

where and 0, 0

0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

000000

100000

0 0 0 0 0 01 0 0 0 0 00 0 0 0 0

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 0

.

The characteristic polynomial of 0, 0 is given by

(4)

The roots of

exp 4

for 0,1, ,7. Now, it is easy to see that | | 1 for all 0,1, ,7. Since all eigenvalues of

Jacobian matrix 0, 0 about (0, 0) lie in open unit disk | | 1. Hence, the equilibrium point (0, 0) is

locally asymptotically stable.

Theorem 4.If 0 1 then equilibrium point (0, 0) of the system (1) is globally asymptotically stable.

Proof. From theorem 3, (0, 0) is locally asymptotically stable. From Lemma 1, every positive solution

, of the system (1) is bounded. Now, it is sufficient to prove that , is decreasing. From

system (1) one has

∏

.

This implies that and . Also

∏

.

This implies that and . So and

. Hence, the subsequences

, , , , , , ,

and

, , , , , , ,

are decreasing. Therefore the sequences and are decreasing. Hence

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lim lim 0.

Theorem 5. Let and . Then, for solution , of system (1) following statements are

true:

(i) If 0, then ∞.

(ii) If 0, then ∞.

3 Rate of Convergence

In this section we will determine the rate of convergence of a solution that converges to the equilibrium point

(0, 0) of the system (1). The following results give the rate of convergence of solutions of a system of

difference equations

, (5)

where is an dimensional vector, is a constant matrix, and : is a

matrix function satisfying

0 (6)

as ∞, where . denotes any matrix norm which is associated with the vector norm

Proposition 1. (Perron’s theorem) (Pituk, 2002) Suppose that condition (6) holds. If is a solution of (5),

then either 0 for all large or

lim (7)

exist and is equal to the modulus of one the eigenvalues of matrix .

Proposition 2. (Pituk, 2002)Suppose that condition (6) holds. If is a solution of (5), then either 0

for all large or

lim (8)

exist and is equal to the modulus of one the eigenvalues of matrix

Assume that lim , lim . First we will find a system of limiting equations for the map

. The error term are given by

∑ ∑ ,

∑ ∑ .

Set and , one has

∑ ∑ ,

∑ ∑ ,

where

∏

∏ ,

∏,

∏ ,

∏,

0for 0,1,2 ,

∏,

0 for 0,1,2 ,

40


IAEES www.iaees.org

∏,

∏

∏,

∏,

∏,

∏.

Taking the limit, we obtain lim for 0,1,2,3 , lim 0 for

0,1,2 , lim , lim 0 for 0,1,2 , lim and

lim for 0,1,2,3 . So, the limiting system of error terms can be written as

, where

and K=

0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

000000

100000

0 0 0 0 0 01 0 0 0 0 00 0 0 0 0

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 0

,

which is similar to linearized system of (1) about the equilibrium point ( , 0, 0 . Using the Proposition

1, one has following result.

Theorem 6.Assume that , be a positive solution of the system (1) such that lim and

lim where , 0,0 . Then, the error term of every solution of (1) satisfies both of the

following asymptotic relations

lim , , lim , ,

where , are the characteristic roots of Jacobian matrix , about 0,0 .

4 Examples

In order to verify our theoretical results and to support our theoretical discussions, we consider several

interesting numerical examples in this section. These examples represent different types of qualitative behavior

of solutions to the system of nonlinear difference equations (1). All plots in this section are drawn with

Mathematica.

Example 1

Consider the system (1) with initial conditions

2.2 , 1.9 , 5.8, , 2.9, 1.8, 3.9 , 2.4, 1.8 .

Moreover, choosing the parameters 116, 117, 0.9, 111, 112, 0.6. Then ,

the system (1) can be written as

. ∏ ,

. ∏, 0,1, , (9)

and with initial condition

41

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45


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. Applied Mathematics and Computation, 158: 303-305

Din Q, Donchev T. 2013.Global character of a host-parasite model. Chaos, Solitons and Fractal,54:1-7

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1847-2013-95

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Din Q. On a system of rational difference equation. Demonstratio Mathematica, in press (b)

Din Q, Khan AQ, Qureshi MN. 2013. Qualitative behavior of a host-pathogen model, Advances in Difference

Equations, doi:10.1186/1687-1847-2013-263

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Advances in Difference Equations, doi:10.1186/1687-1847-2012-216

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equations in the plane. Advances in Difference Equations, 1-30 (132802)

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1261-1267

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Equations and Applications, 8: 201-216

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Kluwer Academic Publishers, Dordrecht, Netherlands

Shojaei M, Saadati R, Adibi H. 2009. Stability and periodic character of a rational third order difference

equation. Chaos, Solitons and Fractals, 39: 1203-1209

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7649-7654

Stevic S. 2012b. On some solvable systems of difference equations. Applied Mathematics and Computation,

218: 5010-5018

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Mathematique de la Societe des Sciences Mathematiques de Roumanie, 2: 217-224

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Difference Equations, 1-8 (doi: 10.1186/1687-1847-2012-136)

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IAEES www.iaees.org

Article

Effect of land uses of Huai Lam Kradon Sub-watershed on quantifying

soil carbon potential with process base model

Chattanong Podong1, Roongreang Poolsiri2

1Department of Conservation, Faculty of Forestry, Kasetsart University, Chatuchak, Bangkok, Thailand 10900 2Department of Sivilculture, Faculty of Forestry, Kasetsart University, Chatuchak, Bangkok, Thailand 10900


Received 19 October 2013; Accepted 25 November 2013; Published online 1 March 2014

Abstract

The study the effect of land use on soil carbon is importantly for the future management of greenhouse gases

and climate change, and soil carbon budget is one activity mention of the United Nations Framework

Convention on Climate Change (UNFCCC) for decreasing effect from climate change. Previous studies based

on field observations have provided direct information about soil carbon storage and fluxes at specific sites,

but soil carbon is highly dynamic in space and time and that is driven by complex combinations of hydrology,

soil vegetation and management condition. The observation results was soil carbon higher in mixed deciduous

forest 17,472.30 Kg C ha-1 than para rubber plantation 8,304.52 Kg C ha-1 at depth 0-5 cm and at depth 5-20

cm 8,304.52 Kg C ha-1 and 6,776.65, respectively. The DNDC model has shown that it can perform well in its

representation of the effects of both land uses change in this study area. Simulation results showed significant

loss of soil carbon from system under both land use types and eight scenarios of land use change from mixed

deciduous forest to para rubber plantation and para rubber tree change to mixed deciduous forest. The annual

50 year soil carbon was 17,960 and 8,300 C ha-1 yr-1 for mixed deciduous forest and para rubber plantation,

respectively. The simulated soil carbon under land uses change scenarios. The result for soil carbon content in

three scenarios for mixed deciduous forest change to para rubber plantation scenarios. The soil carbon decrease

in all scenarios and the mean decrease highest of litter carbon in MDF 10 Year to Para rubber 40 Year scenario

was 8,770.42 C ha-1 yr-1 or 49.79% and mean lowest of soil carbon MDF 40 Year to Para rubber 10 Year

scenario was 4,700.47 ha-1 yr-1 or 26.68 %. The result for soil carbon content in three scenarios for mixed

deciduous forest change to para rubber plantation scenarios. The mean soil carbon and decrease highest of

litter carbon in para rubber plantation 10 year change to mixed deciduous forest 40 year was 6931.22 C ha-1 yr-

1 or 45.57% and mean lowest of soil carbon para rubber plantation 40 year change to mixed deciduous forest

10 year was 3452.57 C ha-1 yr-1 or 22.70%.

Keywords soil carbon; DNDC model; land use change; Northern Thailand.



