Chaotic dynamics of a three species prey–predator competition model with noise in ecology

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Chaotic dynamics of a three species prey–predator competitionmodel with noise in ecology

Kalyan Das a,⇑, K. Shiva Reddy b, M.N. Srinivas c, N.H. Gazi d

a Department of Mathematics, National Institute of Food Technology Entrepreneurship and Management, Plot No. 97, Sector-56, HSIIDC Industrial Estate,Kundli 131028, Haryana, Indiab Department of Mathematics, Anurag Group of Institutions, Hyderabad, Indiac School of Advanced Sciences, Department of Mathematics, VIT University, Vellore 632014, Tamil Nadu, Indiad Department of Mathematics, Aliah University, DN-41, Salt Lake, Sector V, Kolkata 700091, India

a r t i c l e i n f o

Keywords:Prey–predatorTop predatorLocal and global stabilityRouth–Hurwitz criteriaLyapunov functionStochastic delayed perturbationFourier transform methodsChaos

a b s t r a c t

In this paper we investigate a three species ecosystem consisting of a prey, a predator and atop predator. The predator survives on the prey and the top predator survives on both theprey and the predator. The mathematical model consists of non-linear simultaneousdifferential equations. All the eight equilibrium points of the model are identified and byusing Routh–Hurwitz criteria as well as Lyapunov function, we derive the criteria for localand global stabilities. Also the population intensities of fluctuations (variances) around thepositive equilibrium due to noise are computed; their stability is also analyzed withcomputer simulation using Matlab. Some Numerical simulations for justifying the theoret-ical analysis are also provided. The main conclusions are supplied in conclusion.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

In the study of population dynamics, many mathematical models have been proposed since the work of Volterra [1] andLotka [2] assumed significance. In order to reflect the dynamical behavior of the models depending on the existing data andinformation of the system, it is often necessary to incorporate stochastic term with additive noise. For long time, it has beenrecognized that stochastic terms have very complicated impact on the dynamics of a system, which can not cause the insta-bility but also induce some noise, in terms of oscillations and periodic solutions. Many kinds of prey–predator interactionswith mutualism and competitive mechanisms within the same species have been studied extensively by researchers after itwas initiated by Lotka [2] and Volterra [1]. Several mathematicians and ecologists contributed to the growth of this area ofknowledge, and has been extensively reported in the treatises of Meyer [3], Cushing [4], Conlinvaux [5], Freedman [6],Simmons [7] and Kapur [8]. It is natural that two or more species living in a common habitat are often attached to oneanother by interacting in different ways without being isolated. The classification of all kinds of the inter-specificrelationships is not just by their presence but according to the effect produced by the interaction on their size.

Several mathematicians and theoretical ecologists have contributed to the growth of this area of knowledge as reported inthe treatises of Lokta [2], Mayer [3]. The ecological interaction between the species can be broadly classified as prey–preda-tion, competition, commensalism and Ammensalism and so on. Modeling of prey–predation, competition and mutualism arestudied at greater length by several investigators than the studies on commensalism and Ammensalism. Srinivas [9] studiedcompetitive eco-systems of two and three species with limited and unlimited resources. Later, Lakshmi Narayan [10] has

0096-3003/$ - see front matter � 2014 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.12.182

⇑ Corresponding author.E-mail addresses: [email protected] (K. Das), [email protected] (K. Shiva Reddy), [email protected] (M.N. Srinivas),

[email protected] (N.H. Gazi).

Applied Mathematics and Computation 231 (2014) 117–133

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc


investigated the two species prey–predator models. In 1973, Holling [11] emphasized the influence of noise in ecologicaldynamics and resilience. The noise may arise from stochastic disturbances on the external environment. Under the influenceof noise, the species dynamics are always stochastic or seemingly stochastic. In 1996, Ripa et al. [12] examined the noisecolor and the risk of population extinction in a prey–predator system widely. In 2003 Xu et al. [13] investigated white noiseor noise with a white variance spectrum which contains no temporal autocorrelation and is essentially a series of indepen-dent random numbers about the population dynamics and color environmental noise of a three species food chain system.Later, in 2009, Sun et al. [14] emphasized the role of noise in a predator–prey model with Allee effect. In 2011, Wang et al.[15] found the effect of colored noise on spatiotemporal dynamics of biological invasion in a diffusive predator–prey system.Some of the studies motivated us to compute the population intensities [13,16] of fluctuations (variances) around thepositive equilibrium due to noise [12,15]. Some numerical results have been done. The problem ends with a brief descriptionof the principal results obtained here. The present investigation is an analytical study of three species system consisting of aprey, a predator and a top predator. The equilibrium points are identified based on the model equations and these aredefined for: (i) fully washed-out state (ii) partially washed-out state and (iii) co-existent state (Interior equilibrium). Thelocal and global stability of the states have been derived. Based on the work [17–19], we consider the followingpredator–prey model.

