College

ntal fical ithe ptionsprey-Mart

inistic models to,24,10,12])itions, the

http://dx.doi.org/10.1016/j.amc.2014.02.0910096-3003/ 2014 Elsevier Inc. All rights reserved.

q Research supported by NNSF of China Grant Nos. 11271087, 61263006, Scientic research project of Guangxi Education Department, China(No. 2013YB074), Scientic research project of State Ethnic Affairs Commission of China Nos. 12GXZ001 and 2013YJYB03. Corresponding author.

E-mail addresses: liuqun151608@163.com (Q. Liu), yiliangliu100@126.com (Y. Liu), xuepan100@126.com (X. Pan).

Applied Mathematics and Computation 235 (2014) 17

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amccomponent in an ecosystem. Thus many authors studied stochastic models which corresponding to determreveal the inuence of environmental noise on the population dynamics (see e.g. [1,3,9,18,19,1517,21,22over, recently, Liu and Han [14,13] have studied integro-differential equations with impulsive integral condextend some existing models to a great extent of this eld.. More-ir workThe original predatorprey system with BeddingtonDeAngelis type functional response is

dxdt

x b1 a11x a12y1 bx cy

;dydt

y b2 a21y a22x1 bx cy

; 1

where x xt and y yt stand for the preys and the predators densities at time t, respectively. For biological signicanceof each coefcient we refer the reader to [2,4].

On the other hand, population dynamics is always affected by environmental noise (see e.g. [5,6]), which is an importantIn a natural ecosystem, fundameelucidated by empirical and theoretof the predatorprey relationship isresponse can provide better descripstatistical evidence from 19 predatorBeddingtonDeAngelis and Crowleyformed even better.eatures of population interactions, such as predation and competition have beennvestigations of the dynamics between two species. One signicant componentredators functional response. Skalski and Gilliam [23] stated that the functionalof predator feeding over a range of predatorprey abundances by comparing thesystems with the three predator-dependent functional responses (HassellVarley,in), and in some cases, the BeddingtonDeAngelis type functional response per-a r t i c l e i n f o

Keywords:Predatorprey systemGlobal asymptotic stabilityBeddingtonDeAngelis functional responseTime delaysStochastic perturbations

a b s t r a c t

This paper is concerned with the global asymptotic stability of a stochastic delay predatorprey system with BeddingtonDeAngelis functional response. Sufcient criteria for theglobal asymptotic stability of the system are established. Some simulation gures areprovided to show that our model is more realistic than existing models.

2014 Elsevier Inc. All rights reserved.

1. Introductionof Mathematics and Information Science, Yulin Normal University, Yulin, Guangxi 537000, PR ChinaGlobal stability of a stochastic predatorprey system withinnite delaysq

Qun Liu a,b, Yiliang Liu a,, Xue Pan aaCollege of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, PR Chinab

dx x 1 a x b dt r dB t;> k0, dene the stopping time

Dene

Vx; y x 1 0:5 ln x y 1 0:5 ln y:

2 Q. Liu et al. / Applied Mathematics and Computation 235 (2014) 17P sk 6 Tf gP e; kP k1: 4

p psk inf t 2 1; se : xt R 1=k; k or yt R 1=k; kf g;

Obviously, sk is increasing as k!1. Let s1 limk!1sk, then we have s1 6 se a.s. Next, we only need to verify s1 1. Ifthis claim is not true, then there are two constants T > 0 and e 2 0;1 such that Pfs1 6 Tg > e. So there is an integer k1 P k0such thatMx; y 1 bxt cyt ; Nx; y 1 bxt cyt :

viously, if x; y 2 R2, then 0 6 Mx; y 6 a14c ; 0 6 Nx; y 6 b14b .k0 > 0 be large enough such that fih i 1;2; h 2 1;0 lying within the interval 1=k0; k0. For every integerLemma 2.2. Consider system (3), there is a unique global solution xt; yt on R for any given initial valuef1; f2 2 BC1;0; R2 and the solution will remain in R2 with probability 1, where R2 fx 2 R2jxi > 0; i 1;2g.

Proof. The following proof is motivated by the work of Liu and Wang [11]. Let

a14yt b14xtThis paper is organized as follows: in Section 2, we will show that if the noise is small enough and the positive equilib-rium state of the deterministic model is globally stable, the stochastic system will keep the property. In Section 3, we willintroduce some numerical simulations to support our main result. Finally we give some conclusions and discussions.

2. Global stability

As the biological signicance of xt and yt in model (3), we should rstly give some conditions under which system (3)has a unique global positive solution.

