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Abstract and Applied Analysis
Volume 2011, Article ID 518719, 16 pages
http://dx.doi.org/10.1155/2011/518719
Research Article

Analysis on a Stochastic Predator-Prey Model with Modified Leslie-Gower Response

1Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China
2School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received 30 October 2010; Accepted 30 March 2011

Academic Editor: Wing-Sum Cheung

Copyright © 2011 Jingliang Lv and Ke Wang. 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

This paper presents an investigation of asymptotic properties of a stochastic predator-prey model with modified Leslie-Gower response. We obtain the global existence of positive unique solution of the stochastic model. That is, the solution of the system is positive and not to explode to infinity in a finite time. And we show some asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained. Finally we work out some figures to illustrate our main results.

1. Introduction

The dynamic interaction between predators and their prey has been one of the dominant themes in mathematical biology due to its universal existence and importance. Much literature exists on the general problem of food chains in the classical Lotka-Volterra model. In [1, 2], Leslie introduced a predator-prey model where the capacity of the predators environment is proportional to the number of preys. Leslie stresses the fact that there are upper limits to the rates of increase of both prey and predator, which are not recognized in the Lotka-Volterra model. Broer et al. [3] studied the dynamical properties of a predator-prey model with nonmonotonic response function. Reference [4] considered two-species autonomous system which incorporated a modified Leslie-Gower functional response as well as that of the Holling II as follows: 𝑑𝑥𝑑𝑡=𝑥𝑎𝑏𝑥𝑐𝑦𝑥+𝑘1,𝑑𝑦𝑑𝑡=𝑦𝑟𝑓𝑦𝑥+𝑘2,(1.1) where 𝑎,𝑏,𝑐,𝑟,𝑓,𝑘1, and 𝑘2 are all positive constants and 𝑥(𝑡), 𝑦(𝑡) represent the population densities at time 𝑡.

Hsu and Huang [5] studied the global stability property of the following predator-prey system: 𝑑𝑥𝑥𝑑𝑡=𝑟𝑥1𝑘𝑦𝑝(𝑥),𝑑𝑦𝑠𝑑𝑡=𝑦1𝑦𝑥,𝑥0>0,𝑦0>0,𝑟,𝑠,𝑘,>0.(1.2)

Recently, [6] discussed the following model with modified Leslie-Gower response: 𝑑𝑥𝑥𝑑𝑡=𝑟𝑥1𝑘𝑛𝜆𝑥𝑦,𝜆𝑥+𝐴𝑦𝑑𝑦𝑠𝑑𝑡=𝑦1𝑦,𝑥𝜆𝑥+𝑏0>0,𝑦0>0,(1.3) where 𝑟,𝑘,𝑛,𝐴,𝑠,,𝑏,and𝜆 are all positive constants and 𝑟, 𝑠 are the growth rates of prey 𝑥 and predator 𝑦, respectively. Here, we change the form of the predator-prey model above which reads 𝑑𝑥𝑑𝑡=𝑥𝑎𝑏𝑥𝑐𝑦,𝜆𝑥+𝐴𝑦𝑑𝑦𝑑𝑡=𝑦𝑓𝑔𝑦,𝑥𝜆𝑥+0>0,𝑦0>0.(1.4)

As a matter of fact, population systems are often subject to environmental noise. Recently, more and more interest is focused on stochastic systems. Reference [7] investigated the predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation: 𝑑𝑥=𝑥𝑎𝑏𝑥𝑐𝑦𝑥+𝑚𝑑𝑡+𝜎1𝑥𝑑𝐵1(𝑡),𝑑𝑦=𝑦𝑟𝑓𝑦𝑥+𝑚𝑑𝑡𝜎2𝑦𝑑𝐵2(𝑡).(1.5) By virtue of comparison theorem, [7] obtained some interesting results, including globally positive solutions, persistence in mean and extinction. Moreover, [8] continued to consider the stochastic ratio-dependent predator-prey system:𝑑𝑥=𝑥𝑎𝑏𝑥𝑐𝑦𝑥+𝑚𝑦𝑑𝑡+𝜎1𝑥𝑑𝐵1(𝑡),𝑑𝑦=𝑦𝑔+𝑓𝑥𝑥+𝑚𝑦𝑑𝑡𝜎2𝑦𝑑𝐵2(𝑡),(1.6) where 𝐵𝑖(𝑡), 𝑖=1,2, are independent standard Brownian motions. And [8] also obtained some nice conclusions on the stochastic model.

