Persistence and Nonpersistence of a Food Chain Model with Stochastic Perturbation

Abstract

We analyze a three species predator-prey chain model with stochastic perturbation. First, we show that this system has a unique positive solution and its 𝑝th moment is bounded. Then, we deduce conditions that the system is persistent in time average. After that, conditions for the system going to be extinction in probability are established. At last, numerical simulations are carried out to support our results.

1. Introduction

Recently, the dynamical relationship between predator-prey has been one of the dominant themes in both ecology and mathematical ecology due to its universal importance. Especially, the predator-prey chain model is the typical representative. Thereby it significantly changed the biology, the understanding of the existence, and development of the basic law and has made the model become a research hot spot. One of the most famous models for population dynamics is the Lotka-Volterra predator-prey system which has received plenty of attention and has been studied extensively; see [1–4]. Specially persistence and extinction of this model are interesting topics.

The three species predator-prey chain model is described as follows: 𝑥₁̇(𝑡)=𝑥₁(𝑡)𝑎₁−𝑏₁₁𝑥₁(𝑡)−𝑏₁₂𝑥₂(𝑡),𝑥₂̇(𝑡)=𝑥₂(𝑡)−𝑎₂+𝑏₂₁𝑥₁(𝑡)−𝑏₂₂𝑥₂(𝑡)−𝑏₂₃𝑥₃(𝑡),𝑥₃̇(𝑡)=𝑥₃(𝑡)−𝑎₃+𝑏₃₂𝑥₂(𝑡)−𝑏₃₃𝑥₃(𝑡), (1) where 𝑥_𝑖(𝑡)  (𝑖=1,2,3) denotes the population densities of the species at time 𝑡. The parameters 𝑎₁,𝑎₂,𝑎₃,𝑏_𝑖𝑖(𝑖=1,2,3) are positive constants that stand for intrinsic growth rate, predator death rate of the second species, predator death rate of the third species, coefficient of internal competition, respectively. 𝑏₂₁, 𝑏₃₂ represent saturated rate of the second and the third predator and 𝑏₁₂, 𝑏₂₃ represent the decrement rate of predator to prey.

System (1) describes a three species predator-prey chain model in which the latter preys on the former. From a biological viewpoint, we not only require the positive solution of the system but also require its unexploded property in any finite time and stability.

We know that the global asymptotic stability of a positive equilibrium 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,𝑥^∗₃) holds and is global stability if the following condition holds: 𝑎₁−𝑏₁₁𝑏₂₁𝑎₂−𝑏₁₁𝑏₂₂+𝑏₁₂𝑏₂₁𝑏₂₁𝑏₃₂𝑎₃>0, (2) which could refer to [5]. However, population dynamics in the real world is inevitably affected by environmental noise (see, e.g., [6, 7]). Parameters involved in the system are not absolute constants, they always fluctuate around some average values. The deterministic models assume that parameters in the systems are deterministic irrespective of environmental fluctuations which impose some limitations in mathematical modeling of ecological systems. So we cannot omit the influence of the noise on the system. Recently many authors have discussed population systems subject to white noise (see, e.g., [8–15]). May (see, e.g., [16]) pointed out that due to continuous fluctuation in the environment, the birth rates, death rates, saturated rate, competition coefficients, and all other parameters involved in the model exhibit random fluctuation to some extent, and as a result the equilibrium population distribution never attains a steady value but fluctuates randomly around some average value. Sometimes, large amplitude fluctuation in population will lead to the extinction of certain species, which does not happen in deterministic models.

Therefore, Lotka-Volterra predator-prey chain models in random environments are becoming more and more popular. Ji et al. [14, 15] investigated the asymptotic behavior of the stochastic predator-prey system with perturbation. Liu and Chen introduced periodic constant impulsive immigration of predator into predator-prey system and gave conditions for the system to be extinct and permanence. Polansky [17] and Barra et al. [18] have given some special systems of their invariant distribution. After that, Gard [5] analysed that under some conditions the stochastic food chain model exists an invariant distribution. However, seldom people study the persistent and nonpersistent of the food chain model with stochastic perturbation.

In this paper, we introduce the white noise into the intrinsic growth rate of system (1), and suppose 𝑎_𝑖→𝑎_𝑖+𝜎_𝑖𝐵̇_𝑖(𝑡)  (𝑖=1,2,3); then we obtain the following stochastic system: 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁−𝑏₁₁𝑥₁(𝑡)−𝑏₁₂𝑥₂(𝑡)+𝜎₁𝑥₁(𝑡)𝐵̇₁(𝑡),𝑥̇₂(𝑡)=𝑥₂(𝑡)−𝑎₂+𝑏₂₁𝑥₁(𝑡)−𝑏₂₂𝑥₂(𝑡)−𝑏₂₃𝑥₃(𝑡)−𝜎₂𝑥₂(𝑡)𝐵̇₂(𝑡),𝑥̇₃(𝑡)=𝑥₃(𝑡)−𝑎₃+𝑏₃₂𝑥₂(𝑡)−𝑏₃₃𝑥₃(𝑡)−𝜎₃𝑥₃(𝑡)𝐵̇₃(𝑡), (3) where 𝐵_𝑖(𝑡)(𝑖=1,2,3) are independent white noises with 𝐵_𝑖(0)=0, 𝜎²_𝑖>0(𝑖=1,2,3) representing the intensities of the noise.

