Abstract and Applied Analysis

Volume 2013 (2013), Article ID 256249, 13 pages

http://dx.doi.org/10.1155/2013/256249

Research Article

Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

Peiyan Xia,1,2 Xiaokun Zheng,² and Daqing Jiang²

¹School of Basic Sciences, Changchun University of Technology, Changchun, Jilin 130021, China

²School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

Received 22 October 2012; Accepted 29 November 2012

Academic Editor: Jifeng Chu

Copyright © 2013 Peiyan Xia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, a two-species nonautonomous stochastic mutualism system is investigated. The intrinsic growth rates of the two species at time are estimated by respectively. Viewing the different intensities of the noises , as two parameters at time , we conclude that there exists a global positive solution and the th moment of the solution is bounded. We also show that the system is permanent, including stochastic permanence, persistence in mean, and asymptotic boundedness in time average. Besides, we show that the large white noise will make the system nonpersistent. Finally, we establish sufficient criteria for the global attractivity of the system.

1. Introduction

For more than three decades, mutualism of multispecies has attracted the attention of both mathematicians and ecologists. By definition, in a mutualism of multispecies, the interaction is beneficial for the growth of other species. Lotka-Volterra mutualism systems have long been used as standard models to mathematically address questions related to this interaction. Among these, nonautonomous Lotka-Volterra mutualism models are studied by many authors, see [1–7] and references therein. The classical nonautonomous Lotka-Volterra mutualism system can be expressed as follows:

where is the density of the th population at time , is the intrinsic growth rate of the th population at time is the carrying capacity at time , and coefficient describes the influence of the th population upon the th population at time .

It is shown in [1] that if different conditions hold (see conditions (a)–(e) in [1]), then the solution of system (1) is bounded, permanent, extinct, and global attractive, respectively. However, when the intrinsic growth rate and coefficient are periodic, it is shown in [3] that there exists positive periodic solution and almost periodic solutions are obtained.

From another point of view, environmental noise always exists in real life. It is an interesting problem, both mathematically and biologically, to determine how the structure of the model changes under the effect of a fluctuating environment. Many authors studied the biological models with stochastic perturbation, see [8–12] and references therein. In [8] Ji et al. discussed the following two-species stochastic mutualism system

where are mutually independent one dimensional standard Brownian motions with , and are the intensities of white noise. It is shown in [8] that if then there is a unique nonnegative solution of system (2). For small white noise there is a stationary distribution of (2) and it has ergodic property. Biologically, this implies that with small perturbation of environment, the stability of the two species varies with the intensity of white noise, and both species will survive.

However, almost all known stochastic models assume that the growth rate and the carrying capacity of the population are independent of time . In contrast, the natural growth rates of many populations vary with in real situation, for example, due to the seasonality. As a matter of fact, nonautonomous stochastic population systems have recently been studied by many authors, for example, [13–17].

In this paper we consider the system

where , , , are all continuous bounded nonnegative functions on . The objective of our study is to investigate the long-time behavior of system (3). As in [8], we mainly discuss when the system is persistent and when it is not under a fewer conditions. More specifically, we show that there is a positive solution of system (3) and its th moment bounded in Section 2. In Section 3, we deduce the persistence of the system. If the white noise is not large such that , we will prove that the solution of system (3) is a stochastic persistence. In addition, we show that every component of the solution is persistent in mean. We further deduce that every component of the solution of system (3) is an asymptotic boundedness in mean. In Section 4, we show that larger white noise will make system (3) nonpersistent. Finally, we study the global attractivity of system (3).

Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let be the positive cone of , namely, . If , its norm is denoted by . If is a continuous bounded function on , we use the notation

2. Existence and Uniqueness of the Positive Solution

In population dynamics, the first concern is that the solution should be nonnegative. In order to do that a stochastic differential equation can have a unique global (i.e., no explosion at any finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (Mao [18]). However, the coefficients of system (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (3) may explode at a finite time. Following the way developed by Mao et al. [19], we show that there is a unique positive solution of (3).