According to (1.4), taking into account the effect of randomly fluctuating environment, we will consider the corresponding autonomous stochastic system described by the Itô equation 𝑑𝑥(𝑡)=𝑥(𝑡)𝑎𝑏𝑥(𝑡)𝑐𝑦(𝑡)𝜆𝑥(𝑡)+𝐴𝑦(𝑡)𝑑𝑡+𝜎1𝑥(𝑡)𝑑𝐵1(𝑡),𝑑𝑦(𝑡)=𝑦(𝑡)𝑓𝑔𝑦(𝑡)𝜆𝑥(𝑡)+𝑑𝑡+𝜎2𝑦(𝑡)𝑑𝐵2(𝑡),(1.7) where 𝐵𝑖(𝑡), 𝑖=1,2, are independent standard Brownian motions and 𝑎,𝑏,𝑐,𝑓,𝑔,𝜆,𝐴,𝜎1, and 𝜎2 are all positive.

When white noise is taken into account in our model (1.7), we obtain the global existence of positive unique solution of the stochastic model, that is, the solution of the system is positive and not to explode to infinity in a finite time in Section 2. Section 3 shows some fundamental asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained in Section 3. The main contributions of this paper are therefore clear.

Throughout the paper, we use 𝐾 to denote a positive constant whose exact value may be different in different appearances.

2. Positive and Global Solution

As 𝑥(𝑡), 𝑦(𝑡) of the SDE (1.7) are sizes of the species in the system at time 𝑡, it is obvious that the positive solutions are of interest. The coefficients of (1.7) are locally Lipschitz continuous and do not satisfy the linear growth condition, so the solution of (1.7) may explode at a finite time. The following Theorem shows that the solution will not explode at a finite time.

Theorem 2.1. For a given initial value 𝑋0=(𝑥0,𝑦0)𝑅2+, there is a unique positive solution 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡)) to (1.7) on 𝑡0, and the solution will remain in 𝑅2+ with probability one, namely, 𝑋(𝑡)𝑅2+ for all 𝑡0 almost surely.