The aim of this paper is to discuss the long time behavior of system (3). We have mentioned that 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,𝑥^∗₃) is the positive equilibrium of system (1). But, when it suffers stochastic perturbations, there is no positive equilibrium. Hence, it is impossible that the solution of system (3) will tend to a fixed point. In this paper, we show that system (3) is persistent in time average. Furthermore, under certain conditions, we prove that the population of system (3) will die out in probability which will not happen in deterministic system and could reveal that large white noise may lead to extinction.

The rest of this paper is organized as follows. In Section 2, we show that there is a unique nonnegative solution of system (3), and its 𝑝th moment is bounded. In Section 3, we show that system (3) is persistent in time average. While in Section 4, we consider three situations when the population of the system will be extinction. In Section 5, numerical simulations are carried out to support our results.

Throughout this paper, unless otherwise specified, let (Ω,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all 𝑃-null sets). Let 𝑅³₊ denote the positive cone of 𝑅³; namely, 𝑅³₊={𝑥∈𝑅³∶𝑥_𝑖>0,1≤𝑖≤3}, 𝑅³₊={𝑥∈𝑅³∶𝑥_𝑖≥0,1≤𝑖≤3}.

2. Existence and Uniqueness of the Nonnegative Solution

To investigate the dynamical behavior, the first concern thing is whether the solution is global existence. Moreover, for a population model, whether the solution is nonnegative is also considered. Hence, in this section, we show that the solution of system (3) is global and nonnegative. As we have known, in order for a stochastic differential equation to have a unique global (i.e., no explosion at a finite time) solution with any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (see, e.g., [19]). It is easy to see that the coefficients of system (3) are locally Lipschitz continuous, so system (3) has a local solution. By Lyapunov analysis method, we show the global existence of this solution.

Theorem 1. For any given initial value 𝑥(0)=𝑥₀∈𝑅³₊, system (3) has a unique global positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)) for all 𝑡≥0 with probability one.

