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

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: 𝑑𝑥𝑑𝑡𝑑=𝑦𝑥𝑎−𝑏𝑥−𝑥+𝑐𝑦𝑘₁,𝑑𝑡=𝑦𝑟−𝑥+𝑓𝑦𝑘₂,(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: 𝑑𝑥𝑑𝑑𝑡𝑦=𝑟𝑥1−𝑥𝑘−𝑦𝑝(𝑥),𝑥𝑑𝑡=𝑦𝑠1−ℎ𝑥𝑦₀,>0,𝑦₀>0,𝑟,𝑠,𝑘,ℎ>0.(1.2)

Recently, [6] discussed the following model with modified Leslie-Gower response: 𝑑𝑥𝑑𝑑𝑡𝑦=𝑟𝑥1−𝑥𝑘−𝜆𝑥𝑛𝜆+𝑥𝐴𝑦𝑦,𝑑𝑡=𝑥𝑦𝑠1−𝜆𝑥ℎ+𝑦𝑏₀,>0,𝑦₀>0,(1.3) where 𝑟,𝑘,𝑛,𝐴,𝑠,ℎ,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=𝑑𝑦𝑥𝑎−𝑏𝑥−𝜆𝑥𝑐+𝑦𝐴𝑦,𝑑𝑡𝑥=𝑦𝑓−𝜆𝑥𝑔+𝑦ℎ₀,>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=𝑥𝑎−𝑏𝑥−𝑐𝑦𝑥+𝑚𝑑𝑡+𝜎¹𝑑𝑦=𝑦𝑟−𝑓𝑦𝑥𝑑𝐵₁(𝑡),𝑥+𝑚𝑑𝑡−𝜎²𝑦𝑑𝐵₂(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=𝑥𝑎−𝑏𝑥−𝑐𝑦𝑥+𝑚𝑦𝑑𝑡+𝜎¹𝑑𝑦=𝑦−𝑔+𝑓𝑥𝑥𝑑𝐵₁(𝑡),𝑥+𝑚𝑦𝑑𝑡−𝜎²𝑦𝑑𝐵₂(𝑡),(1.6) where 𝐵_𝑖(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)𝑎−𝑏𝑥(𝑡)−𝑐𝑦(𝑡)𝜆𝑥(𝑡)+𝐴𝑦(𝑡)𝑑𝑡+𝜎¹𝑑𝑦(𝑡)=𝑦(𝑡)𝑓−𝑔𝑦(𝑡)𝑥(𝑡)𝑑𝐵₁(𝑡),𝜆𝑥(𝑡)+ℎ𝑑𝑡+𝜎²𝑦(𝑡)𝑑𝐵₂(𝑡),(1.7) where 𝐵_𝑖(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 𝑡≥0, and the solution will remain in 𝑅²⁺ with probability one, namely, 𝑋(𝑡)∈𝑅²⁺ for all 𝑡≥0 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,𝜏_𝑒), where 𝜏_𝑒 is the explosion time. To show that this solution is global, we need to show that 𝜏_𝑒=+∞ 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 𝑘≥𝑘₀, define the stopping time 𝜏_𝑚=inf𝑡∈0,𝜏_𝑒∶𝑥(𝑡)∉1𝑘,𝑘or𝑦(𝑡)∉𝑘1,𝑘,(2.1) where throughout this paper we set inf∅=∞. Obviously, 𝜏_𝑘 is increasing as 𝑘→∞. Let 𝜏_∞=lim_𝑘→∞𝜏_𝑘, whence 𝜏_∞≤𝜏_𝑒 a.s. If we can show that 𝜏_∞=∞ a.s., then 𝜏_𝑒=∞ a.s. and 𝑋(𝑡)∈𝑅²⁺ a.s. for all 𝑡≥0. So we just prove that 𝜏_∞=∞ a.s. If not, there is 𝜖∈(0,1) and 𝑇>0 such that 𝑃𝜏_∞≤𝑇>𝜖.