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1 Introduction

Land uses affect many aspects of ecosystems and agriculture (Zhang et al., 2006). It affect the fixation of CO2

and carbon sequestration mechanism, and carbon is accumulated in plants and soils by growth mechanism of

plant and decay mechanism of microbial. Especially, influence of soil carbon budget in land use is one of the

importance strategies for mitigating the global greenhouse effect (Tan et al., 2004). The capacity of soil to

carbon accumulation in land use is mainly dependent on factors in land use such as: hydrology, soil type,

vegetation type, soil management and the degree to which. The soil carbon dynamics is influenced by soil

erosion on carbon status such as: soil organic carbon (SOC) in soil, CO2 flux in atmosphere and dissolved

organic carbon (DOC) in water (Bajracharya et al., 1998). Therefore, land use change is a major factor to

affect carbon budget in the soil, plant, water and CO2 emission with land use, and it is a continual effect on

climate change.

The study on the effect of land use on soil carbon budget is important for the future management of

greenhouse gases and climate change, and soil carbon budget is one activity mention of the United Nations

Framework Convention on Climate Change (UNFCCC) for decreasing effect from climate change. Previous

studies based on field observations have provided direct information about soil carbon storage and fluxes at

specific sites, but soil carbon is highly dynamic in space and time and that is driven by complex combinations

of hydrology, soil vegetation and management condition. Therefore, quantifying soil carbon dynamics at the

national and regional scale through field measurements is impracticability. Models have been developed trying

to extrapolation from the site scale to help understand the regional scale or site scale (Zhang et al., 2002).

From the problem mentioned above, especially in Thailand the land use change from forest land to

agricultural land in headwater of watershed, and it affects on soil carbon. However, previous study of

evaluation for soil carbon in Thailand was used field observation data. The objectives of this study used data

from field observation at 2 different land uses for evaluation of soil carbon budget to 2 measurements such as:

field observations calculation and Denitrification-Decomposition (DNDC) model. In case of DNDC model is

biogeochemically based model that has limited requirements for input parameters, and this model has been

successfully applied in other counties, especially in Asia (Brown et al., 2002). However, the DNDC model has

not been modified to calibrator, validation and test for the evaluation of soil carbon in Thailand, especially

regard to land use in a watershed.

2 Materials and Methods

2.1 Site description

The study site is located at the Huai Lam Kradon subwatershed where a part of the Wang Thong watershed.

The study area covers forest in the Thung Salang Luang National Park and adjacent some para rubber tree

plantation. This study area is located in lower northern of Thailand, the altitude approximate 700-860 m. The

geological formation of the study area is composed of sedimentary rock and metamorphic rock

(Boonyanuphap et al., 2007). The climate is tropical and sub-tropical with three distinct seasons such as:

winter, summer and rain. March to June are the hottest month mean maximum temperature (29๐C), and

November to February are the coldest months mean minimum temperature (17๐C), and the mean temperature

is 22๐C. The maximum rainfall occurs during the monsoon season May to October with mean rainfall 1,300-

1,700 mm. Monthly rainfall and temperature during study represent in Fig. 1 and site characteristics descript in

Table 1 Two major land uses, namely mixed deciduous forest (MDF) and Para rubber tree (Hevea brasiliensis

Müll. Arg.) plantation (PARA) selected as representative land uses for the study. Study duration on April

2010-March 2011.

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Fig. 1 Monthly rainfall and temperature from April 2009 to March 2011 at Protection Unit 12 of Thung Salang Luang National Park. Data from Royal Irrigation Department telemetering weather station.

Table 1 Site characteristic of Huai Lam Kradon Sub-watershed.

Land uses Mixed deciduous forest Para rubber tree plantation

Location Thung Saleang Lung

National Park

Private owner

Latitude/Longitude 1852004 47Q 0679176

1852077 47Q 0679108

1851955 47Q 0679133

1851988 47Q 0679217

1852151 47Q 0678961

1852220 47Q 0679008

1852285 47Q 1852285

1852290 47Q 1852290

Altitude 458 555

Annual Rainfall 150.06 (mm) 150.06 (mm)

Annual Mean Temperature 27.23 ๐C 27.23 ๐C

Soil Type Clay Loam Sandy Clay Loam

Sand (%) 52.86 % 41.40 %

Silt (%) 23.09 % 16.35 %

Clay (%) 35.70 % 30.38 %

Bulk density (g cm-3) 1.42* 1.53*

pH 5.21 4.62

Soil organic carbon (%) 2.17 % 1.05 %

Data from Boonyanuphap et al (2007).

2.2 Field observation

The field observation at land use types in study area used permanent plots of 50 x 50 m in quadrates involving

four plots per land use types. The field data collected environmental factors for evaluating soil carbon in both

land uses types such as: vegetation census, litter dynamics, soil chemical properties, soil physical properties,

soil respiration, climatic data and hydrological data.

2.3 Process base model with DNDC

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The DNDC model is a general model of carbon (C) and nitrogen (N) biogeochemistry in forestry and

agricultural ecosystems at the site scale or regional scale. For this study at the site scale data is inputted for all

required driving parameters through a user interface. The DNDC model simulates the carbon dynamics for

periods from one year to centuries. A major challenge in applying an ecosystem model at site scale is

assembling adequate data sets needed to initialize and run the model. Applying the DNDC model evaluates the

soil carbon budgets in mixed deciduous forest and para rubber plantation at Huai Lam Kradon subwatershed

that was required data of soil properties, daily climate data and vegetation data. The data was consisted two

sources such as field observation data and record data for process model simulation as:

2.4 Vegetation data

Data of mixed deciduous forest and Para rubber trees used for DNDC model. Input data based on the analysis

of field observation data include: dominant type of tree, dominant type of sapling, dominant type of seedling,

biomass of leave, root and stem, plant N storage, plant C storage and plant C/N ratio.

2.5 Climatic data

Climatic data inputs based on the analysis of field data and record data include: N in rainfall in northern

Thailand amount 0.2±0.1 mg L-1(Moller et al., 2005), atmospheric background CO2 concentration and

meteorological data files (daily air maximum and minimum temperatures, rainfall and solar radiation). These

inputs are derived data from dataset in historical records from 2009 to 2010 interpolation of observed values

with automatic weather data from telemetering.

2.6 Soil data

Soil data inputs based on the analysis of field data include: soil fertility, soil type, thickness of organic layer,

thickness of mineral soil, pH, soil organic carbon content of top soil, soil organic carbon content in profile,

total thickness of the entire soil profile, number of soil layers and soil bulk density.

3 Results and Discussion

3.1 Initial carbon in biomass

Biomass carbon in term mixed deciduous forest, the aboveground biomass of the tree such as stems, branches

and leaves have been estimated using allometric equations by Ogawa et al. (1965). In term para rubber

plantation, The aboveground biomass of the tree such as stems, branches and leaves have been estimated using

equations by Yoonsuk, 2005. All aboveground components were assumed to have 50% C content (Brown and

Lugo, 1984; Levine et al., 1995).The above ground biomass of both land use types represent in Table 2. The

biomass carbon storage of range of forest in Thailand 63,000 Kg C ha-1 (Ogawa et al ., 1965). Compare to

studies in neighboring countries, this results were fairly similar to the natural forest in Malaysia 100,000-

160,000 Kg C ha-1 Philippines 86,000-201,000 Kg C ha-1 (Lasco, 2002).

Table 2 The above ground biomass of mixed deciduous forest and para rubber plantation.

Biomass Carbon Mixed deciduous forest

(MDF) Kg C ha-1

Para rubber plantation

(PARA) Kg C ha-1

Aboveground Carbon 64,850 12,050

Belowground Carbon 32,430 6,030

Vegetation Carbon 97,280 18,350

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3.2 Initial carbon in litter

The amount and quality of litter is one of the important factors that determine how much carbon could be

preserved in soil. Low rate of litter fall, decomposition and incorporation into the amount of litter production

varies from biome to biome. Factors that affect litter fall include plant species, environment, silvicultural

practices, and time factor. Generally, plantation yields more litter fall than the natural stand. This was

attributed to the even-aged condition of plantation rather than stand density (Thaiutsa et al., 1978).Carbon

returns from litterfall are of interest because they can help understand the nutrient uptake in a land use system.