Proof. It is clear that the coefficients of system (3) are locally Lipschitz continuous for the given initial value 𝑥(0)=𝑥₀∈𝑅³₊. So there is a unique local solution 𝑥(𝑡) on 𝑡∈[0,𝜏_𝑒), where 𝜏_𝑒 is the explosion time (see, e.g., [19]). To show that this solution is global, we need to show that 𝜏_𝑒=∞ a.s. Let 𝑚₀≥1 be sufficiently large so that each component of 𝑥₀ all lies within the interval [1/𝑚₀,𝑚₀]. For each integer 𝑚≥𝑚₀, define the stopping time: 𝜏_𝑚=inf𝑡∈0,𝜏_𝑒∶min𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡≤1)𝑚ormax𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)≥𝑚. (4) Throughout this paper, we set inf∅=∞ (as usual ∅ denotes the empty set). Clearly, 𝜏_𝑚 is increasing as 𝑚→∞. Set 𝜏_∞=lim_𝑚→∞𝜏_𝑚; then 𝜏_∞≤𝜏_𝑒 a.s. If we can show that 𝜏_∞=∞ a.s., then 𝜏_𝑒=∞ and (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))∈𝑅³₊ a.s. for all 𝑡≥0. In other words, to complete the proof all we need to show is that 𝜏_∞=∞ a.s. For if this statement is false, then there is a pair of constants 𝑇>0 and 𝜖∈(0,1) such that 𝑃𝜏_∞≤𝑇>𝜖. (5) Hence there is an integer 𝑚₁≥𝑚₀ such that 𝑃𝜏_𝑚≤𝑇≥𝜖∀𝑚≥𝑚₁. (6) Define a 𝐶²-function 𝑉∶𝑅³₊→𝑅₊ by 𝑉𝑥₁,𝑥₂,𝑥₃=𝑏₃₂𝑏₂₁𝑥₁−1−log𝑥₁+𝑏₁₂𝑥₂−1−log𝑥₂+𝑏₂₃𝑏₁₂𝑥₃−1−log𝑥₃; (7) the nonnegativity of this function can be seen from 𝑢−1−(1/2)log𝑢≥0, for all 𝑢>0. Using Itô's formula, we get 𝑑𝑉∶=𝐿𝑉𝑑𝑡+𝑏₃₂𝑏₂₁𝜎₁𝑥₁−1𝑑𝐵₁(𝑡)+𝑏₃₂𝑏₁₂𝜎₂𝑥₂−1𝑑𝐵₂(𝑡)+𝑏₂₃𝑏₁₂𝜎₃𝑥₃−1𝑑𝐵₃(𝑡), (8) where 𝐿𝑉=𝑏₃₂𝑏₂₁𝑥₁−1𝑎₁−𝑏₁₁𝑥₁−𝑏₁₂𝑥₂+𝑏₃₂𝑏₁₂𝑥₂−1−𝑎₂+𝑏₂₁𝑥₁−𝑏₂₂𝑥₂−𝑏₂₃𝑥₃+𝑏₂₃𝑏₁₂𝑥₃−1−𝑎₃+𝑏₃₂𝑥₂−𝑏₃₃𝑥₃+𝑏₃₂𝑏₂₁𝜎²₁2+𝑏₃₂𝑏₁₂𝜎²₂2+𝑏₂₃𝑏₁₂𝜎²₁2=𝑏₃₂𝑏₂₁−𝑎₁+𝜎²₁2+𝑏₃₂𝑏₁₂𝑎₂+𝜎²₂2+𝑏₂₃𝑏₁₂𝑎₃+𝜎²₃2+𝑏₃₂𝑏₂₁𝑎₁+𝑏₃₂𝑏₂₁𝑏₁₁−𝑏₃₂𝑏₁₂𝑏₂₁𝑥₁+𝑏₃₂𝑏₂₁𝑏₁₂+𝑏₃₂𝑏₁₂𝑏₂₂−𝑏₂₃𝑏₁₂𝑏₃₂−𝑏₃₂𝑏₁₂𝑎₂𝑥₂+𝑏₃₂𝑏₁₂𝑏₂₃+𝑏₂₃𝑏₁₂𝑏₃₃−𝑏₂₃𝑏₁₂𝑎₃𝑥₃−𝑏₃₂𝑏₂₁𝑏₁₁𝑥²₁−𝑏₃₂𝑏₁₂𝑏₂₂𝑥²₂−𝑏₂₃𝑏₁₂𝑏₃₃𝑥²₃≤𝑀, (9) where 𝑀 is a constant. Therefore ^𝜏𝑚₀^∧𝑇(𝑥(𝑡≤𝑑𝑉))_{𝜏𝑚0∧𝑇}𝑀𝑑𝑡+^𝜏𝑚₀^∧𝑇𝑏₃₂𝑏₂₁𝜎₁𝑥₁−1𝑑𝐵₁(𝑡)+^𝜏𝑚₀^∧𝑇𝑏₃₂𝑏₁₂𝜎₂𝑥₂−1𝑑𝐵₂(𝑡)+^𝜏𝑚₀^∧𝑇𝑏₂₃𝑏₁₂𝜎₃𝑥₃−1𝑑𝐵₃(𝑡), (10) which implies that 𝐸𝑉𝑥𝜏_𝑚∧𝑇≤𝑉𝑥₀+𝑀𝑇. (11) Set Ω_𝑚={𝜏_𝑚≤𝑇} for 𝑚≥𝑚₁. By (6), we know 𝑃(Ω_𝑚)≥𝜖. Notice that for every 𝜔∈Ω_𝑚, there is at least one of 𝑥_𝑖(𝜏_𝑚,𝜔) equals either 𝑚 or 1/𝑚; then 𝑉𝑥𝜏_𝑚≥𝑚−1−log𝑚∧𝑚⁻¹−1+log𝑚. (12) It then follows from (11) that 𝑉𝑥₀+1𝑀𝑇≥𝐸_Ω𝑚(𝜔)𝑉𝑥𝜏_𝑚≥𝜖𝑚−1−log𝑚∧𝑚⁻¹−1+log𝑚, (13) where 1_Ω𝑚(𝜔) is the indicator function of Ω_𝑚. Letting 𝑚→∞ leads to the contradiction that ∞>𝑉(𝑥₀)+𝑀𝑇=∞. So we must have 𝜏_∞=∞ a.s.