(2.2) Hence, there is an integer 𝑘₁≥𝑘₀ such that 𝑃{𝜏_𝑘≤𝑇}≥𝜖 for all 𝑘≥𝑘₁. Define a function 𝑉∶𝑅²⁺→𝑅₊ by 𝑉(𝑥,𝑦)=(𝑥−1−ln𝑥)+(𝑦−1−ln𝑦). The nonnegativity of this function can be seen from 𝑢−1−ln𝑢≥0,on𝑢>0.(2.3) If 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡))∈𝑅²⁺, by virtue of 𝑢≤2[𝑢−1−ln𝑢]+2 on 𝑢>0, we obtain 𝐿𝑉(𝑥,𝑦)=(𝑥−1)𝑎−𝑏𝑥−𝑐𝑦𝜆𝑥+𝐴𝑦+(𝑦−1)𝑓−𝜆𝑥𝑔+𝑦ℎ+𝜎²¹+𝜎²²=𝑎𝑥−𝑏𝑥₂2−𝑐𝑥𝑦𝜆𝑥+𝐴𝑦−𝑎+𝑏𝑥+𝜆𝑥𝑐+𝑦𝐴𝑦+𝑓𝑦−𝑓−𝜆𝑥𝑔𝑦+²ℎ+𝜆𝑥𝑔+𝑦ℎ+𝜎²¹+𝜎²²≤𝐾₁2𝑉(𝑥,𝑦)+𝐾₂,(2.4) dropping 𝑡 from 𝑥(𝑡) and 𝑦(𝑡). Making use of the Itô formula yields 𝐸𝑉𝑥𝜏_𝑘∧𝑇,𝑦𝜏_𝑘∧𝑇≤𝑉𝑥₀,𝑦₀+𝐾₂𝑇+𝐾₁^𝑇0𝐸𝑉𝑥𝜏_𝑘∧𝑇,𝑦𝜏_𝑘∧𝑇.(2.5) The Gronwall inequality yields 𝐸𝑉𝑥𝜏_𝑘∧𝑇,𝑦𝜏_𝑘∧𝑇≤𝑉𝑥₀,𝑦₀+𝐾₂𝑇𝑒^𝐾1𝑇.(2.6) Set Ω_𝑘=𝜏_𝑘≤𝑇 for 𝑘≥𝑘₁; then 𝑃(Ω_𝑘)≥𝜖. Note that, for every 𝜔∈Ω, there is 𝑥(𝜏_𝑘,𝜔) or 𝑦(𝜏_𝑘,𝜔) equal to either 𝑘 or 1/𝑘, and hence 𝑉(𝑥(𝜏_𝑘,𝜔)) is no less than either 𝑘−1−ln𝑘(2.7) or 1𝑘−1−ln𝑘1=𝑘1−1+ln𝑘.(2.8) Therefore 𝑉𝑥𝜏_𝑘∧𝑇,𝑦𝜏_𝑘∧𝑇≥[𝑘−1−ln𝑘]∧1𝑘−1+ln𝑘.(2.9) So 𝑉𝑥₀,𝑦₀+𝐾₂𝑇𝑒^𝐾1𝑇≥≥𝐸𝜖[1𝑘−_Ω𝑘𝑉1−𝑥ln𝜏𝑘_𝑘]∧∧𝑇1,𝑦𝜏_𝑘∧𝑇𝑘−1+ln𝑘,(2.10) where 1_Ω𝑘 is the indicator function of Ω_𝑘. Letting 𝑘→∞ implies the contradiction ∞>𝑉𝑥₀,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)𝑎−𝑐/𝐴−𝜎²¹/2>0,𝑓−𝜎²²/2>0.

And we list the interesting lemma as follows.