In this study, the carbon return of litterfall was higher in mixed deciduous forest (2,688.3 Kg ha-1 yr-1) than in

the Para rubber plantation (709.1 Kg ha-1 yr-1). The carbon return from litter of both land use types represent in

Table 3.

Table 3 Amount of carbon returns form litter (Kg C ha-1) collected over 1 yr under mixed deciduous forest (MDF) and para rubber plantation (PARA).

Month/

year

Mixed deciduous forest (MDF) (Kg C

ha-1)

Para rubber plantation (PARA)

(Kg C ha-1)

Apr 272.3 11.5

May 171.9 11.1

Jun 88.2 6.3

Jul 75.2 5.5

Aug 120.8 11.8

Sep 153.6 34.9

Oct 272.1 120.5

Nov 179 36

Dec 159.6 60.5

Jan 252.5 59.1

Feb 549.6 244.2

Mar 393.5 107.8

The correlation in MDF between carbon return from litterfall and climate factors such as: maximum

temperature, minimum temperature, average temperature and rainfall. All climate factors were lightly

correlation and there were highest correlation between carbon return from litterfall and average temperature

(R2=0.362). The correlation in MDF between carbon return from litterfall and climate factors represent in Fig.

2.

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Fig. 2 The correlation in MDF between carbon return from litterfall and climate factors such as: maximum temperature, minimum temperature, average temperature and rainfall.

The correlation in PARA between carbon return from litterfall and climate factors such as: maximum

temperature, minimum temperature, average temperature and rainfall. All climate factors were lightly

correlation and there were highest correlation between carbon return from litterfall and average temperature

(R2 = 0.392). The correlation in PARA between carbon return from litterfall and climate factors represent in

Fig. 3.

Fig. 3 The correlation in PARA between carbon return from litterfall and climate factors such as: maximum temperature, minimum temperature, average temperature and rainfall.

3.3 Initial carbon content in soil

The vertical distribution of soil carbon also varied between both land use types. The overall average proportion

of soil carbon was higher in the mixed deciduous forest than para rubber tree plantation. In all land use types,

the deposition of soil carbon was generally higher in the top soil (0–5 cm) and decreased with soil depth (5-20

cm). In both land use types soil carbon highest in mixed deciduous forest. The soil carbon of both land use

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types and two depths represent in Table 4.

Table 4 Percent soil carbon of both land use types and two depths. Data are mean ± SD (n= 20).

Land use types Soil depth (cm) Total carbon (%) Soil organic matter (%)

MDF 0-5 2.51±0.11 4.45 ± 0.24

5-20 1.83±0.10 3.60 ± 0.24

PARA 0-5 1.14±0.08 2.06 ± 0.12

5-20 0.97±0.04 1.68 ± 0.08

According to soil carbon content of both land uses at Huai Lam Kradon sub watershed. Total soil carbon at

depth 0-5 cm higher in mixed deciduous forest (17,472.30 Kg C ha-1) than para rubber plantation (8,304.52 Kg

C ha-1) and total soil carbon at depth. Soil carbon content of both land use types represent in Fig. 4). Compare

to studies in other land use in the mixed deciduous forest, reforestation and agricultureland at Nam Yao sub

watershed soil carbon was 35,762 Kg C ha-1, 19,525 Kg C ha-1 and 10,310 Kg C ha-1 , respectively (Pibumrung

et al., 2008).

Fig. 4 Soil carbon of both land use types and two depths. Vertical lines represent standard deviation from 20 replicated measurements.

The correlation in MDF between soil carbon and soil factors such as: total nitrogen (%), pH, organic matter

(%), sand (%), silt (%), clay (%) and Ks (cm/s). The soil depth 0-5 cm factors were highest correlation

between soil carbon and total nitrogen (%) (R2 = 0.758), soil carbon and organic matter (%) (R2 = 0.724), soil

carbon and pH (R2 = 0.228), soil carbon and Ks (cm/s) (R2 = 0.133), soil carbon and clay (%) (R2 = 0.036)

and soil carbon and sand (%), respectively. The correlation in MDF between soil carbon and soil factors

represent in Fig. 5.

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Fig. 5 The correlation in MDF between % soil carbon and soil factors at depth 0-5 cm.

The correlation in PARA between soil carbon and soil factors such as: total nitrogen (%), pH, organic

matter (%), sand (%), silt (%), clay (%) and Ks (cm/s). The soil depth 0-5 cm factors were highest correlation

between soil carbon and soil organic matter (%) (R2 = 0.778), soil carbon and Ks (cm/s) (R2 = 0.199), soil

carbon and clay (%) (R2 = 0.036), soil carbon and pH ( R2 = 0.021), soil carbon and silt (%) (R2 = 0.016),

respectively. The correlation in MDF between soil carbon and soil factors represent in Fig. 6.

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Fig. 6 The correlation in MDF between % soil carbon and soil factors at depth 0-5 cm

3.4 CO2 Emission from soil surface

The monthly average soil CO2 fluxes from April 2010 to March 2011 in both land use types soil CO2

emissions higher in para rubber tree than mixed deciduous forest. The average CO2 fluxes of both land use

types were 2145.85 Kg CO2 ha-1 in para rubber plantation and 1319.08 Kg CO2 ha-1. The monthly average soil

CO2 fluxes of both land use types higher in wet season than dry season. The monthly average soil CO2 fluxes

in both land use types represent in Fig. 7.

Fig. 7 The monthly average soil CO2 fluxes from both land use types.

3.5 DNDC model simulation

If climate, soil properties and vegetation are kept constant in a relatively long term next 50 year, the litter

carbon will gradually approach to an equilibrium level, on which the litter carbon won’t either increase or

decrease any more. Based on information from field and from literature sources, some model inputs were

modified for use in this study (Table 5).

3.6 Litter carbon return by DNDC model approach under both land uses

The equilibrium level will depend on the climate, soil texture and management conditions but be independent

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on the initial litter carbon return. The results of the DNDC modeling for litter carbon return are presented in

Fig. 8. In mixed deciduous forest area, litter carbon declined severely from the equilibrium value of 2,670 C

ha-1 yr-1 in 2010 to a low of 1,664.73 C ha-1 yr-1 in 2061 and in para rubber plantation area, litter carbon

declined severely from the equilibrium value of 710 kg C ha-1 yr-1 in 2010 to a high of 1397.02 C ha-1 yr-1 in

2061. The litter carbon in 50 year of mixed deciduous forest area decrease 37.65 % or 1005.27 C ha-1 yr-1 and

in para rubber plantation area increase 3.27% or 687.02 C ha-1 yr-1. Litter carbon in mixed deciduous forest

higher than para rubber plantation in all time block periods.

Table 5 Modified DNDC modules in specified site to Huai Lam Kradon sub watershed, northern, Thailand.

Module Name Description

Climate Latitude Latitude at site study Huai Lam Kradon sub watershed, Thailand

Daily maximum-minimum

temperature and rainfall

Maximum-minimum temperature at site study Huai Lam Kradon

sub watershed, Thailand from automatic weather data and run by

Julian day

Nitrogen concentration in

rainfall

Data from Moller et al., 2005 for nitrogen in rainfall at northern,

Thailand

Soil Soil texture Clay loam for MDF and Sandy clay loam for PARA

Bulk density (g cm-3) 1.42 for MDF and 1.53 for PARA

Soil pH 5.21 for MDF and 4.64 for PARA

Hydro conductivity

(m hr-1)

0.008 for MDF and 0.0015 for PARA

Clay content of 1 g soil 0.36 for MDF and 0.31 for PARA

Initial soil carbon content at

0-5 cm (kg C / kg soil )

0.0026 for MDF and 0.0015 for PARA

Crop Land use Tropical forest for MDF and Tree plantation for PARA

Fig. 8 Litter carbon return by DNDC simulation in long term 50 year under both land use types.