2. Mathematical model

The ecological setup of the problem is as follows: the model equations for a three species multi-system are given by thefollowing non-linear differential equations as

dN1

dt¼ a1N1 � a11N2

1 � a12N1N2 � a13N1N3 ð2:1Þ

dN2

dt¼ a2N2 þ a21N1N2 � a22N2

2 � a23N2N3 ð2:2Þ

dN3

dt¼ a3N3 þ a31N1N3 þ a32N2N3 � a33N2

3 ð2:3Þ

Where N1ðtÞ;N2ðtÞ;N3ðtÞ represents population densities of prey, predator and top predator species respectively, ai

represents natural growth rates of three species; i ¼ 1;2;3, aii represents the rate of decrease of species due to its insufficientresources; i ¼ 1;2;3, a12 represents the rate of the prey due to inhibition by the predator, a13 represents the rate of decreaseof the prey due to inhibition by the top predator. a21 represents the rate of increase of the predator due to its successfulattacks on the prey, a23 represents the rate of decrease of the predator due to inhibition by the predator, a31 representsthe increase of the top predator due to its successful attacks on the prey, a32 represents the rate of increase of the toppredator due to its successful attacks on the predator. Throughout this analysis we are assuming that the variables N1, N2

and N3 are non-negative and the model parameters ai;ai j, i = 1,2,3, j = 1,2,3, are also assumed to be non-negative constants.

3. Existence of equilibria

The system dNidt ¼ 0, i = 1,2,3, under investigation has eight equilibrium states given by E1ð0;0; 0Þ, E2ð0;0;K3Þ, E3ð0;K2;0Þ,

E4ðK1;0;0Þ, E5K1�b12K21þb12b21

; K2þb21K11þb12b21

;0� �

, E6 0; K2�b23K31þb23b32

; K3þb32K21þb23b32

� �, E7

K1�b13K31þb13b31

;0; K3þb31K11þb13b31

� �, E8ðN1;N2;N3Þ.

where

N1 ¼ðb23b32 þ 1ÞK1 þ ðb12b23 � b13ÞK3 þ ðb13b32 � b12ÞK2

1þ b23b32 þ b12b21 þ b13b31 þ b13b21b32 � b12b23b31ð3:1Þ

N2 ¼b21K1 þ K2ð1þ b13b31Þ � ðb23b31K1 þ b23K3 þ b13b21K3Þ


N3 ¼ðb21b32 þ b31ÞK1 þ ð1þ b12b21ÞK3 þ ðb32 � b12b31ÞK2


K1 = Carrying capacity of prey species = a1a11

K2 = Carrying capacity of predator species = a2a22

K3 = Carrying capacity of top predator species = a3a33

b12 ¼a12

a11; b13 ¼

a13

a11; b21 ¼

a21

a22; b23 ¼

a23

a22; b31 ¼

a31

a33; b32 ¼

a32

a33

The equilibrium states E5; E6; E7 would exist only when K1 > b12K2, K2 > b23K3, K1 > b13K3 respectively and the interiorequilibrium E8 would exists only when

118 K. Das et al. / Applied Mathematics and Computation 231 (2014) 117–133


1þ b23b32 þ b12b21 þ b13b31 þ b13b21b32 > b12b23b31;

ðb23b32 þ 1ÞK1 þ b12b23K3 þ b13b32K2 > b12K2 þ b13K3;

b21K1 þ K2ð1þ b13b31Þ > ðb23b31K1 þ b23K3 þ b13b21K3Þ;and ðb21b32 þ b31ÞK1 þ ð1þ b12b21ÞK3 þ b32K2 > b12b31K2

ð3:4Þ

4. Local stability analysis of the equilibrium points

Let N ¼ ðN1;N2;N3Þ ¼ N þ U ¼ ðN1 þ u1;N2 þ u2;N3 þ u3Þ where U ¼ ðu1; u2; u3ÞT is a small perturbation over theequilibrium state N. The basic Eqs. (2.1)–(2.3) are linearised to obtain the equations for the perturbed state as

dUdt¼ AU ð4:1Þ

with

A ¼a1 � 2a11N1 � a12N2 � a13N3 �a12N1 �a13N1

a21N2 a2 þ a21N1 � 2a22N2 � a23N3 �a23N2

a31N3 a32N3 a3 þ a31N1 þ a32N2 � 2a33N3

264

375 ð4:2Þ

The equilibrium state is stable only when all the eigenvalues of A negative or have negative real parts according as theyare real or complex. The stability criteria of the system at equilibrium points E6; E7; E8, is being discussed.

