Uncertain Dynamical Systems: Analysis and ApplicationsView this Special Issue
The Asymptotic Behavior of a Stochastic Predator-Prey System with Holling II Functional Response
We discuss a stochastic predator-prey system with Holling II functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we deduce the conditions that there is a stationary distribution of the system, which implies that the system is permanent. At last, we give the conditions for the system that is going to be extinct.
One of the most popular predator-prey model is the one with Michaelis-Menten type (or Holling Type II) functional response [1, 2]: where and are the population densities of prey and predator at time , respectively. The constants , , , , , and are positive constants that stand for prey intrinsic growth rate, carrying capacity, the maximum ingestion rate, half-saturation constant, predator death rate, and the conversion factor, respectively. This model exhibits the well-known but highly controversial “paradox of enrichment” observed by Hairston et al.  and by Rosenzweig  which is rarely reported in nature. It is very important to study the existence and asymptotical stability of equilibria and limit cycle for autonomous predator-prey systems with Holling II functional response. If , then system (1.1) has a unique limit cycle which is stable. If , then system (1.1) has a unique positive equilibrium: which is a stable node or focus (see ).
However, countless organisms live in seasonally or diurnally forced environments. Hence, authors considered models with periodic ecological parameters or perturbations. For example, Liu and Chen  introduced periodic constant impulsive immigration of predator into system (1.1) and gave conditions for the system to be extinct and permanence, respectively. Zhang and Chen  studied a Holling II functional response food chain model with impulsive perturbations. Zhang et al.  further considered system (1.1) with periodic constant impulsive immigration of predator and periodic variation in the intrinsic growth rate of the prey.
On the other hand, the white noise is always present, and we cannot omit the influence of the white noise to the system. May  pointed out that due to continuous fluctuation in the environment, the birth rates, death rates, carrying capacity, competition coefficients, and all other parameters involved with the model exhibit random fluctuation to a great lesser extent, and as a result the equilibrium population distribution never attains a steady value, but fluctuates randomly around some average value. Many authors studied the effect of the stochastic perturbation to the predator-prey system with different functional responses, such as [10–14]. Therefore, in this paper, we also introduce stochastic perturbation system (1.1) and obtain the following stochastic system: where and are mutually independent Brownian motion with , and , are intensities of the white noise.
The aim of this paper is to discuss the long time behavior of system (1.3). As the deterministic population models, we are also interested in the permanence and extinction of the system. The global stability of the positive equilibrium means that the system is permanence. But, for the stochastic system, there is no positive equilibrium. Hence, it is impossible that the solution of system (1.3) will tend to a fixed point. In this paper, we show that there is a stationary distribution of system (1.3) mainly according to the theory of Has’meminskii , if the white noise is small. While if the white noise is large, based on the techniques developed in [16, 17], we prove that the predator population will die out a.s. and the prey population will either extinct or its distribution converges to a probability measure. It does not happen that both the prey population and the predator population in system (1.3) will die out, which is brought by large white noise, such as weather, epidemic disease. From this point, we say the stochastic model is more realistic than the deterministic model.
The rest of this paper is organized as follows. In Section 2, we show that there is a unique nonnegative solution of system (1.3). In Section 3, we show that there is a stationary distribution under small white noise. While in Section 4, we consider the situation when the white noise is large. We prove that the system will be extinct. Finally, we give an appendix containing the stationary distribution theory used in Section 3.
2. Existence and Uniqueness of the Nonnegative Solution
To investigate the dynamical behavior, the first concern is the global existence of the solutions. Hence in this section we show that the solution of system (1.3) is global and nonnegative. It is not difficult to check the uniqueness and global existence of solutions if the coefficients of the equation satisfy the linear growth condition and local Lipschitz condition (cf. ). However, the coefficients of system (1.3) do not satisfy the linear growth condition, but locally Lipschitz continuous, so the solution of system (1.3) may explode at a finite time. In this section, by changing variables, we first show that system (1.3) has a local solution, then show that this solution is global.
Theorem 2.1. For any initial value , there is a unique solution of system (1.3) on , and the solution will remain in with probability 1.
Proof. First, consider the following system, by changing variables, , , It is clear that the coefficients of system (2.1) are locally Lipschitz continuous for the given initial value there is a unique local solution on , where is the explosion time (see ). Hence, by Itô formula, we know is a unique positive local solution of system (1.3). To show that this solution is global, we need to show that a.s. Let be sufficiently large so that all lie 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 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, then there is a pair of constants and such that Hence there is an integer such that Define a -function by where is a positive constant to be determined later. The nonnegativity of this function can be seen from . Using Itô's formula, we get where Choose such that , then where is a positive constant. Therefore which implies that, Set for , then by (2.4), we know that . Note that for every , there is at least one of and equals either or , then It then follows from (2.4) and (2.10) that where is the indicator function of . Letting leads to the contradiction that . So we must therefore have a.s.