Proof. The proof is similar to [9, 10]. Since the coefficients of the equation are locally Lipschitz continuous, for a given initial value 𝑋0=(𝑥0,𝑦0)𝑅2+, there is a unique local solution 𝑋(𝑡) on 𝑡[0,𝜏𝑒), where 𝜏𝑒 is the explosion time. To show that this solution is global, we need to show that 𝜏𝑒=+ a.s. Let 𝑘0>0 be sufficiently large for every component of 𝑥(𝑡) and 𝑦(𝑡) all lying within the interval [1/𝑘0,𝑘0]. For each integer 𝑘𝑘0, define the stopping time 𝜏𝑚=inf𝑡0,𝜏𝑒1𝑥(𝑡)𝑘1,𝑘or𝑦(𝑡)𝑘,𝑘,(2.1) where throughout this paper we set inf=. Obviously, 𝜏𝑘 is increasing as 𝑘. Let 𝜏=lim𝑘𝜏𝑘, whence 𝜏𝜏𝑒 a.s. If we can show that 𝜏= a.s., then 𝜏𝑒= a.s. and 𝑋(𝑡)𝑅2+ a.s. for all 𝑡0. So we just prove that 𝜏= a.s. If not, there is 𝜖(0,1) and 𝑇>0 such that 𝑃𝜏𝑇>𝜖.(2.2) Hence, there is an integer 𝑘1𝑘0 such that 𝑃{𝜏𝑘𝑇}𝜖 for all 𝑘𝑘1. Define a function 𝑉𝑅2+𝑅+ by 𝑉(𝑥,𝑦)=(𝑥1ln𝑥)+(𝑦1ln𝑦). The nonnegativity of this function can be seen from 𝑢1ln𝑢0,on𝑢>0.(2.3) If 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡))𝑅2+, by virtue of 𝑢2[𝑢1ln𝑢]+2 on 𝑢>0, we obtain 𝐿𝑉(𝑥,𝑦)=(𝑥1)𝑎𝑏𝑥𝑐𝑦𝜆𝑥+𝐴𝑦+(𝑦1)𝑓𝑔𝑦+𝜎𝜆𝑥+21+𝜎222=𝑎𝑥𝑏𝑥2𝑐𝑥𝑦𝜆𝑥+𝐴𝑦𝑎+𝑏𝑥+𝑐𝑦𝜆𝑥+𝐴𝑦+𝑓𝑦𝑓𝑔𝑦2+𝜆𝑥+𝑔𝑦+𝜎𝜆𝑥+21+𝜎222𝐾1𝑉(𝑥,𝑦)+𝐾2,(2.4) dropping 𝑡 from 𝑥(𝑡) and 𝑦(𝑡). Making use of the Itô formula yields 𝑥𝜏𝐸𝑉𝑘𝜏𝑇,𝑦𝑘𝑥𝑇𝑉0,𝑦0+𝐾2𝑇+𝐾1𝑇0𝑥𝜏𝐸𝑉𝑘𝜏𝑇,𝑦𝑘.𝑇(2.5) The Gronwall inequality yields 𝑥𝜏𝐸𝑉𝑘𝜏𝑇,𝑦𝑘𝑉𝑥𝑇0,𝑦0+𝐾2𝑇𝑒𝐾1𝑇.(2.6) Set Ω𝑘=𝜏𝑘𝑇 for 𝑘𝑘1; then 𝑃(Ω𝑘)𝜖. Note that, for every 𝜔Ω, there is 𝑥(𝜏𝑘,𝜔) or 𝑦(𝜏𝑘,𝜔) equal to either 𝑘 or 1/𝑘, and hence 𝑉(𝑥(𝜏𝑘,𝜔)) is no less than either 𝑘1ln𝑘(2.7) or 1𝑘11ln𝑘=1𝑘1+ln𝑘.(2.8) Therefore 𝑉𝑥𝜏𝑘𝜏𝑇,𝑦𝑘[]1𝑇𝑘1ln𝑘𝑘.1+ln𝑘(2.9) So 𝑉𝑥0,𝑦0+𝐾2𝑇𝑒𝐾1𝑇1𝐸Ω𝑘𝑉𝑥𝜏𝑘𝜏𝑇,𝑦𝑘[]1𝑇𝜖𝑘1ln𝑘𝑘,1+ln𝑘(2.10) where 1Ω𝑘 is the indicator function of Ω𝑘. Letting 𝑘 implies the contradiction 𝑉𝑥>0,𝑦0+𝐾𝑇=.(2.11) So we have that 𝜏= a.s. The proof is complete.

Theorem 2.1 shows that the solution of the SDE (1.7) will remain in the positive cone 𝑅2+ for any initial value (𝑥0,𝑦0)𝑅2+. The conclusion is fundamental which will be used later.

3. Asymptotic Behavior

3.1. Limit Results

To begin our discussion, we impose the following assumption:(H)𝑎𝑐/𝐴𝜎21/2>0,𝑓𝜎22/2>0.

And we list the interesting lemma as follows.

Lemma 3.1 (see [7, 8]). Consider one-dimensional stochastic differential equation []𝑑𝑥=𝑥𝑎𝑏𝑥𝑑𝑡+𝜎𝑥𝑑𝐵(𝑡),(3.1) where 𝑎,𝑏,and𝜎 are positive and 𝐵(𝑡) is standard Brownian motion. Under condition 𝑎>𝜎2/2, for any initial value 𝑥0>0, the solution 𝑥(𝑡) has the properties lim𝑡ln𝑥(𝑡)𝑡=0,𝑎.𝑠,lim𝑡1𝑡𝑡0𝑥(𝑠)𝑑𝑠=𝑎𝜎2/2𝑏,𝑎.𝑠.(3.2)

To demonstrate asymptotic properties of the stochastic system (1.7), we firstly discuss the long time behavior of ln𝑥(𝑡)/𝑡 and ln𝑦(𝑡)/𝑡.