4.1. Stability of prey washed out equilibrium point (E6)

One of the eigenvalue of variational matrix A is a1 � a12N2 � a13N3 which is positive or negative according to whethera1 > ða12N2 þ a13N3Þ or a1 < ða12N2 þ a13N3Þ. The other two eigenvalues are given by roots of the following quadraticequation

k2 þ ða22N2 þ a33N3Þkþ ða22a33 þ a23a32ÞN2N3 ¼ 0: ð4:1:1Þ

In (4.1.1), the sum of the roots ¼ �ða22N2 þ a33N3Þ which is always negative and the product of the roots¼ ða22a33 þ a23a32ÞN2N3, which is always positive. Therefore the roots of (4.1.1) are real and negative or complex conjugateshaving negative real parts. Thus the state is asymptotically stable only if a1 < ða12N2 þ a13N3Þ. The solutions of theperturbation equations are

u1 ¼ u10ek1t ð4:1:2Þ

u2 ¼ u20 A3ek1t þ B3ek2t þ C3ek3t� �

ð4:1:3Þ


ð4:1:4Þ

where k1ð¼ a1 � a12N2 � a13N3Þ, k2 and k3 are roots of Eq. (4.1.1)

A3 ¼k2

1 þ P3k1 þ Q 3

ðk1 � k2Þðk1 � k3Þ; B3 ¼

k22 þ P3k2 þ Q 3

ðk2 � k1Þðk2 � k3Þ; C3 ¼

k23 þ P3k3 þ Q3

ðk3 � k1Þðk3 � k2Þ

A4 ¼k2

1 þ P4k1 þ Q 4

ðk1 � k2Þðk1 � k3Þ; B4 ¼

k22 þ P4k2 þ Q 4

ðk2 � k1Þðk2 � k3Þ; C4 ¼

k23 þ P4k3 þ Q4

ðk3 � k1Þðk3 � k2Þ

P3 ¼ a33N3 � k1 þ ðu10a21N2 � u30a23N2Þ=u20

Q 3 ¼ �a33N3k1 þ ðu10a33a21N2N3 þ u30a23N2k1 � u10a23a31N2N3Þ=u20:

P4 ¼ a22N2 � k1 þ ðu10a31N3 þ u20a32N3Þ=u30

Q 4 ¼ �a22N2k1 þ ðu10a31a22N2N3 þ u10a21a32N2N3 � u20a32N3k1Þ=u30

4.2. Stability of predator washed out point (E7)

One of the eigenvalue of variational matrix A is a2 þ a21N1 � a23N3 which is positive or negative according to whethera2 þ a21N1 > a23N3 or a2 þ a21N1 < a23N3. The other two eigenvalues are given by roots of the following quadratic equation

K. Das et al. / Applied Mathematics and Computation 231 (2014) 117–133 119


k2 þ ða11N1 þ a33N3Þkþ ða11a33 þ a13a31ÞN1N3 ¼ 0: ð4:2:1Þ

In (4.2.1), the sum of the roots ¼ �ða11N1 þ a33N3Þ which is always negative and the product of the roots¼ ða11a33 þ a13a31ÞN1N3, which is always positive. Therefore, the roots of (4.2.1) are real and negative or complex conjugateshaving negative real parts. Thus the state is asymptotically stable only if a2 þ a21N1 < a23N3. The solution of the perturbationequations are:


ð4:2:2Þ

u2 ¼ u20ek1t ð4:2:3Þ


ð4:2:4Þ

Where k1ð¼ a2 þ a21N1 � a23N3Þ, k2 and k3 are roots of Eq. (4.2.1)

A5 ¼k2

1 þ P5k1 þ Q 5

ðk1 � k2Þðk1 � k3Þ; B5 ¼

k22 þ P5k2 þ Q 5

ðk2 � k1Þðk2 � k3Þ;

C5 ¼k2

3 þ P5k3 þ Q 5


A6 ¼k2

1 þ P6k1 þ Q 6


B6 ¼k2

2 þ P6k2 þ Q6


C6 ¼k2

3 þ P6k3 þ Q 6


P5 ¼ a33N3 � k1 � ðu20a12N1 þ u30a13N1Þ=u10;

P6 ¼ a11N1 � k1 þ ðu20a32N3 þ u10a13N3Þ=u30;

Q5 ¼ �a33N3k1 þ u20a33a12N1N3 þ u30a13N1k1 � u20a13a32N1N3� �

=u10

Q6 ¼ �a11N1k1 � ðu10a31N3k1 þ u20a31a12N1N3 � u20a32a11N1N3Þ=u30:

4.3. Stability of co-existing point (E8)

In this case the characteristic equation is

k3 þ b1k2 þ b2kþ b3 ¼ 0 ð4:3:1Þ

where

b1 ¼ a11N1 þ a22N2 þ a33N3

b2 ¼ ða22a11 þ a21a21ÞN1N2 þ ða11a33 þ a13a31ÞN1N3 þ ða22a33 þ a23a32ÞN2N3

b3 ¼ ða11a22a33 þ a11a23a32 þ a12a21a33 þ a13a31a22 þ a13a21a32 � a12a23a31ÞN1N2N3