Lemma 3.1 (see [7, 8]). Consider one-dimensional stochastic differential equation 𝑑𝑥=𝑥[𝑎−𝑏𝑥]𝑑𝑡+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 lim_𝑡→∞ln𝑥(𝑡)lim_𝑡→∞𝑡=0,𝑎.𝑠,1𝑡^𝑡0𝑥(𝑠)𝑑𝑠=𝑎−𝜎²/2𝑏,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 𝑑𝑥≤𝑥[𝑎−𝑏𝑥]𝑑𝑡+𝜎₁𝑥𝑑𝐵₁(𝑡).(3.3)

Denote by 𝑋₂(𝑡) the solution of the following stochastic equation: 𝑑𝑋₂=𝑋₂𝑎−𝑋𝑏𝑋₂𝑑𝑡+𝜎₁𝑋₂𝑑𝐵₁(𝑡),₂(0)=𝑥₀.(3.4) We have that 𝑥(𝑡)≤𝑋₂(𝑡),𝑡∈[0,+∞),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=𝑋₁𝑎−𝑐𝐴−𝑏𝑋¹𝑑𝑡+𝜎¹𝑋𝑋₁𝑑𝐵₁(𝑡),₁(0)=𝑥₀.(3.6) Consequently 𝑥(𝑡)≥𝑋₁(𝑡),𝑡∈[0,+∞),a.s.(3.7) To sum up, we have that𝑋₁(𝑡)≤𝑥(𝑡)≤𝑋₂(𝑡),𝑡∈[0,+∞),a.s.(3.8) So we have the explicit solutions of 𝑋₁(𝑡) and 𝑋₂(𝑡) as follows: 𝑋₁(𝑡)=𝑒^{[(𝑎−𝑐/𝐴−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]}𝑋1/𝑥₀+𝑏∫^𝑡0𝑒^{[(𝑎−𝑐/𝐴−𝜎21/2)𝑠+𝜎1𝐵1(𝑠)]}𝑑𝑠,(3.9)₂(𝑡)=𝑒^{[(𝑎−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]}1/𝑥₀+𝑏∫^𝑡0𝑒^{[(𝑎−𝜎21/2)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 lim_𝑡→∞ln𝑋₁(𝑡)lim_𝑡→∞𝑡=0,𝑎.𝑠.,ln𝑋₂(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 lim_𝑡→∞ln𝑥(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)𝑓−𝑔𝑦(𝑡)𝜆𝑋₂(𝑡)+ℎ𝑑𝑡+𝜎²𝑦(𝑡)𝑑𝐵₂(𝑡).(3.13)

Denote by 𝑌₂(𝑡) the solution of the stochastic equation as follows: 𝑑𝑌₂=𝑌₂𝑓−𝜆𝑋₂𝑔(𝑌𝑡)²+ℎ𝑑𝑡+𝜎²𝑌𝑌₂𝑑𝐵₂(𝑡),₂(0)=𝑦₀.(3.14) We have that𝑦(𝑡)≤𝑌₂(𝑡),𝑡∈[0,+∞),a.s.(3.15) On the other hand, applying the comparison theorem again, denote by 𝑌₁ the solution of stochastic equation 𝑑𝑌₁=𝑌₁𝑓−𝑔ℎ𝑌¹𝑑𝑡+𝜎²𝑌𝑌₁𝑑𝐵₂(𝑡),₁(0)=𝑦₀.