On the one hand, by the comparison theorem of stochastic equations, it is obvious that []𝑑𝑥𝑥𝑎𝑏𝑥𝑑𝑡+𝜎1𝑥𝑑𝐵1(𝑡).(3.3)

Denote by 𝑋2(𝑡) the solution of the following stochastic equation: 𝑑𝑋2=𝑋2𝑎𝑏𝑋2𝑑𝑡+𝜎1𝑋2𝑑𝐵1𝑋(𝑡),2(0)=𝑥0.(3.4) We have that 𝑥(𝑡)𝑋2[(𝑡),𝑡0,+),a.s.(3.5) On the other hand, by the comparison theorem of stochastic equations, it is obvious that we denote by 𝑋1 the solution of stochastic differential equation 𝑑𝑋1=𝑋1𝑐𝑎𝐴𝑏𝑋1𝑑𝑡+𝜎1𝑋1𝑑𝐵1𝑋(𝑡),1(0)=𝑥0.(3.6) Consequently 𝑥(𝑡)𝑋1[(𝑡),𝑡0,+),a.s.(3.7) To sum up, we have that𝑋1(𝑡)𝑥(𝑡)𝑋2[(𝑡),𝑡0,+),a.s.(3.8) So we have the explicit solutions of 𝑋1(𝑡) and 𝑋2(𝑡) as follows: 𝑋1(𝑒𝑡)=[(𝑎𝑐/𝐴𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]1/𝑥0+𝑏𝑡0𝑒[(𝑎𝑐/𝐴𝜎21/2)𝑠+𝜎1𝐵1(𝑠)],𝑋𝑑𝑠(3.9)2𝑒(𝑡)=[(𝑎𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]1/𝑥0+𝑏𝑡0𝑒[(𝑎𝜎21/2)𝑠+𝜎1𝐵1(𝑠)].𝑑𝑠(3.10)

Theorem 3.2. Under assumption (H), for any initial value 𝑥0>0, the solutions 𝑋1(𝑡) and 𝑋2(𝑡) satisfy lim𝑡ln𝑋1(𝑡)𝑡=0,𝑎.𝑠.,lim𝑡ln𝑋2(𝑡)𝑡=0,𝑎.𝑠.(3.11)

Proof. By assumption (H) and Lemma 3.1, the assertion is straightforward.

Theorem 3.3. Under assumption (H), for any initial value 𝑥0>0, the solution 𝑥(𝑡) satisfies lim𝑡ln𝑥(𝑡)𝑡=0,𝑎.𝑠.(3.12)

Proof. By virtue of (3.8) and Theorem 3.2, we can imply the desired assertion.

Now let us continue to consider the asymptotic behavior of the species 𝑦(𝑡). By the comparison theorem of stochastic equations, we have that 𝑑𝑦(𝑡)𝑦(𝑡)𝑓𝑔𝑦(𝑡)𝜆𝑋2(𝑡)+𝑑𝑡+𝜎2𝑦(𝑡)𝑑𝐵2(𝑡).(3.13)

Denote by 𝑌2(𝑡) the solution of the stochastic equation as follows: 𝑑𝑌2=𝑌2𝑓𝑔𝑌2𝜆𝑋2(𝑡)+𝑑𝑡+𝜎2𝑌2𝑑𝐵2𝑌(𝑡),2(0)=𝑦0.(3.14) We have that𝑦(𝑡)𝑌2[(𝑡),𝑡0,+),a.s.(3.15) On the other hand, applying the comparison theorem again, denote by 𝑌1 the solution of stochastic equation 𝑑𝑌1=𝑌1𝑔𝑓𝑌1𝑑𝑡+𝜎2𝑌1𝑑𝐵2𝑌(𝑡),1(0)=𝑦0.(3.16) Consequently, 𝑦(𝑡)𝑌1[(𝑡),𝑡0,+),a.s.(3.17) To sum up, we have that 𝑌1(𝑡)𝑦(𝑡)𝑌2[(𝑡),𝑡0,+),a.s.(3.18) Moreover, 𝑌1(𝑡) and 𝑌2(𝑡) have the explicit solutions, respectively, 𝑌1(𝑒𝑡)=[(𝑓𝜎22/2)𝑡+𝜎2𝐵2(𝑡)]1/𝑦0+(𝑔/)𝑡0𝑒[(𝑓𝜎22/2)𝑠+𝜎2𝐵2(𝑠)],𝑌𝑑𝑠(3.19)2𝑒(𝑡)=[(𝑓𝜎22/2)𝑡+𝜎2𝐵2(𝑡)]1/𝑦0+𝑡0𝑔/𝜆𝑋2𝑒(𝑠)+[(𝑓𝜎22/2)𝑠+𝜎2𝐵2(𝑠)].𝑑𝑠(3.20)