3.7 Soil carbon by DNDC model approach under both land uses

If climate, soil texture and management condition are kept constant in a relatively long term next 50 year, the

soil carbon content in a soil will gradually approach to an equilibrium level, on which the soil carbon content

won’t either increase or decrease any more. The equilibrium level will depend on the climate, soil texture and

management conditions but be independent on the initial soil carbon content of the soil. The results of the

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DNDC modeling for soil carbon potential are presented in Fig. 9 in mixed deciduous forest and para rubber

plantation. In mixed deciduous forest area, soil carbon content declined severely from the equilibrium value of

17,960 kg C ha-1 yr-1 in 2010 to a high of 27,811 C ha-1 yr-1 in 2061. In para rubber plantation area, soil carbon

content declined severely from the equilibrium value of 8,300 kg C ha-1 yr-1 in 2010 to a high of 24,749 C ha-1

yr-1 in 2061. The soil carbon in 50 year of mixed deciduous forest area increase 35.42 % or 9,851 C ha-1 yr-1

and in para rubber plantation area increase 66.46 % or 16,449 C ha-1 yr-1. The soil carbon in mixed deciduous

forest higher than para rubber plantation in all time block periods and soil carbon of both land use types slowly

increased in next time block periods.

Fig. 9 Soil carbon storage by DNDC simulation in long term 50 year under both land use types.

3.8 CO2 Emission by DNDC model approach under both land uses

The results of the DNDC modeling for CO2 emission are presented in Fig. 10 in mixed deciduous forest and

para rubber plantation. In mixed deciduous forest area, soil carbon content declined severely from the

equilibrium value of 1,319.08 Kg CO2 ha-1 in 2010 to a high of 2,507.72 Kg CO2 ha-1 in 2061. In para rubber

plantation area, CO2 emission declined severely from the equilibrium value of 1319.08 Kg CO2 ha-1. in 2010 to

a high of 2,375.39 Kg CO2 ha-1 in 2061. The average CO2 emission in 50 year of mixed deciduous forest area

increase 47.39 % or 1,188.64 Kg CO2 ha-1and in para rubber plantation area increase 44.47 % or 1,056.31 Kg

CO2 ha-1. The soil CO2 emissions in mixed deciduous forest higher than para rubber plantation in all time

block periods. In fist time block periods CO2 emission rapidly increased and decreased. However, in secondary

to finally time blocks soil CO2 emissions slowly increased.

Fig. 10 CO2 Emission by DNDC simulation in long term 50 year under both land use types

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3.9 Projected under land use change scenarios

Project under land use change scenarios simulated litter carbon, soil carbon and CO2 emissions in time blocks

of 50 years, with 1 time block per 10 years period. Time block 1 covers the period 2011- 2021, time block 2

covers the period 2021-2031, time block 3 covers the period 2031-2041, time block 4 covers the period 2041-

2051 and time block 5 covers the period 2051-2061. Approach allows consideration of actions such as: each

major land use is no change in each block time; forest land is change to para rubber plantation and rubber

plantation change to forest. The assumption condition, which land uses change were 5 year for bare soil and

restore soil condition, 10 year for shrub condition and after 10 for tree condition.

3.10 Mixed deciduous forest change to para rubber plantation scenarios project

The SOC content form litter, soil and CO2 emission under mixed deciduous forest change to para rubber

plantation scenarios project are presented in Figs. 11-13. If climate, soil texture and management condition are

kept constant in a relatively long term next 50 year. The result of 4 scenarios with mixed deciduous forest

change to para rubber plantation scenarios. The litter carbon decrease in all scenarios and the mean decrease

highest of litter carbon in MDF 10 Year to Para rubber 40 Year scenario was 888.04 C ha-1 yr-1 or 52.91 % and

mean lowest of litter carbon in MDF 40 Year to Para rubber 10 Year scenario was 715.93 C ha-1 yr-1 or 37.96%

in Table 6. The result for soil carbon content in 4 scenarios for mixed deciduous forest change to para rubber

plantation scenarios. The soil carbon decrease in all scenarios and the mean decrease highest of litter carbon in

MDF 10 Year to Para rubber 40 Year scenario was 8,770.42 C ha-1 yr-1 or 49.79% and mean lowest of soil

carbon MDF 40 Year to Para rubber 10 Year scenario was 4,700.47 ha-1 yr-1 or 26.68 %. The result for soil

CO2 emission in 4 scenarios for mixed deciduous forest change to para rubber plantation scenarios. The soil

CO2 emission decrease in all scenarios and the mean highest CO2 emission in MDF 10 Year to Para rubber 40

Year scenario was 398.25 C ha-1 yr-1 or 25.42% and mean lowest emission in MDF 40 Year to Para rubber 10

Year was 37.06 C ha-1 yr-1 or 2.55%.

Fig. 11 Litter carbon from mixed deciduous forest change to para rubber plantation scenarios project.

Fig. 12 Soil carbon content from mixed deciduous forest change to para rubber plantation scenarios project.

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Fig. 13 CO2 emission from mixed deciduous forest change to para rubber plantation scenarios project.

Table 6 The results of DNDC model simulation under MDF change to PARA scenario project.

Scenarios Litter carbon

50 year

(C ha-1 yr-1)

Soil carbon

50 year

(C ha-1 yr-1)

CO2 emission

50 year

(C ha-1 yr-1)

Different

Litter carbon

Different

Soil

carbon

Different

CO2

Emission

MDF 50 Year 1,885.82 17,614.20 1,452.21 -

MDF 40 Year

to Para rubber

10 Year

1,169.89 12,913.73

1,415.15 -715.93

(37.96%)

-4,700.47

(26.68%)

37.06

(2.55%)

MDF 30 Year

to Para rubber

20 Year

1,067.30

10,519.22

1,205.55 - 818.52

(43.37%)

-7,094.98

(40.27%)

246.66

(16.98%)

MDF 20 Year

to Para rubber

30 Year

1,032.65 10,166.86

1,164.14 -853.17

(45.24)

-7,447.34

(42.28%)

288.07

(19.83%)

MDF 10 Year

to Para rubber

40 Year

997.78 8,843.78 1,053.96 -888.04

(52.91%)

-8,770.42

(49.79%)

398.25

(25.42%)

3.11 Para rubber plantation to mixed deciduous forest change scenarios project

The SOC content form litter, soil and CO2 emission under para rubber tree change to mixed deciduous forest

scenarios project are presented in Figs. 14-16. If climate, soil texture and management condition are kept

constant in a relatively long term next 50 year. The result for litter carbon return in 3 scenarios for para rubber

plantation change to mixed deciduous forest scenarios. The litter carbon decrease in all scenarios and the mean

decrease highest of litter carbon in para rubber plantation 30 year change to mixed deciduous forest 20 year

was 366.04 C ha-1 yr-1 or 30.25% and mean lowest of litter carbon in para rubber plantation 10 year change to

mixed deciduous forest 40 year was 185.33 C ha-1 yr-1 or 15.32%. The result for soil carbon content in 3

scenarios for mixed deciduous forest change to para rubber plantation scenarios. The mean soil carbon and

decrease highest of litter carbon in para rubber plantation 10 year change to mixed deciduous forest 40 year

was 6,931.22 C ha-1 yr-1 or 45.57% and mean lowest of soil carbon para rubber plantation 40 year change to

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mixed deciduous forest 10 year was 3,452.57 C ha-1 yr-1 or 22.70%. The result for soil CO2 emission in 3

scenarios for mixed deciduous forest change to para rubber plantation scenarios. The soil CO2 emission

decrease in all scenarios and the mean highest CO2 emission in para rubber plantation 20 year change to mixed

deciduous forest 30 year was 711.61 C ha-1 yr-1 or 43.15% and mean lowest emission in para rubber plantation

40 year change to mixed deciduous forest 10 year was 390.92% or 23.70%.