According to Routh–Hurwitz’s criteria, the necessary and sufficient conditions for local stability of co-existent points are

b1 > 0; b3 > 0; and b3ðb1b2 � b3Þ > 0 ð4:3:2Þ

It is evident that b1 > 0 and a11a22a33 þ a11a23a32 þ a12a21a33 þ a13a21a32 þ a13a31a22 > a12a23a31 thus the stability of co-existent state is determined by the sign of b1b2 � b3. By direct calculations we obtain

b1b2 � b3 ¼ ða222a33 þ a23a32a22ÞN2

2N3 þ ða22a233 þ a33a23a32ÞN2

3N2 þ ða211a22 þ a11a12a21ÞN2

1N2

þ ða11a222 þ a22a12a21ÞN1N2

2 þ ða211a33 þ a11a13a31ÞN2

1N3 þ ða11a233 þ a33a13a31ÞN1N2

3

þ ð2a11a22a33 � a13a21a32 þ a12a23a31ÞN1N2N3 > 0 ð4:3:3Þ



Hence the co-existent state is locally asymptotically stable. i.e., all eigenvalues k1; k2; k3 say are negative or negative realpart of complex numbers.

The solutions of the perturbation co-existent system are


ð4:3:4Þ


ð4:3:5Þ


ð4:3:6Þ

where

A7 ¼ �k2

1 þ T1k1 þ U1

ðk1 � k2Þðk1 � k3Þ; B7 ¼

k22 þ T1k2 þ U1

ðk2 � k1Þðk2 � k3Þ; C7 ¼

k23 þ T1k3 þ U1


A8 ¼k2

1 þ T2k1 þ U2

ðk1 � k2Þðk1 � k3Þ; B8 ¼

k22 þ T2k2 þ U2

ðk2 � k1Þðk2 � k3Þ; C8 ¼

k23 þ T2k3 þ U2


A9 ¼k2

1 þ T3k1 þ U3

ðk1 � k2Þðk1 � k3Þ; B9 ¼

k22 þ T3k2 þ U3

ðk2 � k1Þðk2 � k3Þ; C9 ¼

k23 þ T3k3 þ U3


T1 ¼ ða22N2 þ a33N3Þ � ðu20a12N1 þ u30a13N1Þ=u10;

T2 ¼ a33N3 þ a11N1 þ ðu10a21N2 � u30a23N2Þ=u20;

T3 ¼ a11N1 þ a22N2 þ ðu20a32N3 þ u10a13N3Þ=u30;

U1 ¼ ða22a33 þ a23a32ÞN2N3 � ðu20ða12a33 þ a13a32ÞN1N3 þ u30ða12a23 � a13a22ÞN1N2Þ=u10;

U2 ¼ ða11a33 þ a31a13ÞN1N3 � ðu30ða11a23 þ a13a21ÞN1N2 þ u10ða21a33 � a31a23ÞN2N3Þ=u20;

U3 ¼ ða11a22 þ a12a21ÞN1N2 þ ðu20ða11a32 � a12a31ÞN1N3 þ u10ða12a32 þ a31a22ÞN2N3Þ=u30;

5. Global stability analysis of equilibrium points

Theorem 5.1. The Equilibrium point E6ð0;N2;N3Þ is globally asymptotically stable.

Proof. Let us consider the following Lyapunov function

VðN2;N3Þ ¼ N2 � N2 � lnN2

N2

� þ d1 N3 � N3 � ln

N3

N3

� �ð5:1:1Þ

where ‘d1’ is positive constant ,to be chosen laterDifferentiating V w.r. to ‘t’ we get

dVdt¼ N2 � N2

N2

!dN2

dtþ d1

N3 � N3

N3

!dN3

dtð5:1:2Þ

Choosing d1 ¼ a23a32

,

After a little algebraic manipulation yields

dVdt¼ �a22ðN2 � N2Þ

2 � a23

b32ðN3 � N3Þ

2< 0 ð5:1:3Þ

.Therefore, E6ð0;N2;N3Þ is globally asymptotically stable. h

Theorem 5.2. The Equilibrium point E7ðN1;0;N3Þ is globally asymptotically stable.




VðN1;N3Þ ¼ N1 � N1 � lnN1

N1

� þ d1 N3 � N3 � ln

N3

N3

� �ð5:2:1Þ

where ‘d1’ is positive constant ,to be chosen laterDifferentiating V w.r. to ‘t’ we get

dVdt¼ N1 � N1

N1

!dN1

dtþ d1

N3 � N3

N3

!dN3

dtð5:2:2Þ

Choosing d1 ¼ a13a31

, a little algebraic manipulation yields


2 � a13

b31ðN3 � N3Þ

2< 0 ð5:2:3Þ

Therefore, E7ðN1;0;N3Þ is globally asymptotically stable. h

Theorem 5.3. The Equilibrium point E8ðN1;N2;N3Þ is globally asymptotically stable.