Lemma 3.4. Under assumption (H), for any initial value 𝑦0>0, the solutions 𝑌1(𝑡) and 𝑌2(𝑡) satisfy lim𝑡ln𝑌1(𝑡)𝑡=0,𝑎.𝑠.,limsup𝑡ln𝑌2(𝑡)𝑡0,𝑎.𝑠.(3.21)

Proof. The proof is motivated by [7]. Obviously, Lemma 3.1 and assumption (H) yield lim𝑡ln𝑌1(𝑡)𝑡=0,a.s.(3.22)
On the other hand, it follows from (3.20) that 1𝑌2(𝑡)=𝑒[(𝑓(𝜎22/2))𝑡𝜎2𝐵2(𝑡)]1𝑦0+𝑡0𝑔𝜆𝑋2𝑒(𝑠)+[(𝑓𝜎22/2)𝑠+𝜎2𝐵2(𝑠)].𝑑𝑠(3.23) Choose 𝑇 satisfying 𝑒(𝑎(𝜎21/2))𝑡2 for 𝑡𝑇. Thus we have that 𝑒(𝑎𝜎21/2)𝑡/2𝑒(𝑎𝜎21/2)𝑡1 for 𝑡𝑇. Then for 𝑠𝑇, from (3.10), we obtain 𝑋2(𝑒𝑡)=[(𝑎𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]1/𝑥0+𝑏𝑡0𝑒[(𝑎𝜎21/2)𝑠+𝜎1𝐵1(𝑠)]𝑒𝑑𝑠[(𝑎𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]𝑏𝑡0𝑒[(𝑎𝜎21/2)𝑠+𝜎1𝐵1(𝑠)]𝑒𝑑𝑠[(𝑎𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]𝑏𝑒(𝜎1min0𝑠𝑡𝐵1(𝑠))𝑡0𝑒(𝑎𝜎21/2)𝑠=𝑑𝑠𝑎𝜎21/2𝑏𝑒[(𝑎𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]𝑒(𝜎1min0𝑠𝑡𝐵1(𝑠))𝑒(𝑎𝜎21/2)𝑡21𝑎𝜎21/2𝑏𝑒[(𝑎𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]𝑒(𝜎1min0𝑠𝑡𝐵1(𝑠))𝑒(𝑎𝜎21/2)𝑡=2𝑎𝜎21𝑏𝑒𝜎1(𝐵1(𝑡)min0𝑠𝑡𝐵1(𝑠)),𝑡𝑇𝑔𝜆𝑋2𝑒(𝑠)+[(𝑓𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]𝑑𝑠𝑡𝑇𝑔𝜆2𝑎𝜎21𝑒/𝑏𝜎1(𝐵1(𝑠)min0𝑢𝑠𝐵1(𝑢))+×𝑒[(𝑓𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]𝑑𝑠𝑏𝑔𝜆2𝑎𝜎21+𝑏𝑡𝑇𝑒𝜎1(𝐵1(𝑠)min0𝑢𝑠𝐵1(𝑢))×𝑒[(𝑓𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]𝑑𝑠𝑏𝑔𝜆2𝑎𝜎21𝑒+𝑏𝜎1[min0𝑠𝑡𝐵1(𝑠)max0𝑠𝑡𝐵1(𝑠)]+𝜎2min0𝑠𝑡𝐵2(𝑠)×𝑡𝑇𝑒(𝑓𝜎22/2)𝑠𝑑𝑠2𝑏𝑔𝜆2𝑎𝜎21+𝑏2𝑓𝜎22×𝑒(𝑓𝜎22/2)𝑡𝑒(𝑓𝜎22/2)𝑇×𝑒𝜎1[min0𝑠𝑡𝐵1(𝑠)max0𝑠𝑡𝐵1(𝑠)]+𝜎2min0𝑠𝑡𝐵2(𝑠).