(3.16) Consequently, 𝑦(𝑡)≥𝑌₁(𝑡),𝑡∈[0,+∞),a.s.(3.17) To sum up, we have that 𝑌₁(𝑡)≤𝑦(𝑡)≤𝑌₂(𝑡),𝑡∈[0,+∞),a.s.(3.18) Moreover, 𝑌₁(𝑡) and 𝑌₂(𝑡) have the explicit solutions, respectively, 𝑌₁(𝑡)=𝑒^{[(𝑓−𝜎22/2)𝑡+𝜎2𝐵2(𝑡)]}𝑌1/𝑦₀+(𝑔/ℎ)∫^𝑡0𝑒^{[(𝑓−𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]}𝑑𝑠,(3.19)₂(𝑡)=𝑒^{[(𝑓−𝜎22/2)𝑡+𝜎2𝐵2(𝑡)]}1/𝑦₀+∫^𝑡0𝑔/𝜆𝑋₂(𝑠)+ℎ𝑒^{[(𝑓−𝜎22/2)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 lim_𝑡→∞ln𝑌₁(𝑡)limsup_𝑡→∞𝑡=0,𝑎.𝑠.,ln𝑌₂(𝑡)𝑡≤0,𝑎.𝑠.(3.21)

Proof. The proof is motivated by [7]. Obviously, Lemma 3.1 and assumption (H) yield lim_𝑡→∞ln𝑌₁(𝑡)𝑡=0,a.s.(3.22)
On the other hand, it follows from (3.20) that 1𝑌₂(𝑡)=𝑒^{[−(𝑓−(𝜎22/2))𝑡−𝜎2𝐵2(𝑡)]}𝑦1₀+^𝑡0𝑔𝜆𝑋₂(𝑠)+ℎ𝑒^{[(𝑓−𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]}𝑑𝑠.(3.23) Choose 𝑇 satisfying 𝑒^{(𝑎−(𝜎21/2))𝑡}≥2 for 𝑡≥𝑇. Thus we have that 𝑒^{(𝑎−𝜎21/2)𝑡}/2≤𝑒^{(𝑎−𝜎21/2)𝑡}−1 for 𝑡≥𝑇. Then for 𝑠≥𝑇, from (3.10), we obtain 𝑋₂(𝑡)=𝑒^{[(𝑎−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]}≤1/𝑥₀𝑒+𝑏∫^𝑡0𝑒^{[(𝑎−𝜎21/2)𝑠+𝜎1𝐵1(𝑠)]}𝑑𝑠_{[(𝑎−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]}≤𝑏∫^𝑡0𝑒𝑒^{[(𝑎−𝜎21/2)𝑠+𝜎1𝐵1(𝑠)]}𝑑𝑠_{[(𝑎−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]}=𝑎𝑏𝑒−𝜎^{(𝜎1min0≤𝑠≤𝑡𝐵1(𝑠))}₂₁∫^𝑡0𝑒^{(𝑎−𝜎21/2)𝑠}𝑑𝑠/2𝑏𝑒𝑒^{[(𝑎−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)]}_{(𝜎1min0≤𝑠≤𝑡𝐵1(𝑠))}≤2𝑎−𝜎₂₁𝑒^{(𝑎−𝜎21/2)𝑡}−1/2𝑏𝑒𝑒^(𝜎₁^min_{0≤𝑠≤𝑡}^𝐵₁^(𝑠))𝑒^{[(𝑎−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)](𝑎−𝜎21/2