Lemma 2.1. Consider model (3), if ri > 0; i 1;2, then there is a unique positive local solution xt; yt on 1; se almostsurely (a.s.) for any given initial value f1; f2 2 BC1;0; R2, where se denotes the explosion time and BC1;0; R2represents the family of bounded and continuous functions from 1;0 to R2 with the norm kfik suph60jfihj; i 1;2.X;F ;P with a ltration fF tgt2R satisfying the usual conditions, si P 0 and lih is a probability measure on1; 0; i 1;2; x; y is the equilibrium state of system (3). For more biological motivation on this type of modeling wedyt yt r2 b11yt b12yt s2 b13 1 yt hdl2h 141bxtcyt dt r2ytyt y dB2t;:2dxt xt r1 a11xt a12xt s1 a13R 01 xt hdl1h a14yt1bxtcyt

h idt r1xtxt xdB1t;

R 0 b xth i8> 3elays means ignoring reality. Hence it is essential to take time delays into account. Motivated by these, in this paper,ll propose and study a more realistic and complex stochastic delay predatorprey system with BeddingtonDeAngeliswith the initial value x0 x0 > 0; y0 y0 > 0. They showed that if the positive equilibrium point of the deterministicsystem was globally stable, then the stochastic model would preserve the nice property provided the noise was small en-ough. However, like it or not, time delays frequently occur in almost every situation, Kuang [8] has revealed that ignoringdy y 1 211bxcy a22y b2dt r2dB2t;:

11 1bxcy 1 1 1

a xh i> 2a12yh i8In 2011, Liu and Wang [11] studied the following nonlinear stochastic predatorprey system

1

0:125r2 y 2y 2y y 2y dt0:5r1 x1xx dB1t0:5r2 y1yy dB2t

ThNe

Theor

Then tx0; y0

almos

Proofnd ation o

x is t

Dene

Q. Liu et al. / Applied Mathematics and Computation 235 (2014) 17 3Obviously, these two functions are nonnegative on R. We prepare the detailed computation in Appendix A in order toderive our main result directly. If xt; yt 2 R2, making use of Its formula yieldsA < 0; B > 0: 6he equilibrium state x; y of model (3) is globally stochastically asymptotically stable, i.e., for any given initial value, the solution of system (3) has the property thatlimt!1

xt x; limt!1

t y 7

t surely (a.s.)

. The following proof is motivated by the work of Liu and Wang [11]. In order to show our main result, we only need toLyapunov function Vx satisfying LVx 6 0 (see e.g. [20]), where L is the Lyapunov operator and x xt is the solu-f the n-dimensional stochastic functional differential equation

dxt f t; xtdt gt; xtdBt: 8he positive equilibrium state of 8 and

LVx Vxxf t; xt 0:5 trace gTt; xtVxxxgt; xt

:

V1x x x x ln xx ; V2y y y y ln y

y:IfB C2

b11 1 bx cy b12 b13 0:5r2y :A a11 a14by

1 bx cy a12 a13 0:5r21x

;

C1 b14cx 2 EV xsk ^ T; ysk ^ T 6 V x0; y0 K1 K2 T: 5e following proof is similar to Liu and Wang [11] and hence we omit it here.xt, we will give our main result of this paper.

em 2.1. Let

C1 a14bx 1; C2 b14cy 1;where K1 and K2 are positive constants. By the above inequality, we can obtain that6 K1K2dt0:5r1 x1xxdB1t0:5r2 y1yydB2t;b 2 11 12 2 13 2 2

0:5r1x

p 1xxdB1t0:5r2y

p 1yydB2tp pc

0:5 b14y

pr b yb kykkf kb kykkf k

dt0:125r2 y522y22yy322y2

dt60:5x

pr1a11xa14a12kxkkf1ka13kxkkf1k dt0:125r21 x

522x22xx322x2

dt b12yts2b13Z 01

ythdl2hdt0:125r21 x

522x22xx322x2

dt

2 52 2 32 2 p p 60:5 xr1a11xa12xts1a131

xthdl1hFx;y dt0:5y

pGx;yr2b11y8

yy dt0:5r2 y1yy dB2tp Z 0 r222y

p 2 p

0:5r1

x

p 1xxdB1t0:5y

p 1 r2b11yb12yts2b13 ythdl2hGx;y dtdVx;y0:5 x1 r1a11xa12xts1a131

xthdl1hFx;y dtr12 x

8xx2dt

Z 0 Clearly, the function is nonnegative on R2. If xt; yt 2 R2, then we havep Z 0 2 p

2 14

Dene

Then

5 12 12 2

Then

We ob

Set jZ

ObR2 exc

Inf1h b12

4 Q. Liu et al. / Applied Mathematics and Computation 235 (2014) 17this section, we will apply the Milstein method mentioned in Higham [7] to illustrate our main result. Set0:32eh; f2h 0:18eh; l1h l2h eh on h 2 1;0 and r1 0:5; r2 0:4; a11 0:5; b11 0:2; a12 0:3;