(3.24) Thus, 1𝑌2(𝑡)𝑒[(𝑓𝜎22/2)(𝑡𝑇)𝜎2(𝐵2(𝑡)𝐵2(𝑇))]×1+𝑒𝑦(𝑇)2𝑏𝑔(𝑓𝜎22/2)𝑡𝑒(𝑓𝜎22/2)𝑇𝜆2𝑎𝜎21+𝑏2𝑓𝜎22𝑒𝜎1[min0𝑠𝑡𝐵1(𝑠)max0𝑠𝑡𝐵1(𝑠)]+𝜎2min0𝑠𝑡𝐵2(𝑠)2𝑏𝑔𝑒(𝑓𝜎22/2)𝑇𝜎2𝐵2(𝑇)1𝑒(𝑓𝜎22/2)(𝑡𝑇)𝜆2𝑎𝜎21+𝑏2𝑓𝜎22×𝑒𝜎1[min0𝑠𝑡𝐵1(𝑠)max0𝑠𝑡𝐵1(𝑠)]+𝜎2[min0𝑠𝑡𝐵2(𝑠)max0𝑠𝑡𝐵2(𝑠)]=𝐾(𝑡)𝑒𝜎1[min0𝑠𝑡𝐵1(𝑠)max0𝑠𝑡𝐵1(𝑠)]+𝜎2[min0𝑠𝑡𝐵2(𝑠)max0𝑠𝑡𝐵2(𝑠)],(3.25) where 𝐾(𝑡)=2𝑏𝑔𝑒(𝑓𝜎22/2)𝑇𝜎2𝐵2(𝑇)(1𝑒(𝑓𝜎22/2)(𝑡𝑇))/(𝜆(2𝑎𝜎21)+𝑏)(2𝑓𝜎22). So we derive ln𝑌2(𝑡)ln𝐾(𝑡)+𝜎1min0𝑠𝑡𝐵1(𝑠)max0𝑠𝑡𝐵1(𝑠)+𝜎2min0𝑠𝑡𝐵2(𝑠)max0𝑠𝑡𝐵2(𝑠).(3.26) Dividing 𝑡 on both sides yields ln𝑌2(𝑡)𝑡ln𝐾(𝑡)𝑡𝜎1min0𝑠𝑡𝐵1(𝑠)𝑡max0𝑠𝑡𝐵1(𝑠)𝑡𝜎2min0𝑠𝑡𝐵2(𝑠)𝑡max0𝑠𝑡𝐵2(𝑠)𝑡.(3.27) The distributions of max0𝑠𝑡𝐵1(𝑠) and max0𝑠𝑡𝐵2(𝑠) are that same as |𝐵1(𝑡)| and |𝐵2(𝑡)|, respectively, and min0𝑠𝑡𝐵1(𝑠) and min0𝑠𝑡𝐵2(𝑠) have the same distributions as max0𝑠𝑡𝐵1(𝑠) and max0𝑠𝑡𝐵1(𝑠), respectively.
From the representation of 𝐾(𝑡), we can simplify it as follows: 𝐾(𝑡)=𝐴1𝑒𝐴2𝐵2(𝑇)1𝐴3𝑒𝐴4𝑡.(3.28) By assumption (H), constants 𝐴𝑖(𝑖=1,2,3,4) satisfy 𝐴1>0,𝐴2>0,𝐴3>0,and𝐴4<0. Then, ln𝐾(𝑡)=ln𝐴1+𝐴2𝐵2(𝑇)+ln1𝐴3𝑒𝐴4𝑡.(3.29) It follows from (ln𝐵2(𝑡)/𝑡)0,𝑡, that ln𝐾(𝑡)𝑡0,𝑡.(3.30) Hence, letting 𝑡 and by the strong law of large numbers, we have that min0𝑠𝑡𝐵1(𝑠)𝑡0,max0𝑠𝑡𝐵1(𝑠)𝑡0,𝑡,min0𝑠𝑡𝐵2(𝑠)𝑡0,max0𝑠𝑡𝐵2(𝑠)𝑡0,𝑡.(3.31) Then, limsup𝑡ln𝑌2(𝑡)𝑡0,a.s.,(3.32) as desired.