)𝑡}=2𝑎−𝜎²¹𝑏𝑒^{𝜎1(𝐵1(𝑡)−min0≤𝑠≤𝑡𝐵1(𝑠))}_𝑡𝑇,𝑔𝜆𝑋₂(𝑠)+ℎ𝑒^{[(𝑓−𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]}𝑑𝑠≥^𝑡𝑇𝑔×𝑒𝜆2𝑎−𝜎²¹_{[(𝑓−𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]}/𝑏𝑒^{𝜎1(𝐵1(𝑠)−min0≤𝑢≤𝑠𝐵1(𝑢))}+ℎ≥𝑏𝑔𝑑𝑠𝜆2𝑎−𝜎²¹+𝑏ℎ^𝑡𝑇𝑒^{−𝜎1(𝐵1(𝑠)−min0≤𝑢≤𝑠𝐵1(𝑢))}≥×𝑒𝑏𝑔^{[(𝑓−𝜎22/2)𝑠+𝜎2𝐵2(𝑠)]}𝑑𝑠𝜆2𝑎−𝜎²¹+𝑏ℎ𝑒^{𝜎1[min0≤𝑠≤𝑡𝐵1(𝑠)−max0≤𝑠≤𝑡𝐵1(𝑠)]+𝜎2min0≤𝑠≤𝑡𝐵2(𝑠)}≥×^𝑡𝑇𝑒^{(𝑓−𝜎22/2)𝑠}𝑑2𝑠𝑏𝑔×𝜆𝑒2𝑎−𝜎²¹_{(𝑓−𝜎22/2)𝑡}+𝑏ℎ2𝑓−𝜎²²×𝑒−𝑒^{(𝑓−𝜎22/2)𝑇}_{𝜎1[min0≤𝑠≤𝑡𝐵1(𝑠)−max0≤𝑠≤𝑡𝐵1(𝑠)]+𝜎2min0≤𝑠≤𝑡𝐵2(𝑠)}.(3.24) Thus, 1𝑌₂(𝑡)≥𝑒^{[−(𝑓−𝜎22/2)(𝑡−𝑇)−𝜎2(𝐵2(𝑡)−𝐵2(𝑇))]}×⎡⎢⎢⎣1𝑦(𝑇)+2𝑏𝑔𝑒^{(𝑓−𝜎22/2)𝑡}−𝑒^{(𝑓−𝜎22/2)𝑇}𝜆2𝑎−𝜎²¹+𝑏ℎ2𝑓−𝜎²²𝑒^{𝜎1[min0≤𝑠≤𝑡𝐵1(𝑠)−max0≤𝑠≤𝑡𝐵1(𝑠)]+𝜎2min0≤𝑠≤𝑡𝐵2(𝑠)}≥2𝑏𝑔𝑒_{(𝑓−𝜎22/2)𝑇𝜎2𝐵2(𝑇)}⎤⎥⎥⎦1−𝑒^{−(𝑓−𝜎22/2)(𝑡−𝑇)}×𝑒𝜆2𝑎−𝜎²¹+𝑏ℎ2𝑓−𝜎²²_{𝜎1[min0≤𝑠≤𝑡𝐵1(𝑠)−max0≤𝑠≤𝑡𝐵1(𝑠)]+𝜎2[min0≤𝑠≤𝑡𝐵2(𝑠)−max0≤𝑠≤𝑡𝐵2(𝑠)]}∶=𝐾(𝑡)𝑒^{𝜎1[min0≤𝑠≤𝑡𝐵1(𝑠)−max0≤𝑠≤𝑡𝐵1(𝑠)]+𝜎2[min0≤𝑠≤𝑡𝐵2(𝑠)−max0≤𝑠≤𝑡𝐵2(𝑠)]},(3.25) where 𝐾(𝑡)=2𝑏𝑔𝑒^{(𝑓−𝜎22/2)𝑇𝜎2𝐵2(𝑇)}(1−𝑒^{−(𝑓−𝜎22/2)(𝑡−𝑇)})/(𝜆(2𝑎−𝜎²¹)+𝑏ℎ)(2𝑓−𝜎²²). So we derive −ln𝑌₂(𝑡)≥ln𝐾(𝑡)+𝜎₁min_{0≤𝑠≤𝑡}𝐵₁(𝑠)−max_{0≤𝑠≤𝑡}𝐵₁(𝑠)+𝜎₂min_{0≤𝑠≤𝑡}𝐵₂(𝑠)−max_{0≤𝑠≤𝑡}𝐵₂(𝑠).(3.26) Dividing 𝑡 on both sides yields ln𝑌₂(𝑡)𝑡≤−ln𝐾𝑡(𝑡)−𝜎¹min^{0≤𝑠≤𝑡}𝐵₁(𝑠)𝑡−max^{0≤𝑠≤𝑡}𝐵₁(𝑠)−𝜎₂𝑡min^{0≤𝑠≤𝑡}𝐵₂(𝑠)𝑡−max^{0≤𝑠≤𝑡}𝐵₂(𝑠)𝑡.(3.27) The distributions of max_{0≤𝑠≤𝑡}𝐵₁(𝑠) and max_{0≤𝑠≤𝑡}𝐵₂(𝑠) are that same as |𝐵₁(𝑡)| and |𝐵₂(𝑡)|, respectively, and min_{0≤𝑠≤𝑡}𝐵₁(𝑠) and min_{0≤𝑠≤𝑡}𝐵₂(𝑠) have the same distributions as −max_{0≤𝑠≤𝑡}𝐵₁(𝑠) and −max_{0≤𝑠≤𝑡}𝐵₁(𝑠), respectively.