0:2; a13 0:1; b13 0:1; a14 0:7; b14 0:5; b c 1:0.3. Numerical simulationsLVx; y; t 6 jZ Z j0 B

jZ Z j:

viously, if (6) holds then by virtue of the above inequality, we can conclude that LVx; y; t < 0 along all trajectories inept x; y. Then the conclusion follows. T A 0

1 bx cy

C1C2

b11 b14cx

1 bx cy b12 b13 0:5r22y

y y2

Ax x2 By y2:

Zj jx xj; jy yjT , then by the above inequality, we haveLVx; y; t 6 a11 a14by

a12 a13 0:5r21x

x x2LV6t 0:5b13 yt y 2 0:5b131

yt h y 2dl2h:

dene

Vx; y; t V1x C1C2 V2y V3t V4t C1C2

V5t C1C2 V6t

V1x a14bx 1

b14cy 1V2y V3t V4t C1C2

V5t C1C2 V6t:

tain thatZ 0LV4t 0:5a13 xt x 2 0:5a13Z 01

xt h x 2dl1h;

LV t 0:5b yt y 2 0:5b yt s y2;LV3t 0:5a12 xt x 2 0:5a12 xt s1 x 2;V3t 0:5a12ts1

xs x2ds; V4t 0:5a131 th

xs x 2dsdl1h;

V5t 0:5b12Z tts2

ys y2ds; V6t 0:5b13Z 01

Z tth

ys y 2dsdl2h:Z t Z 0 Z t 0:5b131

yt h y dl2h 1 bx cy1 bx cy x xy y:LV2y 6 b11 b14cx1 bx cy 0:5b12 0:5b13 0:5r22y

y y2 0:5b12 yt s2 y 2Z 0 b cy 1LV1x 6 a11 a14by1 bx cy 0:5a12 0:5a13 0:5r21x

x x2 0:5a12 xt s1 x 2

0:5a13Z 01

xt h x2dl1h a14bx 1

1 bx cy1 bx cy x xy y: 9

By the same way, applying Its formula leads to

Consider the following discretization equations:

xk1 xk r1 a11xk a12xks1=Dt 0:16a13ekDt a13ekDtRki1xieiDtDt a14yk

1 bxk cyk

Dt

r1xkxk xDt

pnk r

21

2xkxk xnk2 1Dt;

yk1 yk r2 b11yk b12yks2=Dt 0:09b13ekDt b13ekDtRki1yieiDtDt b14xk

1 bxk cyk

Dt

r2ykyk yDt

pgk r

22

2ykyk ygk2 1Dt;

where nk and gk k 1; . . . ;n are the Gaussian random variables which follow N0;1.In Figs. 1 and 2, we choose r1 0:5; r2 0:4; a11 0:5; a12 0:3; a13 0:1; a 14 0:7; b11 0:2; b12 0:2; b13

0:1; b14 0:5; r1 r2 0:008; b c 1:0 and step size Dt 0:01, then (6) holds and x 2:0392; y 0:3514. The dif-ference in Fig. 1 is that the values of s1 and s2 are different. For populations x1 and y1, we choose s1 1; s2 10, while forpopulations x2 and y2, we choose s1 s2 0. That is to say, the equilibrium state x; y is asymptotically stable. Obviously,from Fig. 1 we can see that our model is more realistic than model (2), so our model owns the better property than model (2).The difference in Fig. 2 is that the values of r1 and r2 are different. For populations x1 and y1, we choose r1 0:18; r2 0:06, while for populations x2 and y2, we choose r1 r2 0, then (6) is satised. In other words, the equilibrium statex; y is globally stochastically asymptotically stable.

Fig. 2.x; y

0 5 10 15 20 25 3010

5

0

5

Time

the population x1the population y1the population x2the population y2

Fig. 1. For populations x1 and y1, we choose s1 1; s2 10, while for populations x2 and y2, we choose s1 s2 0. Then the equilibrium state x ; y is

Q. Liu et al. / Applied Mathematics and Computation 235 (2014) 17 50 5 10 15 20 25 303

2

1

0

Time

the population x1the population y1the population x2the population y2

For populations x1 and y1, we choose r1 0:18; r2 0:06, while for populations x2 and y2, we choose r1 r2 0. Then the equilibrium stateis globally stochastically asymptotically stable.1

2

3

asymptotically stable.

Applying Its formula, we can get

As 1

1

Then

By the

6 Q. Liu et al. / Applied Mathematics and Computation 235 (2014) 17References

[1] C.A. Braumann, Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Math. Biosci. 177(2002) 229245.