Theorem 1. Assume that . Then, there is a unique positive solution of system (3) on for any given initial value , and the solution will remain in with probability 1, namely, for all almost surely.

The proof of Theorem 1 is similar to [8]. But it is skilled in taking the value of . We show it here.

Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial value there is an unique local solution on , where is the explosion time. To show that this solution is global, we need to show that a.s. Let be sufficiently large for every component of lying within the interval . For each integer , define the stopping time

where throughout this paper we set (as usual denotes the empty set). Clearly, is increasing as . Set whence a.s. If we can show that a.s., then a.s. and a.s. for all . In other words, to complete the proof, all we need to show is that a.s. If this statement is false, there is a pair of constant and such that

Hence, there is an integer such that

We define

By Itô's formula, we have

where

According to Young inequality, note that , where , then,

Since , we obtain and . Hence, is a positive constant. Integrating both sides of (9) from 0 to , we therefore obtain

Whence, taking expectations yields

Set for and by (7), . Note that for every , there is or equals either or , and therefore

where . It then follows from (13) that

where is the indicator function of . Letting leads to the contradiction

so we must have a.s. This completes the proof of Theorem 1.

Remark 2. By Theorem 1, we observe that for any given initial value , there is a unique solution of system (3) on and the solution will remain in with probability , no matter how large the intensities of white noise are. So, under the same assumption there is an global unique positive solution of the corresponding deterministic system of system (3).

Next, we show that the th moment of the solution of system (3) is bounded in time average.

Theorem 3. Assume that . Then there exists a positive constant such that the solution of system (3) has the following property:

where satisfy

Proof. By Itô's formula, we have

where , and

where . According to Young inequality, we obtain

Thus, we have

Since , there exist two positive constants which satisfy

Therefore,

From (23) and the values of , we obtain

which implies that and . Let

then we have

Hence, we get

By the comparison theorem, we get

which implies that there is a , such that

Besides, note that is continuous, then there is a such that

Let , then

3. Persistence

Theorem 1 shows that the solution of system (3) will remain in the positive cone if . Studying a population system, we pay more attention on whether the system is persistent. In this section, we first show that the solution is a stochastic permanence. Next we show that the solution is persistent in time average. Moreover, we show that the solution of system (3) is an asymptotic boundedness in time average.

3.1. Stochastic Permanence

Let be the solution of a randomized nonautonomous competitive equation:

where , are independent standard Brownian motions, while is independent of , and , , are all continuous bounded nonnegative functions on .

Lemma 4 (see [15]). Assume that , then for any given initial value , the solution of (36) has the properties

where is a constant, is an arbitrary positive constant satisfying

Let be the solution of a randomized nonautonomous logistic equation

where is a 1-dimensional standard Brownian motion, , and is independent of .

Lemma 5 (see [13]). Assume that , and are bounded continuous functions defined on , and . Then there exists a unique continuous positive solution of (36) for any initial value , which is global and represented by

From Lemma 4 we have the following.

Lemma 6. Assume that , then for any given initial value , the solution of (36) has the properties

where is a constant, is positive constant satisfying

Let be the solution of

where , are independent standard Brownian motions, , and , are all continuous bounded nonnegative functions on . From Lemma 4 it is easy to know the following.

Lemma 7. Assume that , then for any given initial value , the solution of (40) has the properties

where , are two constants, is positive constant satisfying

Lemma 8. Assume that , then for any given initial value , the solution of system (3) has the properties

where are two constants, is positive constant satisfying

Proof. Equation (43) follows directly from the classical comparison theorem of stochastic differential equations (see [20]). Thus, we obtain

Definition 9. System (3) is said to be stochastically permanent if for any , there exists a pair of positive constants and such that for any initial value , the solution obeys

Theorem 10. Assume that , then system (3) is stochastically permanent.

The proof is a simple application of the Chebyshev inequality, we omit it.

