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Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System
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.
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  that if different conditions hold (see conditions (a)–(e) in ), 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  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  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  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 , 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 ). 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. , 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.
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
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 .
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 ). 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
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
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
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
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 . 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