Theorem 2. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡), and let 𝑥₃(𝑡)) be the solution of system (3) with any given initial value 𝑥(0)=𝑥₀∈𝑅³₊, then there exists a positive constant 𝐾(𝑝) such that 𝐸𝑏₃₂𝑏₂₁𝑥₁(𝑡)+𝑏₃₂𝑏₁₂𝑥₂(𝑡)+𝑏₂₃𝑏₁₂𝑥₃(𝑡)^𝑝𝑝()≤𝐾∀𝑡∈0,∞,𝑝>1. (14)

Proof. Let 𝑦(𝑡)=𝑏₃₂𝑏₂₁𝑥₁(𝑡)+𝑏₃₂𝑏₁₂𝑥₂(𝑡)+𝑏₂₃𝑏₁₂𝑥₃(𝑡); then 𝑑𝑦(𝑡)=𝑎₁𝑏₂₁𝑏₃₂𝑥₁−𝑎₂𝑏₁₂𝑏₃₂𝑥₂−𝑎₃𝑏₁₂𝑏₂₃𝑥₃−𝑏₁₁𝑏₂₁𝑏₃₂𝑥²₁−𝑏₁₂𝑏₂₂𝑏₃₂𝑥²₂−𝑏₁₂𝑏₂₃𝑏₃₃𝑥²₃𝑑𝑡+𝜎₁𝑏₂₁𝑏₃₂𝑥₁𝑑𝐵₁(𝑡)+𝜎₂𝑏₁₂𝑏₃₂𝑥₂𝑑𝐵₂(𝑡)+𝜎₃𝑏₁₂𝑏₂₃𝑥₃𝑑𝐵₃(𝑡), (15) and so 𝑑𝑦^𝑝=𝑝𝑦^𝑝−1𝑎₁𝑏₂₁𝑏₃₂𝑥₁−𝑎₂𝑏₁₂𝑏₃₂𝑥₂−𝑎₃𝑏₁₂𝑏₂₃𝑥₃−𝑏₁₁𝑏₂₁𝑏₃₂𝑥²₁−𝑏₁₂𝑏₂₂𝑏₃₂𝑥²₂−𝑏₁₂𝑏₂₃𝑏₃₃𝑥²₃𝑑𝑡+𝑝𝑦^𝑝−1𝜎₁𝑏₂₁𝑏₃₂𝑥₁𝑑𝐵₁(𝑡)+𝜎₂𝑏₁₂𝑏₃₂𝑥₂𝑑𝐵₂(𝑡)+𝜎₃𝑏₁₂𝑏₂₃𝑥₃𝑑𝐵₃(𝑡)+12𝑝𝑝−1𝑦^𝑝−2×𝜎²₁𝑏²₂₁𝑏²₃₂𝑥²₁+𝜎²₂𝑏²₁₂𝑏²₃₂𝑥²₂+𝜎²₃𝑏²₁₂𝑏²₂₃𝑥²₃𝑑𝑡. (16) Note that 𝑏₁₁𝑏₂₁𝑏₃₂𝑥²₁+𝑏₁₂𝑏₂₂𝑏₃₂𝑥²₂+𝑏₁₂𝑏₂₃𝑏₃₃𝑥²₃≥min𝑏₁₁,𝑏₂₂,𝑏₃₃𝑏₃₂𝑏₂₁+𝑏₃₂𝑏₁₂+𝑏₂₃𝑏₁₂×𝑏₃₂𝑏₂₁𝑥₁+𝑏₃₂𝑏₁₂𝑥₂+𝑏₂₃𝑏₁₂𝑥₃²=min𝑏₁₁,𝑏₂₂,𝑏₃₃𝑏₃₂𝑏₂₁+𝑏₃₂𝑏₁₂+𝑏₂₃𝑏₁₂𝑦². (17) Then 𝑑𝑦^𝑝=𝑝𝑦^𝑝−1𝑎₁𝑏𝑦−min₁₁,𝑏₂₂,𝑏₃₃𝑏₃₂𝑏₂₁+𝑏₃₂𝑏₁₂+𝑏₂₃𝑏₁₂𝑦²+1𝑑𝑡2𝑝𝑝−1𝑦^𝑝−2max𝜎²₁,𝜎²₂,𝜎²₃𝑦²𝑑𝑡+𝑝𝑦^𝑝−1𝜎₁𝑏₂₁𝑏₃₂𝑥₁𝑑𝐵₁(𝑡)+𝜎₂𝑏₁₂𝑏₃₂𝑥₂𝑑𝐵₂(𝑡)+𝜎₃𝑏₁₂𝑏₂₃𝑥₃𝑑𝐵₃(𝑡≤𝑝𝑎₁)+𝑝2max𝜎²₁,𝜎²₂,𝜎²₃𝑦^𝑝−𝑝min𝑏₁₁,𝑏₂₂,𝑏₃₃𝑏₃₂𝑏₂₁+𝑏₃₂𝑏₁₂+𝑏₂₃𝑏₁₂𝑦^𝑝+1𝑑𝑡+𝑝𝑦^𝑝−1𝜎₁𝑏₂₁𝑏₃₂𝑥₁𝑑𝐵₁(𝑡)+𝜎₂𝑏₁₂𝑏₃₂𝑥₂𝑑𝐵₂(𝑡)+𝜎₃𝑏₁₂𝑏₂₃𝑥₃𝑑𝐵₃(𝑡). (18) Hence 𝑑𝐸𝑦^𝑝(𝑡)𝑎𝑑𝑡≤𝑝₁𝜎+𝑝max²₁,𝜎²₂,𝜎²₃2𝐸𝑦^𝑝(𝑡𝑏)−𝑝min₁₁,𝑏₂₂,𝑏₃₃𝑏₃₂𝑏₂₁+𝑏₃₂𝑏₁₂+𝑏₂₃𝑏₁₂𝐸𝑦^𝑝+1(𝑡𝑎)≤𝑝₁𝜎+𝑝max²₁,𝜎²₂,𝜎²₃2𝐸𝑦^𝑝(𝑡𝑏)−𝑝min₁₁,𝑏₂₂,𝑏₃₃𝑏₃₂𝑏₂₁+𝑏₃₂𝑏₁₂+𝑏₂₃𝑏₁₂𝐸𝑦^𝑝(𝑡)^{(𝑝+1)/𝑝}. (19) Therefore, by comparison theorem, we get limsup_𝑡→∞𝐸𝑦^𝑝(𝑡≤𝑎₁)+𝑝max𝜎²₁,𝜎²₂,𝜎²₃𝑏₃₂𝑏₂₁+𝑏₃₂𝑏₁₂+𝑏₂₃𝑏₁₂min𝑏₁₁,𝑏₂₂,𝑏₃₃^𝑝. (20) Besides, note that 𝐸[𝑦^𝑝(𝑡)] is continuous; then there is a positive constant 𝐾(𝑝) such that 𝐸𝑦^𝑝(𝑡)≤𝐾𝑝,∀𝑡∈[0,∞). (21)