VðN1;N2;N3Þ ¼ N1 � N1 � lnN1

N1

� þ d1 N2 � N2 � ln

N2

N2

� �þ d2 N3 � N3 � ln

N3

N3

� �ð5:3:1Þ

where ‘d1, d2’ are positive constants, to be chosen later onDifferentiating V w.r. to ‘t’ we get

dVdt¼ N1 � N1

N1

!dN1

dtþ d1

N2 � N2

N2

!dN2

dtþ d2

N3 � N3

N3

!dN3

dtð5:3:2Þ


2 � a12ðN1 � N1ÞðN2 � N2Þ � a13ðN1 � N1ÞðN3 � N3Þ þ d1fa21ðN1 � N1ÞðN2 � N2Þ � a22ðN2 � N2Þ2

� a23ðN2 � N2ÞðN3 � N3Þg þ d2fa31ðN1 � N1ÞðN3 � N3Þ þ a32ðN2 � N2ÞðN3 � N3Þ � a33ðN3 � N3Þ2g

dVdt

<� a11ðN1 � N1Þ2 � ða12 � a21d1Þ

ðN1 � N1Þ2

2þ ðN2 � N2Þ

2

2

" #� ða13 � a31d2Þ

ðN1 � N1Þ2

2þ ðN3 � N3Þ

2

2

" #

� d1a22ðN2 � N2Þ2 � ðd1a23 � a32d2Þ

ðN2 � N2Þ2

2þ ðN3 � N3Þ

2

2

" #� d2a33ðN3 � N3Þ

2< 0 ð5:3:3Þ

provided d1 6a12a21

, d2 6a13a31

and d1d2

P a32a23

Therefore, E8ðN1;N2;N3Þ is globally asymptotically stable if the lumping parametric conditions d1 6a12a21

,d2 6a13a31

andd1d2

P a32a23

hold. h

6. Stochastic mathematical model

The main assumption that leads us to extend the deterministic model (2.1)–(2.3) to a stochastic counterpart is that it isreasonable to conceive the open sea as a noisy environment. There are many number of ways in which environmental noisemay be incorporated in system (2.1)–(2.3). Note that environmental noise should be distinguished from demographic orinternal noise, for which the variation over time is due. External noise may arise either form random fluctuations of oneor more model parameters around some known mean values or from stochastic fluctuations of the population densitiesaround some constant values.

In this section, we compute the population intensities of fluctuations (variances) around the positive equilibrium E8 dueto noise, according to the method introduced by Nisbet and Gurney [20] and Carletti [21]. Now we assume the presence ofrandomly fluctuating driving forces on the deterministic growth of the prey, predator and top predator populations at time‘t’, so that the system (2.1)–(2.3) results in the stochastic system with ‘additive noise’

dN1

dt¼ a1N1 � a11N2

1 � a12N1N2 � a13N1N3 þ a1n1ðtÞ ð6:1Þ



dN2

dt¼ a2N2 þ a21N1N2 � a22N2

2 � a23N2N3 þ a2n2ðtÞ ð6:2Þ

dN3

dt¼ a3N3 þ a31N1N3 þ a32N2N3 � a33N2

3 þ a3n3ðtÞ ð6:3Þ

where N1ðtÞ represents prey species, N2ðtÞ represents predator species, N3ðtÞ represents top predator species, a1;a2;a3 arereal constants and nðtÞ ¼ ½n1ðtÞ; n2ðtÞ; n3ðtÞ� is a 3D Gaussian White noise process satisfying

E niðtÞ½ � ¼ 0; i ¼ 1;2;3 ð6:4Þ

E niðtÞnjðt0Þ� �

¼ dijdðt � t0Þ; i ¼ j ¼ 1;2;3 ð6:5Þ

where dij is the Kronecker symbol; d is the d-dirac function.

Let N1ðtÞ ¼ u1ðtÞ þ S�; N2ðtÞ ¼ u2ðtÞ þ P�; N3ðtÞ ¼ u3ðtÞ þ T�; ð6:6Þ

dN1

dt¼ du1ðtÞ

dt;

dN2

dt¼ du2ðtÞ

dt;

dN3

dt¼ du3ðtÞ

dt; ð6:7Þ

Using (6.6) and (6.7), Eq. (6.1) becomes

Fig. 1(a). Variations in the growth rate of the populations for a1 = 2, a11 = 0.1, a12 = 0.2, a13 = 0.01, a2 = 3, a21 = 0.3, a22 = 0.1, a23 = 0.2, a3 = 4, a31 = 0.01,a32 = 0.1, a33 = 0.2, N1 = 15, N2 = 20 and N3 = 30.