Fig. 14 Litter carbon from para rubber plantation change to mixed deciduous forest scenarios project.

Fig. 15 Soil carbon content from para rubber plantation change to mixed deciduous forest scenarios project.

Fig. 16 CO2 emissions from para rubber plantation change to mixed deciduous forest scenarios project.

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Table 7 The results of DNDC model simulation under PARA change to MDF scenario project.

Scenarios Litter

carbon

50 year

(C ha-1 yr-1)

Soil carbon

50 year

(C ha-1 yr-1)

CO2 emission

50 year

(C ha-1 yr-1)

Different

Litter carbon

Different

Soil

carbon

Different

CO2

Emission

Para50 Year 1,209.88 1,5210.90 1,649.15 - - -

Para 40 year +

mdf 10 year

979.21

11,758.33

1,258.23

-230.67

(19.07%)

-3452.57

(22.70%)

-390.92

(23.70%)

Para 30 year +

mdf 20 year

843.84

8882.76

969.87

-366.04

(30.25%)

-6328.14

(41.60%)

-679.28

(41.19%)

Para 20 year +

mdf 30 year

927.574

8,417.078

937.544

-282.31

(23.33%)

-6793.82

(44.66%)

-711.61

(43.15%)

Para 10

year+mdf 40

year

1,024.55

8,279.69

948.99

-185.33

(15.32%)

-6931.22

(45.57%)

-700.16

(42.46%)

4 Conclusion

The study on the effect of land use on soil carbon budget is importantly for the future management of

greenhouse gases and climate change, and soil carbon budget is one activity mention of the UNFCCC for

decreasing effect from climate change. The study used data from field observation at 2 different main land uses

for quantifying soil carbon to 3 measurements such as: field observations calculation and Denitrification-

Decomposition (DNDC) model. The field observation results of soil carbon of both land uses at Huai Lam

Kradon sub watershed. Total soil carbon higher in mixed deciduous forest (17,472.30 Kg C ha-1) than para

rubber plantation (8,304.52 Kg C ha-1) at depth 0-5 cm and at depth 5-20 cm 8,304.52 Kg C ha-1 and 6,776.65,

respectively. The DNDC model has shown that it can perform well in its representation of the effects of both

land uses in this study area. Simulation results showed significant loss of soil carbon from system under two

land use types and eight scenarios of land use change from mixed deciduous forest to para rubber plantation

and para rubber tree change to mixed deciduous forest. The results indicated that the simulated soil carbon of

mixed deciduous forest was strongly affected by climate and soil properties. The annual soil carbon was

17,960 and 8,300 C ha-1 yr-1 for mixed deciduous forest and para rubber plantation, respectively. The simulated

soil carbon under land uses change scenarios. The result for soil carbon content in 4 scenarios for mixed

deciduous forest change to para rubber plantation scenarios. The soil carbon decrease in all scenarios and the

mean decrease highest of litter carbon in MDF 10 Year to Para rubber 40 Year scenario was 8,770.42 C ha-1 yr-

1 or 49.79% and mean lowest of soil carbon MDF 40 Year to Para rubber 10 Year scenario was 4,700.47 ha-1

yr-1 or 26.68 %. The result for soil carbon content in 4 scenarios for para rubber plantatio change to para rubber

plantation scenarios. The mean soil carbon and decrease highest of litter carbon in para rubber plantation 10

year change to mixed deciduous forest 40 year was 6931.22 C ha-1 yr-1 or 45.57% and mean lowest of soil

carbon para rubber plantation 40 year change to mixed deciduous forest 10 year was 3452.57 C ha-1 yr-1 or

22.70%.

This result is essential for estimating soil carbon capacity to quantifying soil carbon dynamics in two land

use types such as mixed deciduous forest and para rubber plantation. Since soil carbon dynamics is determined

by a complex systems, in which the carbon input through litter as well the carbon output through

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decomposition are collectively and simultaneously controlled by a pattern of natural and management factors

(Li et al., 1994). In Thailand, especially in Northern Thailand, soil carbon loss from land used change from

forest to agriculture land. The model results indicated that soil carbon sequestration potential can be

substantially elevated if the people protected forest area before change and manage agriculture are after change

because from the model result soil carbon dynamics in both land use little different in long time 50 year.

Acknowledgement

The research was supported by a CHE-Ph.D-SW-NEWU scholarship from the Commission on Higher

Education. The Department of National Parks, Wildlife and Plant Conservation is thanked for giving

permission to access and collect data from the study area.

References

Bajracharya RM, Lal R, Kimble JM. 1998. Long-term tillage effect on soil organic carbon distribution in

aggregate and primary particle fractions of two Ohio soils. In: Management of Carbon Sequestration in

Soil (Lal R, Kimble JM, Follett RF, et al., eds). 113-123, CRC Press, Boca Raton, USA

Boonyanuphap J, Sakurai K, Tanaka S. 2007. Soil nutrient status under upland farming practice in the Lower

Northern Thailand. Tropics, 16(3): 215-231

Brown S, Lugo AE. 1984. Biomass of tropical forests: A new estimate based on forest volumes. Science, 223:

1290-1293

Lasco RD. 2002. Forest carbon budgets in Southeast Asia following harvesting and land cover change. Science

in China (Series C), 45: 55-64

Levine JS. 1995. Biomass burning, a driver for global change. Environmental Science and Technology Letters,

120: 120-125

Li C, Frolking SF, Harriss RC. 1994. Modeling carbon biogeochemistry in agriculture soils. Global

Biogeochemical Cycles, 8: 237-254

Moller A, Kaiser K, Guggenberger G. 2005. Dissolved organic carbon and nitrogen in precipitation,

through fall, soil solution, and stream water of the tropical highlands in northern Thailand. Journal of Plant

Nutrition and Soil Science, 168: 649-659

Ogawa H, Yoda K, Ogino K, et al. 1965. Comparative ecological studies on three main types of forest

vegetation in Thailand II. Plant Biomass. Nature and Life in Southeast Asia, 4: 49-80

Pibumrung P, Gajaseni N, Popan A. 2008. Profiles of carbon stocks in forest, reforestation and agricultural

land, Northern Thailand. Journal of Forest Research, 19: 11-18

Tan ZX, Lal R, Izaurralde RC, et al. 2004. Biochemically protected soil organic carbon at the Appalachian

experimental watershed. Soil Science, 169(6): 423-433

Thaiutsa B, Suwannapinunt W, Kaitpraneet W. 1978. Production and Chemical Composition of Forest Litter

in Thailand. Forest Research Bulletin. Kasetsart University, Bangkok, Thailand

Yoosuk S. 2005. Carbon Sink in Rubber Plantation of Klaeng District, Rayong Province. MSc Thesis, Mahidol

University, Thailand

Zhang WJ, Qi YH, Zhang ZG. 2006. A long-term forecast analysis on worldwide land uses. Environmental

Monitoring and Assessment, 119: 609-620

Zhang Y, Li CS, Zhou X, et al. 2002. A simulation model linking crop growth and soil biogeochemistry for

sustainable agriculture. Ecological Modelling, 151: 75-108

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Article

Analytical treatment of system of KdV equations by Homotopy

Perturbation Method (HPM) and Homotopy Analysis Method (HAM)

Hafiz Abdul Wahab1, Tahir Khan1, Muhammad Shakil1, Saira Bhatti2, Muhammad Naeem3 1Department of Mathematics, Hazara University, Mansehra, Pakistan 2Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 3Department of Information Technology, Hazara University, Mansehra, Pakistan


Received 16 September 2013; Accepted 20 October 2013; Published online 1 March 2014

Abstract

In this article the Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM) are applied

to obtain analytic approximate solution to three system of nonlinear wave equations, namely two component

evolutionary system of a homogeneous KdV Equations of order three (system-I) as well as (system-II) and the

generalized coupled Hirota Satsuma KdV (System-III).