3. Persistent in Time Average

There is no equilibrium of system (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 first show that this system is persistent in mean. Before we give the result, we should do some prepared work.

L. S. Chen and J. Chen in [20] proposed the definition of persistence in mean for the deterministic system. Here, we also use this definition for the stochastic system.

Definition 3. System (3) is said to be persistent in mean, if liminf_𝑡→∞1𝑡^𝑡₀𝑥₃(𝑠)𝑑𝑠>0,a.s. (22)

Lemma 4 (see [21, Lemma 17]). Let 𝑓∈𝐶([0,+∞)×Ω,(0,+∞)) and 𝐹∈𝐶([0,+∞)×Ω,𝑅). If there exist positive constants 𝜆₀, 𝜆, such that log𝑓(𝑡)≥𝜆𝑡−𝜆₀^𝑡₀𝑓(𝑠)𝑑𝑠+𝐹(𝑡),𝑡≥0a.s., (23) and lim_𝑡→∞(𝐹(𝑡)/𝑡)=0a.s., then liminf_𝑡→∞1𝑡^𝑡₀𝑓(𝑠)𝜆𝑑𝑠≥𝜆₀,a.s. (24)

From Lemma 4, it is easy to see that we could get Lemmas 5 and 6 with the same method.

Lemma 5. Let 𝑓∈𝐶([0,+∞)×Ω,(0,+∞)) and 𝐹∈𝐶([0,+∞)×Ω,𝑅). If there exist positive constants 𝜆₀, 𝜆, such that log𝑓(𝑡)≤𝜆𝑡−𝜆₀^𝑡₀𝑓(𝑠)𝑑𝑠+𝐹(𝑡),𝑡≥0a.s., (25) and lim_𝑡→∞(𝐹(𝑡)/𝑡)=0a.s., then limsup_𝑡→∞1𝑡^𝑡₀𝑓(𝑠)𝜆𝑑𝑠≤𝜆₀,a.s. (26)

Lemma 6. Let 𝑓∈𝐶([0,+∞)×Ω,(0,+∞)) and 𝐹∈𝐶([0,+∞)×Ω,𝑅). If there exist positive constants 𝜆₀, 𝜆, such that log𝑓(𝑡)=𝜆𝑡−𝜆₀^𝑡₀𝑓(𝑠)𝑑𝑠+𝐹(𝑡),𝑡≥0a.s., (27) and lim_𝑡→∞(𝐹(𝑡)/𝑡)=0a.s., then lim_𝑡→∞1𝑡^𝑡₀𝑓(𝑠)𝜆𝑑𝑠=𝜆₀,a.s. (28)

From the stochastic comparison theorem [11], it is easy to get the following result.