3.2. Persistence in Time Average

Theorem 10 shows that if the white noise is not large, the solution of system (3) is survive with large probability. In this part, we show is persistence in mean.

Lemma 11. Assume that , then for any given initial value , the solution of (40) has the properties

where is the solution of

Proof. From Lemma 5, we know

Similarly, we have

Lemma 12. Assume that , then for any given initial value , the solution of (49) has the following properties

where are the solutions of the two equations, respectively,

Proof. Let , are the solutions of SDE (53) and (54), respectively, with the positive initial value . By Lemma 5, we know

Thus,

By the classical comparison theorem of ordinary differential equations, we know

Lemma 13. Assume that , then for any given initial value , the solution of (40) has the properties

Proof. By Lemma 12, we know

So, we have

Let then

Since are bounded, then

By the strong law of large numbers, we know

Thus,

Then from (60) we obtain

Lemma 14. Assume that , then for any given initial value , the solution of (40) has the properties

Proof. By Itô's formula, we have

Integrating both sides of this equation from 0 to yields

By Lemma 13, we know that

Hence,

Definition 15. System (3) is said to be persistent in time average if

Theorem 16. Assume that and , then the solution of system (3) with any initial value has the following property:

and so system (3) is persistent in time average.

Proof. By Lemma 8, we know that

where is the solution of system (40). Moreover, by Lemma 14 we know that

Hence, by Lemma 13 we know that

3.3. Asymptotic Boundedness of Integral Average

Theorem 16 shows that every component of the solution of system (3) will survive forever in time average, if the white noise is not large. In this part, we further deduce that every component of of system (3) will be an asymptotic boundedness in time average. Before we give the result, we do some preparation work.

Lemma 17. Let , . If there exist positive constants and such that

and a.s., then

Proof. The proof is similar to the proof of Lemma in [21]. Let

Since is differentiable on and

Substituting and into (76), we obtain the following:

thus

Note that ., then for , and such that and . Then we have

Integrating inequality (82) from to results in the following:

This inequality can be rewritten into

Taking the logarithm of both sides and dividing both sides by yields

Then,

Letting yields

This finishes the proof of the Lemma.

Theorem 18. Assume that and , then the solution of system (3) with any initial value has the property

where

Proof. To prove the results, we only need to prove

By Itô's formula, we have

First, we prove (91). Integrating both sides of (92) from to yields

where . Since , hence

So we have

By Theorem 16, we know that

Obviously,

Hence, we have

Similarly, we have

Next, we prove that (90) is true. Taking integration both sides of (92) from to , we have

By Theorem 16 we know that

then for any , there is a such that

for . It follows from (100) that, for ,

From Lemma 17, we have

Similarly, we have

Continuing this process, we obtain two sequences such that

By induction, we can easily show that , that is, sequences and are nondecreasing. Moreover, note that (98) and (99), then the sequences and , have upper bounds. Therefore, there are two positive such that

which together with (106) implies

Letting yields

Hence,

which is as required.

4. Nonpersistence

In this section, we discuss the dynamics of system (3) as the white noise is getting larger. We show that system (3) will be nonpersistent if the white noise is large, which does not happen in the deterministic system.

Definition 19. System (3) is said to be nonpersistent, if there are positive constants such that

Theorem 20. Assume that and , then system (3) is nonpersistent, where .

Proof. Since and , from (93) we have

where which together with

implies

If , then there must be

Hence, system (3) is nonpersistent.

Theorem 21. Assume that and , then system (3) is nonpersistent, where .

Here we omit the proof of Theorem 21 which is similar to the proof of Theorem 20.

Remark 22. If , then the conditions in Theorems 20 and 21 are obviously satisfied, respectively. That is to say, the large white noise will lead to the population system being non-persistent.

5. Global Attractivity

In this section, we turn to establishing sufficient criteria for the global attractivity of stochastic system (3).

Definition 23. Let , be two arbitrary solutions of system (3) with initial values , respectively. If

then we say system (3) is globally attractive.

Theorem 24. Assume that , then system (3) is globally attractive.