Keywords Homotopy Perturbation Method (HPM); Homotopy Analysis Method (HAM); Systems of KdV

equations.

1 Introduction

In different field the nonlinear phenomena is very important and it played a tremendous role, especially in the

field of applied mathematics, engineering and physics etc. Now for mechanism of physical model described by

differential equations, different types of effective methods have been discovered for helping the engineers,

scientist and physicist to know about the problem and its application, because in most cases it is still difficult

to obtain the exact solution.

Like other nonlinear analytic technique Homotopy Perturbation Method (HPM) and Homotopy Analysis

Method (HAM) are two well known methods for obtaining the analytic approximate solutions to differential

equations. In He (1999), the Homotopy Perturbation Method (HPM) was first presented. The Homotopy

Perturbation Method (HPM) applied by many authors (Shakil et al, 2013; Siddiqui et al., 2014; Wahab et al.,

2013; Wahab et al., 2014), to find the solution of nonlinear problems in the field of science and engineering.

This method have the ability to solve linear and nonlinear problems (Alquran and Muhammad, 2011; Hemeda,

2012). Homotopy Perturbation Method (HPM) provides an opportunity that is no requirement of small

parameter like perturbation methods, in the equations. Homotopy Perturbation Method (HPM) also applicable



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to different types of equations like Volterra’s equation, integro equation, nonlinear oscillator equation,

bifurcation equation, nonlinear wave equation etc (Hemeda, 2012). In most cases Homotopy Perturbation

Method (HPM) provides a very rapid and fast convergence (Hemeda, 2012). Thus a method which have the

ability to solve different types of nonlinear equations is known as Homotopy Perturbation Method (HPM).

Liao has presented another analytic approximation in 1992 (Liao, 1992). This method is based on a

interesting property called homotopy, a fundamental concept of differential geometry and topology (Taiwo et

al., 2012). Homotopy Analysis Method (HAM) is an analytic approximated method through which we can find

the solution of nonlinear problem. Since perturbation techniques are often non valid in case of strong non

linearity, but Homotopy Analysis Method (HAM) is valid in non-linearity case (Liao, 2003). By using one

interesting property of homotopy, the non-linear problem can be changed into an infinite number of linear

problems, no matter comes from small or large parameter. If a non-linear problem has even a single solution,

then through this method namely Homotopy Analysis Method (HAM), there exist an infinite number of

disjoint solution expression whose the region of convergence and rate of convergence dependent on an axillary

parameter (Liao, 2003).

The purpose of this paper is to present analytic approximate solution to system of KdV equations by using

Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM).

2 Analysis of Homotopy Perturbation Method (HPM)

Let us consider a general nonlinear differential equation of the form,

,L x x g z z (1)

subject to the boundary condition,

, 0,x

x zn

(2)

In equation (1)-(2), " "L is the linear operator. " "N is the nonlinear operator. " " is defined to be the

boundary operator. Boundary of the domain " " is " " . The known function is define to be the function

" "g z . Now by Homotopy Analysis Method (HAM) constructing a homotopy, such that,

, : [0,1]r q R ,

which satisfies,

0, 1 ( ) ( ) 0H L L x L g z (3)

Equation (3), becomes

0 0, 0H L L x L x N g z (4)

where " "z and " 0,1 " is known to be the embedding parameter, 0" "x is define to be the, so

called initial approximation, must satisfies the boundary condition. Now setting " 0" and " 1" in

equation (4), then

0,0 0H L L x (5)

,1 0H L g z (6)

Equation (5) and equation (6), are called homotopic equations and the value of " " changing from

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"0" to unity is called deformation in topology (Hemeda, 2012). Now according to Homotopy Perturbation

Method (HPM), the basic assumption of the for the solution of equation (3)-(4) the basic assumption is that,

the " " can be expressed in a power series, such that,

0 1, ; , , ...x t v x t v x t (7)

and thus the analytical approximate solution for equation (1), can be derived through Homotopy Perturbation

Method (HPM) by setting " 1" in equation (7), which becomes,

0 1 21lim , ; ...q

x x t v v v

(8)

which will be the required approximated solution of the given nonlinear problem derived by Homotopy

Perturbation Method (HPM).

3 Analysis of Homotopy Analysis Method (HAM)

Homotopy Analysis Method (HAM) is a straight forward and very simple method. This method was presented

by means of homotopy (Hilton, 1953; Liao, 2003), which is a fundamental concept of topology.

Consider a differential equation, such that

, 0x t (9)

In equation (9), " " is non-linear operator and " , "x t is the unknown function. Now the

generalization of the traditional homotopy by Homotopy Perturbation Method (HPM) presented by Liao

(2003), construct a new type of homotopy called deformation equation of zero-order, such that,

01 , ; , , ( , ; )L x t x t x t x t (10)

In equation (10), 0" , "x t is known to be the initial approximation of the given unknown function that

is, " , "x t . " , ; "x t is a function, which is not known. " " is the embedding parameter, " "

and " , "H x t are the non-zero auxiliary parameter and non-zero auxiliary function respectively, " " is

the operator called non-linear and " "L is the auxiliary operator called linear operator. In this method it is

very important that we can easily and with great freedom chose the auxiliary materials (Hemeda, 2012).

Now if 0 and 1 , then equation (10), becomes,

0, ;0 ,x t x t and ( , ;1) ( , )x t x t (11)

Thus equation (11), shows that the variation of the embedding parameter varies from zero to unity make the

solution " , ; "x t from the initial approximation to the exact solution. The variation of these kind is

called deformation in the manner of topology (Liao, 2003).

According to Homotopy Analysis Method (HAM), expending " " in a power series with respect to

" " . Such that,

20 1 2

00

, ; ...

, ; mm

m

x t

x t

(12)

In equation (12), , ;" , "

!

m

m m

x tx t

m

at " 0" . Now if the auxiliary elements that is, the

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auxiliary linear-operator, auxiliary parameter, the initial-approximation and the auxiliary function are chosen

through proper way, then the above series that is series in equation (12), is convergent at " 1" . Thus the

solution of the original non-linear problem becomes,

01

mm

m

(13)

which is one-solution of the original nonlinear problem. Now according to the fundamental theorem of

Homotopy Analysis Method (HAM), consider a vector, such that,

0 1, ,..., n

(14)

Then the deformation equation of order " "m is given by

1 1. , .m m m m mL x t

(15)

In equation (15),

0, 1

1, 1m

m

m

and

1

1 1

, ;

1 !

m

m m m

x t

m

(16)

Now if " 1" and " , 1"H x t in equation (10), then it become homotopy constructed in

Homotopy Perturbation Method (HPM), which shows that Homotopy Perturbation Method (HPM) is a

specified case of Homotopy Analysis Method (HAM). The genialized homotopy only not depend on the

parameter " " , but it also dependent on" " and " , "x t called auxiliary parameter and auxiliary function

respectively. Thus the generalized homotopy give us a family of approximation series whose the region of

convergence depend upon on " " and " , "x t . Also the generalized homotopy provide us a straight

forward way to control and adjust the convergence region and rate of approximation series (Liao, 2003).

Homotopy Analysis Method (HAM) is more general and valid for non-linear and linear Differential equations

in many types.

This is a very brief introduction, for details, we refer to Liao (2003), Hemeda (2012), Zedan and El Adrous

(2012).

4 Applications

Taking the following systems of nonlinear KdV equations.

System-I

Consider the first system is a system of KdV equations of order three, such that

3

3,

t x x x

(17)

3

32 ,

t x x

(18)

subject to,

20 3 6 tanh ,

2

x

(19)

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1

220 3 2 tanh .