Lemma 7. Let 𝑥(𝑡)∈𝑅³₊ be a solution of system (3) with 𝑥(0)=(𝑥₁(0),𝑥₂(0),𝑥₃(0)). Then one has 𝑥(𝑡)≤Φ(𝑡); (29) that is, 𝑥_𝑖(𝑡)≤Φ_𝑖(𝑡),𝑖=1,2,3, (30) where Φ(𝑡)=Φ₁(𝑡),Φ₂(𝑡),Φ₃(𝑡)^⊤, (31)Φ_𝑖(𝑡) is solutions of the following stochastic differential equations: 𝑑Φ₁(𝑡)=Φ₁(𝑡)𝑎₁−𝑏₁₁Φ₁(𝑡)𝑑𝑡+𝜎₁Φ₁(𝑡)𝑑𝐵₁(𝑡),Φ₁(0)=𝑥₁(0),𝑑Φ₂(𝑡)=Φ₂(𝑡)−𝑎₂+𝑏₂₁Φ₁(𝑡)−𝑏₂₂Φ₂(𝑡)𝑑𝑡−𝜎₂Φ₂(𝑡)𝑑𝐵₂(𝑡),Φ₂(0)=𝑥₂(0),𝑑Φ₃(𝑡)=Φ₃(𝑡)−𝑎₃+𝑏₃₂Φ₂(𝑡)−𝑏₃₃Φ₃(𝑡)𝑑𝑡−𝜎₃Φ₃(𝑡)𝑑𝐵₃(𝑡),Φ₃(0)=𝑥₃(0). (32)

Assumption 8. Consider 𝑟₁−𝑏₁₁𝑏₂₁𝑟₂−𝑏₁₁𝑏₂₂+𝑏₁₂𝑏₂₁𝑏₂₁𝑏₃₂𝑟₃𝑟>0,₁=𝑎₁−𝜎²₁2>0,𝑟_𝑖=𝑎_𝑖+𝜎²_𝑖2𝑖=2,3. (33)

Lemma 9. If Assumption 8 is satisfied, the solution Φ(𝑡) of system (32) with any initial value Φ(0)∈𝑅³₊ has the following property: lim_𝑡→∞logΦ_𝑖(𝑡)𝑡=0,lim_𝑡→∞1𝑡^𝑡₀Φ_𝑖(𝑠)𝑑𝑠=𝑀_𝑖,a.s., (34) where 𝑀₁=𝑟₁𝑏₁₁,𝑀₂=𝑟₁𝑏₂₁−𝑟₂𝑏₁₁𝑏₁₁,𝑀₃=𝑟₁𝑏₂₁𝑏₃₂−𝑟₂𝑏₁₁𝑏₃₂−𝑟₃𝑏₁₁𝑏₂₂𝑏₁₁𝑏₂₂𝑏₃₃. (35)

Proof. From the result in [14] and Assumption 8 being satisfied, we know lim_𝑡→∞logΦ₁(𝑡)𝑡=0,liminf_𝑡→∞1𝑡^𝑡₀Φ₁(𝑠)𝑎𝑑𝑠=¹−𝜎²₁/2𝑏₁₁=𝑟₁𝑏₁₁, (36) Besides, according to Itô's formula, the second population of system (32) is changed into 𝑑logΦ₂(𝑡)=−𝑟₂+𝑏₂₁Φ₁(𝑡)−𝑏₂₂Φ₂(𝑡)𝑑𝑡−𝜎₂𝑑𝐵₂(𝑡). (37) It then follows logΦ₂(𝑡)=logΦ₂(0)−𝑟₂𝑡+𝑏₂₁^𝑡₀Φ₁(𝑠)𝑑𝑠−𝑏₂₂^𝑡₀Φ₂(𝑠)𝑑𝑠−𝜎₂𝐵₂(𝑡), (38) With Lemma 6 and Assumption 8, we could get lim_𝑡→∞1𝑡^𝑡₀Φ₂(𝑠)=𝑑𝑠−𝑟₂+𝑏₂₁𝑟₁/𝑏₁₁𝑏₂₂=𝑟₁𝑏₂₁−𝑟₂𝑏₁₁𝑏₁₁𝑏₂₂>0. (39) Let (38) divide 𝑡, and 𝑡→∞, together with (36) and (39), consequently lim_𝑡→∞logΦ₂(𝑡)𝑡=0. (40) Similarly, according to Itô's formula, the third population of system (25) is changed into 𝑑logΦ₃(𝑡)=−𝑟₃+𝑏₃₂Φ₂(𝑡)−𝑏₃₃Φ₃(𝑡)𝑑𝑡−𝜎₃𝑑𝐵₃(𝑡); (41) it then follows logΦ₃(𝑡)=logΦ₃(0)−𝑟₃𝑡+𝑏₃₂^𝑡₀Φ₂(𝑠)𝑑𝑠−𝑏₃₃^𝑡₀Φ₃(𝑠)𝑑𝑠−𝜎₃𝐵₃(𝑡),lim_𝑡→∞1𝑡^𝑡₀Φ₃(𝑠)=𝑑𝑠−𝑟₃+𝑏₃₂𝑟₁𝑏₂₁−𝑟₂𝑏₁₁/𝑏₁₁𝑏₂₂𝑏₃₃>0,lim_𝑡→∞logΦ₃(𝑡)𝑡=0. (42)

From this, together with Lemmas 7 and 9, the following result is obviously true.