From the representation of 𝐾(𝑡), we can simplify it as follows: 𝐾(𝑡)=𝐴₁𝑒^{𝐴2𝐵2(𝑇)}1−𝐴₃𝑒^𝐴4𝑡.(3.28) By assumption (H), constants 𝐴_𝑖(𝑖=1,2,3,4) satisfy 𝐴₁>0,𝐴₂>0,𝐴₃>0,and𝐴₄<0. Then, ln𝐾(𝑡)=ln𝐴₁+𝐴₂𝐵₂(𝑇)+ln1−𝐴₃𝑒^𝐴4𝑡.(3.29) It follows from (ln𝐵₂(𝑡)/𝑡)→0,𝑡→∞, that ln𝐾(𝑡)𝑡⟶0,𝑡⟶∞.(3.30) Hence, letting 𝑡→∞ and by the strong law of large numbers, we have that min_{0≤𝑠≤𝑡}𝐵₁(𝑠)𝑡⟶0,max^{0≤𝑠≤𝑡}𝐵₁(𝑠)min_{0≤𝑠≤𝑡}𝑡⟶0,𝑡⟶∞,𝐵₂(𝑠)𝑡⟶0,max^{0≤𝑠≤𝑡}𝐵₂(𝑠)𝑡⟶0,𝑡⟶∞.(3.31) Then, limsup_𝑡→∞ln𝑌₂(𝑡)𝑡≤0,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 lim_𝑡→∞ln𝑦(𝑡)𝑡=0,𝑎.𝑠.(3.33)

Proof. It follows from (3.18) and Lemma 3.4 that 0≤liminf_𝑡→∞ln𝑌₁(𝑡)𝑡≤liminf^𝑡→∞ln𝑦(𝑡)𝑡≤limsup_𝑡→∞ln𝑦(𝑡)𝑡≤limsup_𝑡→∞ln𝑌₂(𝑡)𝑡≤0,a.s.(3.34) Consequently, lim_𝑡→∞ln𝑦(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 liminf_𝑡→∞∫^𝑡0𝑥(𝑡)𝑑𝑠𝑡>0,liminf^𝑡→∞∫^𝑡0𝑦(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=𝑎−𝜎²¹2𝑡−𝑏^𝑡0𝑥(𝑠)𝑑𝑠−^𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎¹𝐵₁(𝑡).(3.37) That is, 𝑏^𝑡0𝑥(𝑠)𝑑𝑠=−ln𝑥(𝑡)+ln𝑥₀+𝑎−𝜎²¹2𝑡−^𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎¹≥−ln𝑥(𝑡)+ln𝑥₀𝐵₁(𝑡)+𝑎−𝜎²¹2𝑡−𝑐𝐴𝑡+𝜎¹𝐵₁(𝑡).(3.38) Dividing 𝑡 on both sides and using the strong law of large numbers, it follows from Theorem 3.3 that liminf_𝑡→∞∫^𝑡0𝑥(𝑡)𝑑𝑠𝑡≥𝑎−𝑐/𝐴−𝜎²¹/2𝑏>0,a.s.(3.39)
Moreover, define the function 𝑉=ln𝑦; using the Itô formula again, we have that ln𝑦(𝑡)−ln𝑦₀=𝑓−𝜎²²2𝑡−^𝑡0𝑔𝑦(𝑠)𝜆𝑥(𝑠)+ℎ𝑑𝑠+𝜎²𝐵₂(𝑡).