2

xi

(20)

The closed form solution given in Alquran and Muhammad (2011), is as,

23 6 tanh ,2closed

x t

23 2 tanh .2closed

x ti

Homotopy Perturbation Method (HPM) Solution

Through Homotopy Perturbation Method (HPM) the approximate solution of the system-I is derived as,

2 2 2 4

3 2

3 tanh 6 3 6 tanh tanh sec 24sec2 2 2 2 2

,

tanh 24 tanh sec ...2 2 2

HPM

x x x x xt h h

x tx x x

t h

2 4

2 3

9 2 sec tanh 24 2 sec tanh 12 22 2 2 2

,

sec tanh ...2 2

HPM

x x x xit h it h it

x tx x

h

Solution by Homotopy Analysis Method (HAM)

To solve system-I by Homotopy Analysis Method (HAM), and keeping in the view the given conditions, we

define a linear-operator as, " "Lt

and the inverse of linear operator is define as, 1

0

" () "t

L dt . Now

by definition of Homotopy Analysis Method (HAM), the deformation equation of order-zero for the given

system of non-linear partial differential equations becomes,

1 0 11 , ; . . , . , ; ,L x t x t x t (21)

2 0 21 , ; . . , . , ; ,L x t x t x t (22)

where 0 0" , " are the given initial approximation define as,

20 ,0 3 6 tanh ,

2

xx

(23)

1

220 ,0 3 2 tanh .

2

xx i

(24)

Now since " " is the embedding parameter, so the deformation process gives us,

1 0, ;0 ,x t (25)

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2 0, ;0 ,x t (26)

1 , ;1 ,x t (27)

2 , ;1 .x t (28)

Now suppose that, the solution of the original equation can be expressed in the power of the embedding

parameter, such that,

1 01

, ; ,nn

n

x t

(29)

2 01

, ; .nn

n

x t

(30)

In equation (29)-(30), 1 , ;1

!

n

n n

x t

n

and 2 , ;1

!

n

n n

x t

n

at " 0" and exist for

" 1"n also converges at " 1" . Then the solution of the given problem becomes,

01

,nn

(31)

01

.nn

(32)

Now the fundamental theorem of Homotopy Analysis Method (HAM) provide us, that the deformation

equation of nth-order for the given system of nonlinear PDEs becomes,

1 1, ,n n n n nL x t

(33)

1 1, .n n n n nL x t

(34)

Applying the inverse operator that is, 1

0

" . "t

L dt , on equation (33)-(34), we obtain,

1 1

0

, ,t

n n n n nx t dt

(35)

1 1

0

, .t

n n n n nx t dt

(36)

Starting with the initial approximation,

20 , 3 6 tanh ,2

xx t (37)

20 , 3 2 tanh .2

xx t i (38)

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Now substituting " 1"n and also for simplicity using " ( , ) 1"x t in equation (35)-(36), we get,

2 2 4

1

3 2

6 3 6 tanh tanh sec 24 sec2 2 2 2

, ,

tanh 24 tanh sec2 2 2

x x x xt h t h

x tx x x

t h

(39)

2 4

12 3

9 2 sec tanh 24 2 sec2 2 2

, .

tanh 12 2 sec tanh2 2 2

x x xi t h i t h

x tx x x

i t h

(40)

Similarly for obtaining 2 2 3 3, , , , and so on, using " 2,3, 4,..."n in equation (35)-(36). Now to

obtain the analytic approximate solution of the given system of KdV equation by Homotopy Analysis Method

(HAM), since

0 1 2, ( , ) ( , ) ( , ) ...x t x t x t x t (41)

0 1 2( , ) ( , ) ( , ) ( , ) ...x t x t x t x t (42)

Using the initial gauss, 1 1" , " and also the 2nd components that is 2 2" , " obtained through Maple

package given in appendix, in equation (41)-(42), we will get the solution by Homotopy Analysis Method

(HAM). After the derivation of the solution by Homotopy Analysis Method (HAM), if we use the non-zero

auxiliary parameter that is," 1" in the obtained solution by Homotopy Analysis Method (HAM), it will

give us the Homotopy Perturbation Method (HPM) solution, which shows that Homotopy Perturbation Method

(HPM) is a specified case of Homotopy Analysis Method (HAM).

System-II

In this system, we consider two component evolutionary system of KdV equation of order three, such that,

3

32 0,

t x x x

(43)

0,t x

(44)

initial condition,

0 1

2

tanh ,3

x

(45)

2

0 1

2

1 1tanh .

6 2 3

x

(46)

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The closed form solution given in Alquran and Muhammad (2011), that is

1

2

tanh ,3

closed

x t

2

1

2

1 1tanh .

6 2 3closed

x t

Homotopy Perturbation solution

The approximated analytic solution obtained by Homotopy Perturbation Method (HPM) is as,

2 2 21 1 1 1

2 2 2 2

2 2 41 1 1

2 2 2

tanh sec tanh sec3 3 33 3 3 3

2tanh sec sec ...

3 3 3 33 3 3

HPM

x t x t x xh h

t x x t xh h

(47)

2 2

1 1 1

2 2 2

1 1tanh tanh sec ...

6 2 33 3 3HPM

x t x xh

(48)

Solution by Homotopy Analysis Method

Now we want to find out the analytic approximated solution of the system-II, by Homotopy Analysis Method

(HAM), and then to compare it with the the result obtained by Homotopy Perturbation Method (HPM), and

also it tendency to the closed form solution, thus we have,

Since we need to define a linear operator , which is already define in Homotopy Perturbation Method

(HPM), that is " "Lt

with 1

0

" . "t

L dt . Now according to Homotopy Analysis Method (HAM), the

zero-order deformation equations becomes, such that,

1 0 11 , ; , , ; ,L x t x t x t (49)

2 0 21 , ; , , ; .L x t x t x t (50)

In equation (40)-(50), 0" " and 0" " denote the initial approximation and define as

0 1

2

,0 tanh ,3

xx

(51)

20 1

2

1 1,0 tanh .

6 2 3

xx

(52)

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Now using the deformation process, the zero order deformation equation becomes,

1 , ;1 ,x t (53)

2 , ;1 ,x t (54)

1 0, ;0 ,x t (55)

2 0, ;0 .x t (56)

In equation (53)-(56), " , " and 0 0" , " are the exact solution and initial approximation of the

system-ii respectively. Now assume that the solution can be expressed in a series of " " , such that,

1 01

, ; ,nn

n

x t

(57)

2 01

, ; .nn

n

x t

(58)

In equation (57)-(58),

1 2, ; , ;1 1" , "

! !

n n

n nn n

x t x t

n n

at " 0" and exist for " 1"n also converges at

" 1" . Then the solution of the original problem becomes,

01

,nn

(59)

01

.nn

(60)

Now to find the component of equation (59)-(60), that is 1 2, ,... and 1 2, ,... , according to the

fundamental theorem of Homotopy Analysis Method (HAM), the nth-order deformation equation for the given

system of non-linear PDE,s becomes,

1 1, , 1,n n n n nL x t n

(67)

1 1, , 1.n n n n nL x t n

(68)

Now we starting with the initial approximation

0 1

2

tanh ,3

x

(69)

2

0 1

2

1 1tanh .