Theorem 10. If Assumption 8 is satisfied, the solution 𝑥(𝑡) of system (3) with any initial value 𝑥(0)∈𝑅³₊ has the following property: limsup_𝑡→∞log𝑥_𝑖(𝑡)𝑡≤0,𝑖=1,2,3. (43)

Above all, we could get.

Theorem 11. If Assumption 8 is satisfied, the the solution 𝑥(𝑡) of system (3) with any initial value 𝑥(0)∈𝑅³₊ has the following property: liminf_𝑡→∞1𝑡^𝑡₀𝑥₃(𝑠)𝑑𝑠≥𝑥^∗₃,a.s., (44) where 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,𝑥^∗₃) is the only nonnegative solution of the following equation: 𝑟₁−𝑏₁₁𝑥₁−𝑏₁₂𝑥₂=0,−𝑟₂+𝑏₂₁𝑥₁−𝑏₂₂𝑥₂−𝑏₂₃𝑥₃=0,−𝑟₃+𝑏₃₂𝑥₂−𝑏₃₃𝑥₃=0. (45)

Proof. From system (3), such that 𝑑𝑐₁log𝑥₁(𝑡)+𝑐₂log𝑥₂(𝑡)+𝑐₃log𝑥₃(𝑡)=𝑟₁𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃+−𝑏₁₁𝑐₁+𝑏₂₁𝑐₂𝑥₁+−𝑏₁₂𝑐₁−𝑏₂₂𝑐₂+𝑏₃₂𝑐₃𝑥₂−𝑏₂₃𝑐₂+𝑏₃₃𝑐₃𝑥₃𝑑𝑡+𝑐₁𝜎₁𝑑𝐵₁(𝑡)−𝑐₂𝜎₂𝑑𝐵₂(𝑡)−𝑐₃𝜎₃𝑑𝐵₃(𝑡). (46) Let 𝑐₁=𝑏₂₁, 𝑐₂=𝑏₁₁, and 𝑐₃=(𝑏₁₁𝑏₂₂+𝑏₁₂𝑏₂₁)/𝑏₃₂, together with Assumption 8, we know 𝑟₁𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃>0; (47) hence 𝑐₁log𝑥₁(𝑡)−log𝑥₁(0)+𝑐₂log𝑥₂(𝑡)−log𝑥₂(0)+𝑐₃log𝑥₃(𝑡)−log𝑥₃(0)×(𝑡)⁻¹=𝑟₁𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃−𝑐₂𝑏₂₃+𝑐₃𝑏₃₃1𝑡^𝑡₀𝑥₃(𝑠)+𝑐𝑑𝑠₁𝜎₁𝐵₁(𝑡)−𝑐₂𝜎₂𝐵₂(𝑡)−𝑐₃𝜎₃𝐵₃(𝑡)𝑡. (48) According to Theorem 10, where limsup_𝑡→∞log𝑥_𝑖(𝑡)𝑡≤0,𝑖=1,2,3, (49) and lim_𝑡→∞(𝐵_𝑖(𝑡)/𝑡)=0, 𝑖=1,2,3, liminf_𝑡→∞1𝑡^𝑡₀𝑥₃(𝑠)𝑟𝑑𝑠≥¹𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃𝑐₂𝑏₂₃+𝑐₃𝑏₃₃=𝑥^∗₃, (50) where 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,𝑥^∗₃) is the only nonnegative solution of the following equation when Assumption 8 is satisfied: 𝑟₁−𝑏₁₁𝑥₁−𝑏₁₂𝑥₂=0,−𝑟₂+𝑏₂₁𝑥₁−𝑏₂₂𝑥₂−𝑏₂₃𝑥₃=0,−𝑟₃+𝑏₃₂𝑥₂−𝑏₃₃𝑥₃=0. (51)

4. Nonpersistence

In this section, we show the situation when the population of system (3) will be extinction in three cases.