There is no equilibrium of system (1.3). Hence we cannot show the permanence of the system by proving the stability of the positive equilibrium as the deterministic system. In this section we show that there is a stationary distribution of system (1.3).
Remark 3.1. Theorem 2.1 shows that there exists a unique positive solution of system (1.3) with any initial value . From the proof of Theorem 2.1, we obtain that . Define , then , and it is clear that as , where . Hence by Remark 2 of Theorem 4.1 of Has’meminskii, 1980, page 86 in , we obtain that the solution is a homogeneous Markov process in .
Theorem 3.2. If and such that and where is the positive equilibrium of system (1.1) and is defined as in the proof. Then system (1.3) has a stationary ergodic solution.
Proof. Since , then there is a positive equilibrium of system (1.1), and Let where is a positive constant to be determined later. Let be the generating operator of system (1.3). Then Choose such that and yields Let Note that then where is also the generating operator of system (1.3). Note that then Now define where is a positive constant to be determined later. Then Choose such that , then it follows from (3.12) that Note that then the ellipsoid lies entirely in . We can take to be a neighborhood of the ellipsoid with , so that for , ( is a positive constant), which implies condition () in Lemma A.1 is satisfied. Hence the solution is recurrent in the domain , which together with Lemma A.3 and Remark 3.1 implies that is recurrent in any bounded domain . Besides, for all , there is an such that which implies that condition () is also satisfied. Therefore, system (1.3) has a stable a stationary distribution and it is ergodic.
Note that then Hence by comparison theorem, we get by which together with the continuity of , we have that there exists a positive constant such that
By Doob's martingale inequality, together with the (3.20), for , we have In view of the well-known Borel-Cantelli lemma, we see that for almost all , holds for all but finitely many . Hence there exists an , for all excluding a -null set, for which (3.22) holds whenever . Consequently, letting , we have, for almost all ,
By the ergodic property, for any given constant , we have On the other hand, by dominated convergence theorem and (3.20), we get which together with (3.24) implies Letting , we get That is to say, the function is integrable with respect to the measure . Therefore, by ergodicity property again, we get Besides, Let which is a martingale with and then by the strong law of large numbers, we get which together with (3.23) and (3.28) implies (3.29) that
Hence from these arguments, we get the following result.
Theorem 3.3. Assume the same conditions as in Theorem 3.2. Then one has
In this section, we show the situation when system (1.3) will be extinct.
Case 1. .
Obviously, then when , That is to say, for all , there exist and a set such that and for and . Then and so
Case 2. .
Note that then if , we have In this situation, for all , there exist and a set such that and for and . Then Consider the following equation: If , (4.9) has the density such that Therefore from (4.8) and the arbitrary of , we get that the distribution of converges weakly to the probability measure with density . Thus, from (4.10), we obtain that the distribution of converges weakly to the probability measure with density , where . Besides, from the ergodic theorem and (4.10), it follows that
Therefore, by the above arguments, we obtain the following.
Theorem 4.1. Let be the solution of system (1.3) with the initial value . Then,(i)if , then (ii)if , then the distribution of converges weakly to the probability measure with density , where , and
For the completeness of the paper, in this section, we list some theories about stationary distribution (see ).
Let be a homogeneous Markov process in ( denotes euclidean -space) described by The diffusion matrix is .
Assumption B. There exists a bounded domain with regular boundary , having the following properties. (B.1) In the domain and some neighbourhood thereof, the smallest eigenvalue of the diffusion matrix is bounded away from zero. (B.2) If , the mean time at which a path issuing from reaches the set is finite, and for every compact subset .
Lemma A.1 (see ). If (B) holds, then the Markov process has a stationary distribution . Let be a function integrable with respect to the measure . Then for all .
Remark A.2. The proof is given in . Exactly, the existence of stationary distribution with density is referred to Theorem 4.1, Page 119, and Lemma 9.4, Page 138, in . The weak convergence and the ergodicity is obtained in Theorem 5.1, Page 121, and Theorem 7.1, Page 130, in .
To validate (), it suffices to prove that is uniformly elliptical in any bounded domain , where ; that is, there is a positive number such that (see Chapter 3, Page 103 of  and Rayleigh's principle in [20, Chapter 6, Page 349]). To verify (), it is sufficient to show that there exists some neighborhood and a nonnegative -function such that and for any is negative (for details refer to [21, Page 1163]).
Lemma A.3. Let be a regular temporally homogeneous Markov process in . If is recurrent relative to some bounded domain , then it is recurrent relative to any nonempty domain in .
The work was supported by the Ministry of Education of China (no. 109051), the Ph.D. Programs Foundation of Ministry of China (no. 200918), NSFC of China (no. 10971021), and the Graduate Innovative Research Project of NENU (no. 09SSXT117), Youth Fund of Jiangsu Province (no. BK2012208), and the Tian Yuan Special Funds of the National Natural Science Foundation of China (no. 11226205).
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