Theorem 3.5. Under assumption (H), for any initial value 𝑦0>0, the solution 𝑦(𝑡) of (1.7) has the property lim𝑡ln𝑦(𝑡)𝑡=0,𝑎.𝑠.(3.33)

Proof. It follows from (3.18) and Lemma 3.4 that 0liminf𝑡ln𝑌1(𝑡)𝑡liminf𝑡ln𝑦(𝑡)𝑡limsup𝑡ln𝑦(𝑡)𝑡limsup𝑡ln𝑌2(𝑡)𝑡0,a.s.(3.34) Consequently, lim𝑡ln𝑦(𝑡)𝑡=0,a.s.(3.35) The proof is complete.

3.2. Persistent in Mean and Extinction

As we know, the property of persistence is more desirable since it represents the long-term survival to a population dynamics. Now we present the definition of persistence in mean proposed in [7, 11].

Definition 3.6. System (1.7) is said to be persistent in mean if liminf𝑡𝑡0𝑥(𝑡)𝑑𝑠𝑡>0,liminf𝑡𝑡0𝑦(𝑡)𝑑𝑠𝑡>0,a.s.(3.36)

Theorem 3.7. Assume that condition (H) holds. Then system (1.7) is persistent in mean.

Proof. Define the function 𝑉=ln𝑥; by the Itô formula, we get ln𝑥(𝑡)ln𝑥0=𝜎𝑎212𝑡𝑏𝑡0𝑥(𝑠)𝑑𝑠𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎1𝐵1(𝑡).(3.37) That is, 𝑏𝑡0𝑥(𝑠)𝑑𝑠=ln𝑥(𝑡)+ln𝑥0+𝜎𝑎212𝑡𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎1𝐵1(𝑡)ln𝑥(𝑡)+ln𝑥0+𝜎𝑎212𝑡𝑐𝑡𝐴+𝜎1𝐵1(𝑡).(3.38) Dividing 𝑡 on both sides and using the strong law of large numbers, it follows from Theorem 3.3 that liminf𝑡𝑡0𝑥(𝑡)𝑑𝑠𝑡𝑎𝑐/𝐴𝜎21/2𝑏>0,a.s.(3.39)
Moreover, define the function 𝑉=ln𝑦; using the Itô formula again, we have that ln𝑦(𝑡)ln𝑦0=𝜎𝑓222𝑡𝑡0𝑔𝑦(𝑠)𝜆𝑥(𝑠)+𝑑𝑠+𝜎2𝐵2(𝑡).(3.40) Thus, 𝑔𝑡0𝑦(𝑠)𝑑𝑠𝑡0𝑔𝑦(𝑠)𝜆𝑥(𝑠)+𝑑𝑠=ln𝑦(𝑡)+ln𝑦0+𝜎𝑓222𝑡+𝜎2𝐵2(𝑡).(3.41) Dividing both sides by 𝑡 and letting 𝑡 and also by the strong law of large numbers and Theorem 3.5, we have that liminf𝑡1𝑡𝑡0𝑦(𝑠)𝑑𝑠𝑓𝜎22/2𝑔>0,a.s.(3.42) So the system is persistent in mean and we complete the proof.

Under condition (H), we show that the system is persistent in mean. To a large extent, (H) is the condition that stands for small environmental noises. That is, small stochastic perturbation does not change the persistence of the system. Here, we will consider that large noises may make the system extinct.

Theorem 3.8. Assume that condition 𝑎𝜎21/2<0,𝑓𝜎22/2<0 holds. Then system (1.7) will become extinct exponentially with probability one.

Proof. Define the function 𝑉=ln𝑥; by the Itô formula, we get ln𝑥(𝑡)ln𝑥0=𝜎𝑎212𝑡𝑏𝑡0𝑥(𝑠)𝑑𝑠𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎1𝐵1(𝑡).(3.43) Then, ln𝑥(𝑡)ln𝑥0𝜎𝑎212𝑡+𝜎1𝐵1(𝑡).(3.44) By the strong law of large numbers of martingales, we have that lim𝑡𝐵1(𝑡)𝑡=0,a.s.(3.45) Therefore, limsup𝑡ln𝑥(𝑡)𝑡𝜎𝑎212<0,a.s.(3.46)
On the other hand, by the Itô formula again, we derive ln𝑦(𝑡)=ln𝑦0+𝜎𝑓222𝑡𝑡0𝑔𝑦(𝑠)𝜆𝑥(𝑠)+𝑑𝑠+𝜎2𝐵2(𝑡)ln𝑦0+𝜎𝑓222𝑡+𝜎2𝐵2(𝑡).(3.47) Applying the strong law of large numbers of martingales, we obtain limsup𝑡ln𝑦(𝑡)𝑡𝜎𝑓222<0,a.s.(3.48) The proof is complete.

We continue to discuss the asymptotic behaviors of the stochastic system (1.7).

Theorem 3.9. Assume that condition 𝑎𝑐/𝐴𝜎21/2>0,𝑓𝜎22/2<0 holds. Then the prey 𝑥(𝑡) of system (1.7) is persistent in mean; however, the predator 𝑦(𝑡) will become extinct exponentially with probability one.

