The Asymptotic Behavior of a Stochastic Predator-Prey System with Holling II Functional Response

and Applied Analysis 3 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. 18 . 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 x 0 , y 0 ∈ R2 , there is a unique solution x t , y t of system 1.3 on t ≥ 0, and the solution will remain in R2 with probability 1. Proof. First, consider the following system, by changing variables, x t e t , y t e t , du t ( a − σ 2 1 2 − be t − αe v t 1 βeu t ) dt σ1dB1 t , dv t ( − e − σ 2 2 2 kαe t 1 βeu t ) dt σ2dB2 t . 2.1 It is clear that the coefficients of system 2.1 are locally Lipschitz continuous for the given initial value logx 0 , logy 0 ∈ R2 there is a unique local solution u t , v t on t ∈ 0, τe , where τe is the explosion time see 18 . Hence, by Itô formula, we know e t , e t , t ∈ 0, τe is a unique positive local solution of system 1.3 . To show that this solution is global, we need to show that τe ∞ a.s. Let m0 ≥ 1 be sufficiently large so that x 0 , y 0 all lie within the interval 1/m0, m0 . For each integer m ≥ m0, define the stopping time: τm inf { t ∈ 0, τe : min { x t , y t } ≤ 1 m or max { x t , y t } ≥ m } , 2.2 Where, throughout this paper, we set inf ∅ ∞ as usual ∅ denotes the empty set . Clearly, τm is increasing as m → ∞. Set τ∞ limm→∞τm, whence τ∞ ≤ τe a.s. If we can show that τ∞ ∞ a.s., then τe ∞ and x t , y t ∈ R2 a.s. for all t ≥ 0. In other words, to complete the proof all we need to show is that τ∞ ∞ a.s. If this statement is false, then there is a pair of constants T > 0 and ∈ 0, 1 such that P{τ∞ ≤ T} > . 2.3 4 Abstract and Applied Analysis Hence there is an integer m1 ≥ m0 such that P{τm ≤ T} ≥ ∀m ≥ m1. 2.4 Define a C2-function V : R2 → R by V ( x, y ) ( x − c − c log x c ) 1 k ( y − 1 − logy), 2.5 where c is a positive constant to be determined later. The nonnegativity of this function can be seen from u − 1 − logu ≥ 0, for all u > 0. Using Itô’s formula, we get dV : LVdt σ1 x − c dB1 t σ2 k ( y − 1)dB2 t , 2.6


Introduction
One of the most popular predator-prey model is the one with Michaelis-Menten type or Holling Type II functional response 1, 2 : ẋ t x t a − bx t − αy t 1 βx t , ẏ t y t −e kαx t 1 βx t , 1.1 where x t and y t are the population densities of prey and predator at time t, respectively.The constants a, b/a, α, β, e, and k 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. 3 and by Rosenzweig 4 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 kaαβ > aeβ 2 kbα beβ, then system 1.1 has a unique limit cycle which is stable.If akα > aeβ be, then system 1.1 has a unique positive equilibrium: x * e kα − eβ , y * kα akα − aeβ − be kα − eβ which is a stable node or focus see 5 .
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 6 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 7 studied a Holling II functional response food chain model with impulsive perturbations.Zhang et al. 8 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 9 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: 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 15 , 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.

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.18 .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 x 0 , y 0 ∈ R 2 , there is a unique solution x t , y t of system 1.3 on t ≥ 0, and the solution will remain in R 2 with probability 1.
Proof.First, consider the following system, by changing variables, x t e u t , y t e v t ,

2.1
It is clear that the coefficients of system 2.1 are locally Lipschitz continuous for the given initial value log x 0 , log y 0 ∈ R 2 there is a unique local solution u t , v t on t ∈ 0, τ e , where τ e is the explosion time see 18 .Hence, by It ô formula, we know e u t , e v t , t ∈ 0, τ e is a unique positive local solution of system 1.3 .To show that this solution is global, we need to show that τ e ∞ a.s.Let m 0 ≥ 1 be sufficiently large so that x 0 , y 0 all lie within the interval 1/m 0 , m 0 .For each integer m ≥ m 0 , define the stopping time: Where, throughout this paper, we set inf ∅ ∞ as usual ∅ denotes the empty set .Clearly, s.If we can show that τ ∞ ∞ a.s., then τ e ∞ and x t , y t ∈ R 2 a.s.for all t ≥ 0. In other words, to complete the proof all we need to show is that τ ∞ ∞ a.s.If this statement is false, then there is a pair of constants T > 0 and ∈ 0, 1 such that Hence there is an integer m 1 ≥ m 0 such that where c is a positive constant to be determined later.The nonnegativity of this function can be seen from u − 1 − log u ≥ 0, for all u > 0. Using It ô's formula, we get where