Proof. Let , be two arbitrary solutions of system (3) with initial values . By the Itô's formula, we have

Then,

Since , there exist two positive constants , which satisfy

Thus, , .

Consider a Lyapunov function defined by

A direct calculation of the right differential of along the ordinary differential equation (119) leads to

where . Integrating both sides of (122) form 0 to , we have

Let , we obtain

Note that . Clearly, . It is straightforward to see from (124) that

Next, we prove that

By Theorem 3 we obtain that the th moment of the solution of system (3) is bounded, the following proof is similar to the proof of Theorem 6.2 in [15] and hence is omitted.

Acknowledgments

The work was supported by the Ph.D. Programs Foundation of Ministry of China (no. 200918), NSFC of China (no. 10971021), and Program for Changjiang Scholars and Innovative Research Team in University.

References

R. Redheffer, “Nonautonomous Lotka-Volterra systems. I,” Journal of Differential Equations, vol. 127, no. 2, pp. 519–541, 1996.

R. Redheffer, “Nonautonomous Lotka-Volterra systems. II,” Journal of Differential Equations, vol. 132, no. 1, pp. 1–20, 1996.

A. Tineo, “On the asymptotic behavior of some population models. II,” Journal of Mathematical Analysis and Applications, vol. 197, no. 1, pp. 249–258, 1996.

Z. D. Teng, “Nonautonomous Lotka-Volterra systems with delays,” Journal of Differential Equations, vol. 179, no. 2, pp. 538–561, 2002.

F. Y. Wei and W. Ke, “Global stability and asymptotically periodic solution for nonautonomous cooperative Lotka-Volterra diffusion system,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 161–165, 2006.

Y. Nakata and Y. Muroya, “Permanence for nonautonomous Lotka-Volterra cooperative systems with delays,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 528–534, 2010.

X. Abdurahman and Z. Teng, “On the persistence of a nonautonomous -species Lotka-Volterra cooperative system,” Applied Mathematics and Computation, vol. 152, no. 3, pp. 885–895, 2004.

C. Y. Ji, D. Q. Jiang, H. Liu, and Q. S. Yang, “Existence, uniqueness and ergodicity of positive solution of mutualism system with stochastic perturbation,” Mathematical Problems in Engineering, vol. 2010, Article ID 684926, 18 pages, 2010.

C. Y. Ji and D. Q. Jiang, “Persistence and non-persistence of a mutualism system with stochastic perturbation,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 3, pp. 867–889, 2012.

X. R. Mao, S. Sabanis, and E. Renshaw, “Asymptotic behaviour of the stochastic Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 141–156, 2003.

X. R. Mao, C. G. Yuan, and J. Z. Zou, “Stochastic differential delay equations of population dynamics,” Journal of Mathematical Analysis and Applications, vol. 304, no. 1, pp. 296–320, 2005.

M. Liu and K. Wang, “Survival analysis of a stochastic cooperation system in a polluted environment,” Journal of Biological Systems, vol. 19, no. 2, pp. 183–204, 2011.

D. Q. Jiang and N. Z. Shi, “A note on nonautonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 164–172, 2005.

D. Q. Jiang, N. Z. Shi, and X. Y. Li, “Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 588–597, 2008.

X. Y. Li and X. R. Mao, “Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 2, pp. 523–545, 2009.

M. Liu and K. Wang, “Persistence and extinction in stochastic non-autonomous logistic systems,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 443–457, 2011.

M. Liu and K. Wang, “Asymptotic properties and simulations of a stochastic logistic model under regime switching II,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 405–418, 2012.

X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.

X. Mao, G. Marion, and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stochastic Processes and their Applications, vol. 97, no. 1, pp. 95–110, 2002.

I. N. Wantanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, The Netherlands, 1981.

H. P. Liu and Z. E. Ma, “The threshold of survival for system of two species in a polluted environment,” Journal of Mathematical Biology, vol. 30, no. 1, pp. 49–61, 1991.