Fig. 1(b). Phase-space trajectories corresponding to the stabilities of the population for a1 = 2, a11 = 0.1, a12 = 0.2, a13 = 0.01, a2 = 3, a21 = 0.3, a22 = 0.1, a3 = 4,a31 = 0.01, a23 = 0.2, a32 = 0.1, a33 = 0.2, N1 = 15, N2 = 20 and N3 = 30.



du1ðtÞdt

¼ a1u1ðtÞ þ a1S� � a11u21ðtÞ � a11ðS�Þ2 � 2a11u1ðtÞS� � a12u1ðtÞu2ðtÞ

a12u1ðtÞP� � a12u2ðtÞS� � a12S�P� � a13u1ðtÞu3ðtÞ � a13u1ðtÞT� � a13u3ðtÞS�

a13S�T� þ a1n1ðtÞ ð6:8Þ

The linear part of (6.8) is

du1ðtÞdt

¼ �a11u1ðtÞS� � a12u2ðtÞS� � a13u3ðtÞS� þ a1n1ðtÞ ð6:9Þ

Again using (6.6) and (6.7), Eq. (6.2) becomes

du2ðtÞdt

¼ a2u2ðtÞ þ a2P� � a22u22ðtÞ � a22ðP�Þ2 � 2a22u2ðtÞP� þ a21u1ðtÞu2ðtÞ

a21u1ðtÞP� þ a21u2ðtÞS� þ a21S�P� � a23u2ðtÞu3ðtÞ � a23u2ðtÞT� � a23u3ðtÞP�

a23P�T� þ a2n2ðtÞ ð6:10Þ

Fig. 2(a). Variations in the growth rate of the populations for a1 = 6, a11 = 0.01, a12 = 0.2, a13 = 0.01, a2 = 3, a21 = 0.43, a22 = 0.1, a23 = 0.32, a3 = 2.5, a31 = 0.01,a32 = 0.6, a33 = 0.05, N1 = 15, N2 = 30 and N3 = 25.

Fig. 2(b). Phase-space trajectories corresponding to the stabilities of the population for a1 = 6, a11 = 0.01, a12 = 0.2, a13 = 0.01, a2 = 3, a21 = 0.43, a22 = 0.1,a23 = 0.32, a3 = 2.5, a31 = 0.01, a32 = 0.6, a33 = 0.05, N1 = 15, N2 = 30 and N3 = 25.




du2ðtÞdt

¼ �a22u2ðtÞP� þ a21u1ðtÞP� � a23u3ðtÞP� þ a2n2ðtÞ ð6:11Þ

Using Eqs. (6.6) and (6.7), Eq. (6.3) becomes

du3ðtÞdt

¼ a3u3ðtÞ þ a3T� � a33u23ðtÞ � a33ðT�Þ2 � 2a33u3ðtÞT� þ a31u1ðtÞu3ðtÞ

a31u1ðtÞT� þ a31u3ðtÞS� þ a31S�T� þ a32u2ðtÞu3ðtÞ þ a32u2ðtÞT� þ a32u3ðtÞP�

a32P�T� þ a3n3ðtÞ ð6:12Þ


du3ðtÞdt

¼ �a33u3ðtÞT� þ a31u1ðtÞT� þ a32u2ðtÞT� þ a3n3ðtÞ ð6:13Þ

Taking the Fourier transform on both sides of 6.9, 6.11, 6.13 we get,

a1~n1ðxÞ ¼ ixþ a11S�ð Þ~u1ðxÞ þ a12S�~u2ðxÞ þ a13S�~u3ðxÞ ð6:14Þ

Fig. 3(a). Variations in the growth rate of the populations for a1 = 6, a11 = 0.01, a12 = 0.2, a13 = 0.01, a2 = 2.5, a21 = 0.43, a22 = 0.1, a23 = 0.32, a3 = 3, a31 = 0.01,a32 = 0.12, a33 = 0.23, N1 = 15, N2 = 20 and N3 = 30.

Fig. 3(b). Phase-space trajectories corresponding to the stabilities of the population for a1 = 6, a11 = 0.01, a12 = 0.2, a13 = 0.01, a2 = 2.5, a21 = 0.43, a22 = 0.1,a23 = 0.32, a3 = 3, a31 = 0.01, a32 = 0.12, a33 = 0.23, N1 = 15, N2 = 20 and N3 = 30.



a2~n2ðxÞ ¼ a21P�~u1ðxÞ þ ðixþ a22P�Þ~u2ðxÞ þ a23P�~u3ðxÞ ð6:15Þ

a3~n3ðxÞ ¼ �a31T�~u1ðxÞ � a32T�~u2ðxÞ þ ðixþ a33T�Þ~u3ðxÞ ð6:16Þ

The matrix form of (6.14)–(6.16) is

MðxÞ~uðxÞ ¼ ~nðxÞ ð6:17Þ

where

MðxÞ ¼a1ðxÞ a2ðxÞ a3ðxÞb1ðxÞ b2ðxÞ b3ðxÞc1ðxÞ c2ðxÞ c3ðxÞ

0B@

1CA; ~uðxÞ ¼

~u1ðxÞ~u2ðxÞ~u3ðxÞ

264

375; ~nðxÞ ¼

~n1ðxÞ~n2ðxÞ~n3ðxÞ

264

375;

a1ðxÞ ¼ ixþ a11S�; a2ðxÞ ¼ a12S�; a3ðxÞ ¼ a13S�; b1ðxÞ ¼ a21P�;

b2ðxÞ ¼ ixþ a22P�; b3 ¼ a23P�; c1ðxÞ ¼ �a31T�; c2ðxÞ ¼ �a32T�;

c3ðxÞ ¼ ixþ a33T� ð6:18Þ





Eq. (6.17) can also be written as ~uðxÞ ¼ ½MðxÞ��1~nðxÞLet ½MðxÞ��1 ¼ KðxÞ, therefore,