(3.40) Thus, 𝑔ℎ^𝑡0𝑦(𝑠)𝑑𝑠≥^𝑡0𝑔𝑦(𝑠)𝜆𝑥(𝑠)+ℎ𝑑𝑠=−ln𝑦(𝑡)+ln𝑦⁰+𝑓−𝜎²²2𝑡+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 liminf_𝑡→∞1𝑡^𝑡0𝑦(𝑠)𝑑𝑠≥ℎ𝑓−𝜎²²/2𝑔>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 𝑎−𝜎²¹/2<0,𝑓−𝜎²²/2<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=𝑎−𝜎²¹2𝑡−𝑏^𝑡0𝑥(𝑠)𝑑𝑠−^𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎¹𝐵₁(𝑡).(3.43) Then, ln𝑥(𝑡)−ln𝑥₀≤𝑎−𝜎²¹2𝑡+𝜎¹𝐵₁(𝑡).(3.44) By the strong law of large numbers of martingales, we have that lim_𝑡→∞𝐵₁(𝑡)𝑡=0,a.s.(3.45) Therefore, limsup_𝑡→∞ln𝑥(𝑡)𝑡≤𝑎−𝜎2²¹<0,a.s.(3.46)
On the other hand, by the Itô formula again, we derive ln𝑦(𝑡)=ln𝑦₀+𝑓−𝜎²²2𝑡−^𝑡0𝑔𝑦(𝑠)𝜆𝑥(𝑠)+ℎ𝑑𝑠+𝜎²≤ln𝑦₀𝐵₂(𝑡)+𝑓−𝜎²²2𝑡+𝜎²𝐵₂(𝑡).(3.47) Applying the strong law of large numbers of martingales, we obtain limsup_𝑡→∞ln𝑦(𝑡)𝑡≤𝑓−𝜎2²²<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 𝑎−𝑐/𝐴−𝜎²¹/2>0,𝑓−𝜎²²/2<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=𝑎−𝜎²¹2𝑡−𝑏^𝑡0𝑥(𝑠)𝑑𝑠−^𝑡0𝑐𝑦(𝑠)𝜆𝑥(𝑠)+𝐴𝑦(𝑠)𝑑𝑠+𝜎¹𝐵₁(𝑡).(3.49) Thus, 𝑏^𝑡0𝑥(𝑠)𝑑𝑠≥−ln𝑥(𝑡)+ln𝑥₀+𝑎−𝜎²¹2−𝐴𝑐𝑡+𝜎¹𝐵₁(𝑡).(3.50) Under condition 𝑎−𝑐/𝐴−𝜎²¹/2>0, it follows from the proof of Theorem 3.3 that lim_𝑡→∞ln𝑥(𝑡)𝑡=0,a.s.(3.51) So liminf_𝑡→∞∫^𝑡0𝑥(𝑡)𝑑𝑠𝑡≥𝑎−𝑐/𝐴−𝜎²¹/2𝑏>0,a.s.(3.52) That is, the prey 𝑥(𝑡) is persistent in mean. However, under condition 𝑓−𝜎²²/2<0, from the proof of Theorem 3.8, we have that limsup_𝑡→∞ln𝑦(𝑡)𝑡≤𝑓−𝜎2²²<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 𝑥_𝑘+1=𝑥_𝑘+𝑥_𝑘𝑎−𝑏𝑥^𝑘−𝑐𝑦^𝑘𝜆𝑥_𝑘+𝐴𝑦_𝑘Δ𝑡+𝜎¹𝑥_𝑘√Δ𝑡𝜉_𝑘+𝜎²¹2𝑥^𝑘𝜉^2𝑘𝑦−1Δ𝑡,_𝑘+1=𝑦_𝑘+𝑦_𝑘𝑓−𝜆𝑥𝑔_𝑘𝑦+^𝑘ℎΔ𝑡+𝜎²𝑦_𝑘√Δ𝑡𝜂_𝑘+𝜎²²2𝑦^𝑘𝜂^2𝑘−1Δ𝑡,(4.1) where 𝜉_𝑘 and 𝜂_𝑘 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.

Figure 1 

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).

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).