2.7
Choose c e/αk such that e/k − αc 0, then where K is a positive constant.Therefore which implies that,

2.10
Set Ω m {τ m ≤ T } for m ≥ m 1 , then by 2.4 , we know that P Ω m ≥ .Note that for every ω ∈ Ω m , there is at least one of x τ m , ω and y τ m , ω equals either m or 1/m, then

2.11
It then follows from 2.4 and 2.10 that 12 where 1 Ω m ω is the indicator function of Ω m .Letting m → ∞ leads to the contradiction that ∞ > V x 0 , y 0 KT ∞.So we must therefore have τ ∞ ∞ a.s.

Permanence
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 x t , y t of system 1.3 with any initial value x 0 , y 0 ∈ R 2 .From the proof of Theorem 2.1, we obtain that Hence by Remark 2 of Theorem 4.1 of Has'meminskii, 1980, page 86 in 15 , we obtain that the solution x t , y t is a homogeneous Markov process in R 2 .

3.1
where x * , y * is the positive equilibrium of system 1.1 and l 2 is defined as in the proof.Then system 1.3 has a stationary ergodic solution.
Proof.Since akα > aeβ be, then there is a positive equilibrium x * , y * of system 1.1 , and where l 1 is a positive constant to be determined later.Let L be the generating operator of system 1.3 .Then .

3.6
Note that where L is also the generating operator of system 1.3 .Note that 3.9 then

3.10
Now define where l 2 is a positive constant to be determined later.Then

3.13
Note that 3.14 then the ellipsoid lies entirely in R 2 .We can take U to be a neighborhood of the ellipsoid with U ⊆ E l R 2 , so that for x, y ∈ U \ E l , LV ≤ −C C is a positive constant , which implies condition B.2 in Lemma A.1 is satisfied.Hence the solution x t , y t is recurrent in the domain U, which together with Lemma A.3 and Remark 3.1 implies that x t , y t is recurrent in any bounded domain D ⊂ R 2 .Besides, for all D, there is an which implies that condition B.1 is also satisfied. 3.33

Extinction
In this section, we show the situation when system 1.3 will be extinct.
Case 1. a < σ  Lemma A.3.Let X t be a regular temporally homogeneous Markov process in E l .If X t is recurrent relative to some bounded domain U, then it is recurrent relative to any nonempty domain in E l .

4 ,
Page 138, in 4 .The weak convergence and the ergodicity is obtained in Theorem 5.1, Page 121, and Theorem 7.1, Page 130, in 4 .To validate B.1 , it suffices to prove that F is uniformly elliptical in any bounded domain D, where Fu b x •u x 1/2 tr A x u xx ; that is, there is a positive number M such that k i,j 1 a ij x ξ i ξ j ≥ M|ξ| 2 , x ∈ D, ξ ∈ R k see Chapter 3, Page 103 of 19 and Rayleigh's principle in 20, Chapter 6, Page 349 .To verify B.2 , it is sufficient to show that there exists some neighborhood U and a nonnegative C 2 -function such that and for any E l \ U, LV is negative for details refer to 21, Page 1163 .
That is to say, for all 0 < 1 < e σ 2 2 /2, there exist T 1 T 1 ω and a set Ω 1 such that P Ω 1 > 1 − 1 and kαx t ≤ 1 for t ≥ T 1 and ω ∈ Ω 1 .Then −ey t dt σ 2 y t dB 2 t ≤ dy t ≤ y t −e 1 dt σ 2 y t dB 2 t , l f x μ dx } 1 for all x ∈ E l .Remark A.2.The proof is given in 15 .Exactly, the existence of stationary distribution with density is referred to Theorem 4.1, Page 119, and Lemma 9.
Lemma A.1 see 15 .If (B) holds, then the Markov process X t has a stationary distribution μ • .Let f • be a function integrable with respect to the measure μ.Then P x {lim T → ∞ 1/T T 0 f X t dt E