~uðxÞ ¼ KðxÞ~nðxÞ ð6:19Þ

where

KðxÞ ¼ AdjMðxÞjMðxÞj ð6:20Þ

if the function YðtÞ has a zero mean value , then the fluctuation intensity (variance) of it is components in the frequencyinterval ½x;xþ dx� is SY ðxÞdx. Where SYðxÞ is spectral density of Y and is defined as

SYðxÞ ¼ lim~T!1

j~YðxÞj2~T

ð6:21Þ

If Y has a zero mean value, the inverse transform of SYðxÞ is the auto covariance function

CYðsÞ ¼1

2p

Z 1

�1SYðxÞeixsdx ð6:22Þ

The corresponding variance of fluctuations in YðtÞ is given by





r2Y ¼ CYð0Þ ¼

12p

Z 1

�1SY ðxÞdx ð6:23Þ

and the auto correlation function is the normalized auto covariance

PYðsÞ ¼CY ðsÞCYð0Þ

ð6:24Þ

For a Gaussian white noise process, it is

SninjðxÞ ¼ lim

T̂!þ1

E½~niðxÞ~njðxÞ�T̂

¼ limT̂!þ1

1

T̂

Z T̂2

�T̂2

Z T̂2

�T̂2

E½~niðtÞ~njðt0Þ�e�ixðt�t0Þdt dt0 ¼ dij ð6:25Þ

From (6.19), we have

~uiðxÞ ¼X3

j¼1

KijðxÞ~njðxÞ; i ¼ 1;2;3 ð6:26Þ

From (6.21) we have

SuiðxÞ ¼

X3

j¼1

gj KijðxÞ�� 2; i ¼ 1;2;3 ð6:27Þ





0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Time t

Popu

latio

ns

PreyPredatorTop Predator

Fig. 7(a). The stable population growth under random environment with very low intensity.

05

1015

20

0

10

20

3015

20

25

30

Prey

Predator

Top

Pred

ator

Fig. 7(b). The stable phase-portrait of population growth under random environment with very low intensity.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Time t

Popu

latio

ns


Fig. 8(a). The oscillatory population growth (almost periodic) under random environment when the intensity is increased by 100 times.



05

1015

20

0

10

20

3015

20

25

30

Prey

Predator

Top

Pred

ator

Fig. 8(b). The oscillatory (almost periodic) phase-portrait of population growth under random environment when the intensity is increased by 100 times.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Time t

Popu

latio

ns


Fig. 9(a). The chaotic population growth under random environment with high intensity.

05

1015

2025

0

10

20

3010

15

20

25

30

Prey

Predator

Top

Pred

ator

Fig. 9(b). The chaotic phase-portrait of population growth under random environment with high intensity.



Hence by (6.23) and (6.27), the intensities of fluctuations in the variable ui; i ¼ 1;2;3 are given by

r2ui¼ 1

2pX3

j¼1

Z 1

�1aj KijðxÞ�� 2dx; i ¼ 1;2;3 ð6:28Þ

and by (6.20), we obtain

r2u1¼ 1

2p

Z 1

�1a1

A1

jMðxÞj

��

2

dxþZ 1

�1a2

B1

jMðxÞj

��

2

dxþZ 1

�1a3

C1

jMðxÞj

��2

dx

( )

r2u2¼ 1

2p

Z 1

�1a1

A2

jMðxÞj

��

2

dxþZ 1

�1a2

B2

jMðxÞj

��

2

dxþZ 1

�1a3

C2

jMðxÞj

��2

dx

( )

r2u3¼ 1

2p

Z 1

�1a1

A3

jMðxÞj

��

2

dxþZ 1

�1a2

B3

jMðxÞj

��

2

dxþZ 1

�1a3

C3

jMðxÞj

��2

dx

( )ð6:29Þ

where jMðxÞj ¼ jRðxÞj þ ijIðxÞj and Real part of jMðxÞj is

R2ðxÞ ¼ �a33x2T� � a22x2P� � a11x2S� þ a11a22a33S�T�P� þ a11a32a23S�T�P� � a12a21a33S�T�P�

� a12a31a23S�T�P� þ a13a21a32S�T�P� þ a13a31a22S�T�P� ð6:30Þ

Imaginary part of

jMðxÞj ¼ I2ðxÞ¼ �x3 þxa22a33P�T� þxa11a33S�T� þxa11a22S�P� þxa32a23P�T� �xa12a21S�P� þxa13a31S�T� ð6:31Þ

and

jA1j2 ¼ X21 þ Y2

1Þ; jB1j2 ¼ X22 þ Y2

2; jC1j2 ¼ X23 þ Y2

3; jA2j2 ¼ X24 þ Y2

4;

jB2j2 ¼ X25 þ Y2

5; jC2j2 ¼ X26 þ Y2

6; jA3j2 ¼ X27 þ Y2

7; jB3j2 ¼ X28 þ Y2

8; jC3j2 ¼ X29 þ Y2

9;

where

X1 ¼ �x2 þ a32a23T�P�; Y1 ¼ xa33T� þxa22P�;

X2 ¼ �a12a33S�T� � a13a32S�T�;

Y2 ¼ �xa12S�; X3 ¼ a12a23S�P� � a13a22S�P�;

Y3 ¼ �xa13S�;

X4 ¼ �a21a33P�T� � a23a31P�T�; Y4 ¼ �xa21P�;

X5 ¼ �x2 þ a11a33S�T� þ a31a13S�T�; Y5 ¼ xa33T� þxa11S�;

X6 ¼ a21a13S�P� � a11a23S�P�; Y6 ¼ �xa23P�;

X7 ¼ a31a22P� � a21a32P�T�; Y7 ¼ xa31T�;

X8 ¼ a11a32T�S� � a12a31S�T�;

Y8 ¼ xa32T�;

X9 ¼ �x2 þ a11a22S�P� � a12a21S�P�; Y9 ¼ xa22P� þxa11S�;

Thus (6.29) becomes

r2u1¼ 1

2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

a1 X21 þ Y2

1

� �þ a2 X2

2 þ Y22

� �þ a3 X2

3 þ Y23

� �h idx

( )

r2u1¼ 1

2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

a1 X24 þ Y2

4

� �þ a2 X2

5 þ Y25

� �þ a3 X2

6 þ Y26

� �h idx

( )



r2u1¼ 1

2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

g1 X27 þ Y2

7

� �þ g2 X2

8 þ Y28

� �þ g3 X2

9 þ Y29

� �h idx

( )

If we are interested in the dynamics of system (6.1)–(6.3) with either a1 ¼ 0 or a2 ¼ 0 or a3 ¼ 0, then the populationvariances are

If a1 = 0, a2 = 0, then

r2u1¼

a3 X23 þ Y2

3

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

r2u2¼

a3 X26 þ Y2

6

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

r2u3¼

a3 X29 þ Y2

9

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

If a2 = 0, a3 = 0, then,

r2u1¼

a1 X21 þ Y2

1

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

r2u2¼

a1 X24 þ Y2

4

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

r2u3¼

a1 X27 þ Y2

7

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

If a3 = 0, a1 = 0, then,

r2u1¼

a2 X22 þ Y2

2

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

r2u2¼

a2 X25 þ Y2

5

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

r2u3¼

a2 X28 þ Y2

8

� �2p

Z 1

�1

1R2ðxÞ þ I2ðxÞ

dx

Finally we conclude that Fourier transform method which has been used to study the effects of stochasticity on thepositive equilibrium of our model leading to chaos in realistic ecological situation.

7. Computer simulation and numerical examples

In this segment we authenticate as well as enhance our analytical findings which are compared through numericalsimulations in view of change of the following sensitive parameters in real word situation.

8. Discussion and concluding remarks

In this paper, we studied a model of one prey and two predator system with effect of stochastic perturbation. Initially wehave discussed about the model without stochasticity and investigated the existence of equilibrium points, local stability byemploying Routh–Hurwitz criteria, global analysis by constructing Lyapunov function. Later, we investigated the effects ofadditive noise on the positive equilibrium of our model by evaluating fluctuations of the population densities around theirvalues at the equilibrium point. Also Numerical simulations justified the analytical results. The analytical results and numer-ical simulation of deterministic model suggest that the deterministic prey–predator model is stable in nature. The stability ofthe system shows in Figs. 1-6. Besides this, for stochastic system, population variances has a great role for the stochastic sta-bility of the system. The conclusion is that the noise on the equation results in a big variances of fluctuations around the



equilibrium point which suggest that our system is chaotic with respect to a noisy environment. So in our model standarddeviations which acts as a fluctuating driving force induces large fluctuations for intensities around the equilibrium point. Inthis model, noise just contributes even more fluctuations of intensities to the system causes chaos. Numerical simulationsexhibit that the trajectories of the system oscillate randomly with remarkable variance of amplitudes with the increasingvalue of the strength of noises initially but ultimately fluctuating which are observed in Figs. 7-9. Hence we conclude thatdue to inclusion of stochastic perturbation which creates a significant change of intensity in our considering dynamicalsystem for a small change of sensitive parameters which causes large environmental fluctuations leading to chaos in realisticecological arena.

Acknowledgments

We are very grateful to the anonymous referee and the editor for their careful reading, constructive criticisms, helpfulcomments and suggestions, which have helped us to improve the presentation of this work significantly.

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Chaotic dynamics of a three species prey–predator competition model with noise in ecology

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