Proof. Define the function 𝑉=ln𝑥; by the Itô formula, we get ln𝑥(𝑡)ln𝑥0=𝜎𝑎212𝑡𝑏𝑡0𝑥(𝑠)𝑑𝑠𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎1𝐵1(𝑡).(3.49) Thus, 𝑏𝑡0𝑥(𝑠)𝑑𝑠ln𝑥(𝑡)+ln𝑥0+𝜎𝑎212𝑐𝐴𝑡+𝜎1𝐵1(𝑡).(3.50) Under condition 𝑎𝑐/𝐴𝜎21/2>0, it follows from the proof of Theorem 3.3 that lim𝑡ln𝑥(𝑡)𝑡=0,a.s.(3.51) So liminf𝑡𝑡0𝑥(𝑡)𝑑𝑠𝑡𝑎𝑐/𝐴𝜎21/2𝑏>0,a.s.(3.52) That is, the prey 𝑥(𝑡) is persistent in mean. However, under condition 𝑓𝜎22/2<0, from the proof of Theorem 3.8, we have that limsup𝑡ln𝑦(𝑡)𝑡𝜎𝑓222<0,a.s.(3.53) That is, the predator 𝑦(𝑡) will become extinct exponentially with probability one.

4. Numerical Simulations

In this section, some simulation figures are introduced to support the main results in our paper.

For model (1.7), we consider the discretization equations 𝑥𝑘+1=𝑥𝑘+𝑥𝑘𝑎𝑏𝑥𝑘𝑐𝑦𝑘𝜆𝑥𝑘+𝐴𝑦𝑘Δ𝑡+𝜎1𝑥𝑘Δ𝑡𝜉𝑘+𝜎212𝑥𝑘𝜉2𝑘𝑦1Δ𝑡,𝑘+1=𝑦𝑘+𝑦𝑘𝑓𝑔𝑦𝑘𝜆𝑥𝑘+Δ𝑡+𝜎2𝑦𝑘Δ𝑡𝜂𝑘+𝜎222𝑦𝑘𝜂2𝑘1Δ𝑡,(4.1) where 𝜉𝑘 and 𝜂𝑘 are Gaussian random variables that follow 𝑁(0,1).

In Figure 1, we choose 𝑎=0.4, 𝑓=0.3, 𝑐=0.1, 𝐴=0.5, 𝜎1(𝑡)2/2=𝜎22(𝑡)/2=0.01, and (𝑥0,𝑦0)=(0.5,0.2). By virtue of Theorem 3.7, the system will be persistent in mean. What we mentioned above can be seen from Figure 1. The difference between the conditions of Figure 1 and Figure 2 is that the values of 𝜎1 and 𝜎2 are different. In Figure 1, we choose 𝜎21/2=𝜎22/2=0.01. In Figure 2, we choose 𝜎21/2=𝜎22/2=1. In view of Theorem 3.8, both species 𝑥 and 𝑦 will go to extinction. Figure 2 confirms this.

518719.fig.001
Figure 1
518719.fig.002
Figure 2

In Figure 3, we choose 𝑎=0.4, 𝑓=0.3, 𝑐=0.1, 𝐴=0.5, 𝜎1(𝑡)2/2=0.01,𝜎22/2=1, and (𝑥0,𝑦0)=(0.5,0.2). Then the prey 𝑥(𝑡) is persistent in mean; however, the predator 𝑦(𝑡) will become extinct. Figure 3 confirms the assertion (Theorem 3.9).

518719.fig.003
Figure 3

By comparing Figures 1 and 2, with Figure 3, we can observe that small environmental noise can retain the stochastic system permanent; however, sufficiently large environmental noise makes the stochastic system extinct.

Remark 4.1. White noise is taken into account in our model in this paper. It tells us that, when the intensities of environmental noises are not too big, some nice properties such as nonexplosion and permanence are desired. However, Theorem 3.8 reveals that a large white noise will force the population to become extinct while the population may be persistent under a relatively small white noise. To some extent, Theorem 3.9 shows that, though the predator 𝑦(𝑡) has plenty of food 𝑥(𝑡), they may be extinct because of large environmental noise.

Acknowledgment

This research is supported by China Postdoctoral Science Foundation (no. 20100481000).

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