[2] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efciency, J. Anim. Ecol. 44 (1975) 331341.[3] J.R. Beddington, R.M. May, Harvesting natural populations in a randomly uctuating environment, Science 197 (1977) 463465.[4] D.L. DeAngelis, R.A. Goldsten, R. Neill, A model for trophic interaction, Ecology 56 (1975) 881892.[5] T.C. Gard, Persistence in stochastic food web models, Bull. Math. Biol. 46 (1984) 357370.[6] T.C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal. 10 (1986) 14111419.[7] D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43 (2001) 525546.[8] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.[9] X. Li, X. Mao, Population dynamical behavior of non-autonomous LotkaVolterra competitive system with random perturbation, Discrete Contin. Dyn.

Syst. 24 (2009) 523545.[10] M. Liu, K. Wang, Global asymptotic stability of a stochastic LotkaVolterra model with innite delays, Commun. Nonlinear Sci. Numer. Simul. 17 (2012)

31153123.[11] M. Liu, K. Wang, Global stability of a nonlinear stochastic predatorprey system with BeddingtonDeAngelis functional response, Commun. Nonlinear

Sci. Numer. Simul. 16 (2011) 11141121.1 bx cy

0:5a13Z 01

xt h xdl1h 2

a14bx 1x xy y

1 bx cy1 bx cy :

the desired assertion (9) follows from the Hlders inequality:

Z 01

xt h x dl1h 2

6Z 01

xt h x 2dl1h:

same way, we compute that

LV2y 6 b11 0:5b12 0:5b13 b14cx

1 bx cy 0:5r22y

y y2 0:5b12 yt s2 y 2

0:5b13Z 01

yt h y 2dl2h b14cy 1x xy y1 bx cy1 bx cy :LV1x 6 a11 a14by

0:5a12 0:5a13 0:5r2x

x x2 0:5a12 xt s1 x 26a11xx 0:5a12xx 0:5a12xts1x 0:5a13xx 0:5a131

xthx dl1h

a14by

1bxcy1bx cyxx2 a14bx

11bxcy1bx cyxx

yy0:5r21xxx2:

bx cyP 1, one can obtain that 2 2 2 2Z 0

21

a14by

1bxcy1bx cyxx2 a14bx

11bxcy1bx cyxx

yy0:5r21xxx2a11xx2a12xxxts1xa13Z 0

xxxthxdl1hLV1x xx r1a11xa12xts1a13Z 01

xthdl1ha14y

1bxcy

0:5r21xxx2

xx a11x xa12x xts1a13Z 01

x xthdl1hFx;yFx;y

0:5r21xxx24. Conclusions and discussions

In this paper, we consider the global asymptotic stability of a stochastic delay predatorprey system with BeddingtonDeAngelis functional response. Sufcient criteria for the global asymptotic stability of the system are established. The resultis interesting and important because from the biological point of view, a globally stable positive equilibrium means that thecommunity is stable in which all species could coexist. The simulation gures show that our model has the better propertythan some existing models.

Some interesting topics are deserved further investigations. One may propose some more realistic systems, such as con-sidering the effects of impulsive perturbations on the systems. It is also interesting to investigate the models with jumps andso on. We will leave these investigations for future work.

Appendix A. Calculations of LV1x; LV2y and LVx; y; t

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[13] Z.H. Liu, J.F. Han, L.J. Fang, Integral boundary value problems for rst order integro-differential equations with impulsive integral conditions, Comput.Math. Appl. 61 (2011) 30353043.

[14] Z.H. Liu, J.F. Han, Boundary value problems for second order impulsive functional differential equations, Dyn. Syst. Appl. 20 (2011) 369382.[15] Z.H. Liu, BrowderTikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Prob. 21 (2005) 1320.[16] Z.H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differ. Equ. 244 (2008) 13951409.[17] Z.H. Liu, Anti-periodic solutions to nonlinear evolution equations, J. Funct. Anal. 258 (6) (2010) 20262033.[18] Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 6984.[19] X. Mao, Stochastic stabilisation and destabilisation, Syst. Control Lett. 23 (1994) 279290.[20] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997.[21] X. Mao, C. Yuan, J. Zou, Stochastic differential delay equations of population dynamics, J. Math. Anal. Appl. 304 (2005) 296320.[22] R. Rudnicki, K. Pichor, Inuence of stochastic perturbation on preypredator systems, Math. Biosci. 206 (2007) 108119.[23] G.T. Skalski, J.F. Gilliam, Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology 82 (2001) 3083

3092.[24] C. Zhu, G. Yin, On hybrid competitive LotkaVolterra ecosystems, Nonlinear Anal. 71 (2009) 13701379.

Q. Liu et al. / Applied Mathematics and Computation 235 (2014) 17 7

Global stability of a stochastic predatorprey system with infinite delays1 Introduction2 Global stability3 Numerical simulations4 Conclusions and discussionsAppendix A Calculations of ? and ? References