6 2 3

x

(70)

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Applying the inverse operator that is 1

0

" . "t

L dt on equation (67)-(68), we obtain,

1 1

0

. , . ,t

n n n n nx t dt

(71)

1 1

0

. , . ,t

n n n n nx t dt

(72)

Using " 1"n and " ( , ) 1"x t in equation (71)-(72), we have,

4 2 21 1 1

2 2 2

1

2 2 21 1 1

2 2 2

2sec tanh sec

3 3 3 33 3 3,

sec tan sec3 3 33 3 3

t x t x xh h

t x t x xh h h

(73)

2

1 1 1

2 2

tan sec .3 3 3

t x xh h

(74)

Similarly putting 2,3, 4,...n , in equation (71)-(72), and by using Maple Package we get

2 2 3 3,, , , and so on. Now since

0 1 2, ( , ) ( , ) ( , ) ...x t x t x t x t (75)

0 1 2( , ) ( , ) ( , ) ( , ) ...x t x t x t x t (76)

Hence using the initial gauss 1 1" , " and 2 2" , " obtained through Maple package, included in

Appendix in equation (75)-(76), we will get the solution by Homotopy Analysis Method (HAM). Then by

using the auxiliary parameter that is " 1" in the obtained solution by Homotopy Analysis Method

(HAM), then it become the solution derived by Homotopy Perturbation Method (HPM).

System-III

In system-iii, we consider a system of three nonlinear wave equation, which is also called the generalized KdV

system of coupled Hirota Satsuma, such that,

3

3

13 3 0,

2t x x x

(77)

3

33 0,

t x x

(78)

3

33 0,

t x x

(79)

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subject to initial condition,

20

12 tanh ,

3x (80)

0 tanh ,x (81)

0

8tanh .

3x (82)

The closed form solution to the system-III, were seen in Alquran and Muhammad (2011), which is given as

212 tanh ,

3x t

tanh ,x t

8tanh .

3x t

and obtained the solution of the above system by Homotopy Perturbation Method (HPM), given as,

Solution by Homotopy Perturbation Method (HPM)

2 2

2 3 2

12 tanh 20 sec tanh

3 ,16 sec tanh 16 sec tanh ...

HPM

x t h x x

t h x x t h x x

(83)

4 4

4 2

tanh 2 sec 2 sec,

tanh sec ...HPM

x t h x t h x

x t h x

(84)

4 2

2

8 16 8tanh sec sec

3 3 3 .16

sec tanh ...3

HPM

t x t h x t h x

t h x x

(85)

Solution by Homotopy Analysis Method (HAM)

To find the solution of the generalized KdV system of coupled Hirota Satsuma by Homotopy Analysis Method

(HAM), first we need to define a linear operator, which is already define in Homotopy Perturbation Method

(HPM), that is, " "L t with the inverse define as, 1

0

" . "t

L dt . Now using the definition of

Homotopy Analysis Method (HAM), the zero-order deformation equation for the generalized KdV system of

coupled Hirota Satsuma becomes,

1 0 1(1 ) , ; , , ; ,L x t x t x t (86)

2 0 21 , ; , , ; ,L x t x t x t (87)

3 0 31 , ; , , ; .L x t x t x t (88)

In equation (86)-(88), 0 0" ," " and 0" " denote the initial approximation, define as,

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20

1,0 2 tanh ,3x x

0 ,0 tanh ,x x

0 ,0 tanh .x x

Using the deformation process, then equations (86)-(88), becomes

1 0, ;0 ,x t (89)

2 0, ;0 ,x t (90)

3 0, ;0 ,x t (91)

1 , ;1 ,x t (92)

2 , ;1 ,x t (93)

3 , ;1 .x t (94)

In equation (89)-(94), 0 0 0" , , " and " , , " are the initial approximation and exact solution of

the given system respectively. Now assume the solution is of the form, such that,

1 01

, ; ,nn

n

x t

(95)

2 01

, ; ,nn

n

x t

(96)

3 01

, ; .nn

n

x t

(97)

In equation (95)-(97),

1 2 3, ; , ; , ;1 1 1" , , , , , "

! ! !

n n n

n n nn n n

x t x t x tx t x t x t

n n n

at " 0" , and exist for " 1"n also converges at " 1" . Then the solution of the original problem

takes the form,

01

,nn

(98)

01

,nn

(99)

01

.nn

(100)

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Now according to the fundamental theorem of Homotopy Analysis Method (HAM), the higher order

deformation equations for the generalized KdV system of coupled Hirota Satsuma becomes,

1 1, , 1,n n n n nL x t n

1 1, , 1,n n n n nL x t n

1 1, , 1.n n n n nL x t n

Applying 1" "L we get the following equation, such that,

1 1

0

, ,t

n n n n nx t dt

(101)

1 1

0

, ,t

n n n n nx t dt

(102)

1 1

0

, .t

n n n n nx t dt

(103)

Here we starting with the initial gauss,

20

1 2 tanh ,3 x (104)

0 tanh ,x (105)

0

8tanh .

3x (106)

By using " 1"n and " ( , ) 1"x t in equation (101)-(103), we obtain,

4 2 3

1 2

16 sec tanh 16 sec tanh 20,

sec tanh

t h x x t h x x t

h x x

(107)

2 2 2 41 sec 2 sec tanh 2 sec ,t h x t h x x t h x (108)

2 2 2 41

8 16 16sec sec tanh sec ,

3 3 3t h x t h x x t h x

(109)

Now since solution of the given system is given by,

0 1 2, ( , ) ( , ) ( , ) ...x t x t x t x t

0 1 2( , ) ( , ) ( , ) ( , ) ...x t x t x t x t

0 1 2( , ) ( , ) ( , ) ( , ) ...x t x t x t x t

Using the calculated components, that is initial gauss, 1 1" , " and 2 2" , " , which is included in

Appendix, we will get the solution of the system-III by Homotopy Analysis Method (HAM). Which will be the

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required obtain solution of generalized KdV system of coupled Hirota Satsuma equation by Homotopy

Analysis Method (HAM).

5 Conclusion

The Homotopy Perturbation Method (HPM), and Homotopy Analysis Method (HAM) was successfully

applied to linear and nonlinear problem and do not require any small or large parameter like perturbation

methods and avoid the difficulties arising in the perturbation and non perturbation technique. Also the

calculation is very simple and straight forward in these methods, but still Homotopy Perturbation Method

(HPM) method is not a perfect tool for the solution of nonlinear problem. Homotopy Perturbation Method

(HPM) give us a divergent result even for a linear problem some time. Thus it is clear that this method is also

not a perfect tool.

But the method which have the ability to cover such types of deficiencies appears in the above mentioned

technique is known as Homotopy Analysis Method (HAM), this method has successfully applied to all types

of equations and give us an opportunity to apply for every type of problem. Also the effectiveness of this

method is that, the series obtained by this method is more accurate than numerical solution in many cases.

This method provides us a convergence control parameter known auxiliary parameter. Also Homotopy

Perturbation Method (HPM) is a special case of Homotopy Analysis Method (HAM). Thus from all the above

it is clear that Homotopy Analysis Method (HAM) is valid in all cases and have a great potential for nonlinear

problem.

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Appendix

System-I

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System-II

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System-III

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References

Alquran M, Muhammad M. 2011. Approximate solution to system of nonlinear partial differential equation

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178(3-4): 257-262

Hemeda AA. 2012. Homotopy Perturbation Method for solving system of nonlinear couped equation, Applied

Mathematical Sciences, 96: 4787-4800

Hilton PJ. 1953. An Introduction to Homotopy Theory. Cambridge University Pess, UK

Liao SJ. 2003. Beyond Perturbation, Introduction to Homotopy Analysis Method. Chapman and Hall/CRC,

USA

Shakil M, Khan T, Wahab HA, Bhatti S. 2013. IMPACT: 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

Taiwo OA, Oadewumi A, Raji RA. 2012. Application of new homotopy analysis method for first and second

order integro-differential equations. International Journal of Science and Technology, 2(5): 328-332

Wahab HA, Bhatti S, Naeem M, Qureshi MT, Afzal M. 2014. A mathematical model for the rods with heat

generation and convective cooling. Journal of Basic and Applied Scientific Research, 4(4)

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

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Global behavior of an anti-competitive system of fourth-order rational difference equations

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Analytical treatment of system of KdV equations by Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM)

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