Case 1 (𝑟₁<0). According to Itô's formula, the first population of system (25) is changed into 𝑑logΦ₁(𝑡)≤𝑟₁−𝑏₁₁Φ₁(𝑡)𝑑𝑡−𝜎₁𝑑𝐵₁(𝑡). (52) If 𝑟₁<0, we could get limsup_𝑡→∞logΦ₁(𝑡)𝑡≤𝑟₁𝑏₁₁<0a.s. (53) From the stochastic comparison theorem, we have limsup_𝑡→∞log𝑥₁(𝑡)𝑡≤𝑟₁𝑏₁₁<0a.s., (54) hence lim_𝑡→∞𝑥₁(𝑡)=0,a.s. (55) From the second population of system (25), we have limsup_𝑡→∞logΦ₂(𝑡)𝑡≤−𝑎₂+𝑏₂₁limsup_𝑡→∞1𝑡^𝑡₀Φ₁(𝑠)𝑑𝑠≤−𝑎₂a.s.; (56) similarly limsup_𝑡→∞logΦ₃(𝑡)𝑡≤−𝑎₃a.s.,lim_𝑡→∞𝑥_𝑖(𝑡)=0a.s.𝑖=2,3. (57)

Case 2 (𝑟₁>0,𝑟₁−(𝑏₁₁/𝑏₂₁)𝑟₂<0). It is clear that from the proof section of Case 1, we get logΦ₂(𝑡)−logΦ₂(0)𝑡≤−𝑟₂+𝑏₂₁1𝑡^𝑡₀Φ₁(𝑠)𝜎𝑑𝑠−²𝑑𝐵₂(𝑡)𝑡a.s., (58) hence limsup_𝑡→∞logΦ₂(𝑡)𝑡≤−𝑟₂+𝑏₂₁𝑀₁=−𝑟₂+𝑏₂₁𝑟₁𝑏₁₁<0a.s. (59) Similarly limsup_𝑡→∞logΦ₃(𝑡)𝑡≤−𝑟₃+𝑏₃₂limsup_𝑡→∞1𝑡^𝑡₀Φ₂(𝑠)𝑑𝑠≤−𝑎₃<0a.s.; (60) thus, lim_𝑡→∞𝑥_𝑖(𝑡)=0a.s.,𝑖=2,3. (61)

Above all, and from the conclusion in [22], we could easily know that the distribution of 𝑥₁(𝑡) converges weekly to the probability measure with density: 𝑓^∗(𝜁)=𝐶₀𝜁^{2𝑟1/𝜎21−1}𝑒^{−2𝑏11𝜁/𝜎21}, (62) where 𝐶₀=(2𝑏₁₁/𝜎²₁)^{2𝑟1/𝜎21}/Γ(2𝑟₁/𝜎²₁) and lim_𝑡→∞1𝑡^𝑡₀𝑥₁(𝑠)𝑟𝑑𝑠=¹𝑏₁₁,a.s. (63)

Case 3 (𝑟₁−(𝑏₁₁/𝑏₂₁)𝑟₂−((𝑏₁₁𝑏₂₂+𝑏₁₂𝑏₂₁)/𝑏₂₁𝑏₃₂)𝑟₃<0). It is clear that 𝑑𝑐₁log𝑥₁(𝑡)+𝑐₂log𝑥₂(𝑡)+𝑐₃log𝑥₃(𝑡)=𝑟₁𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃+−𝑏₁₁𝑐₁+𝑏₂₁𝑐₂𝑥₁+−𝑏₁₂𝑐₁−𝑏₂₂𝑐₂+𝑏₃₂𝑐₃𝑥₂−𝑏₂₃𝑐₂+𝑏₃₃𝑐₃𝑥₃𝑑𝑡+𝑐₁𝜎₁𝑑𝐵₁(𝑡)−𝑐₂𝜎₂𝑑𝐵₂(𝑡)−𝑐₃𝜎₃𝑑𝐵₃(𝑡). (64) Since 𝑐₁=𝑏₂₁, 𝑐₂=𝑏₁₁, 𝑐₃=(𝑏₁₁𝑏₂₂+𝑏₁₂𝑏₂₁)/𝑏₃₂, we get 𝑐₁log𝑥₁(𝑡)+𝑐₂log𝑥₂(𝑡)+𝑐₃log𝑥₃(𝑡)≤𝑟₁𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃𝑡+𝑐₁log𝑥₁(0)+𝑐₂log𝑥₂(0)+𝑐₃log𝑥₃(0)+𝑐₁𝜎₁𝐵₁(𝑡)−𝑐₂𝜎₂𝐵₂(𝑡)−𝑐₃𝜎₃𝐵₃(𝑡), (65) Moreover, log𝑥^𝑐1₁(𝑡)𝑥^𝑐2₂(𝑡)𝑥^𝑐3₃(𝑡)𝑡≤𝑟₁𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃+𝑐₁log𝑥₁(0)+𝑐₂log𝑥₂(0)+𝑐₃log𝑥₃(0)𝑡+𝑐₁𝜎₁𝐵₁(𝑡)𝑡−𝑐₂𝜎₂𝐵₂(𝑡)𝑡−𝑐₃𝜎₃𝐵₃(𝑡)𝑡. (66) And lim_𝑡→∞(𝐵_𝑖(𝑡)/𝑡)=0, 𝑖=1,2,3, implies limsup_𝑡→∞log𝑥^𝑐1₁(𝑡)𝑥^𝑐2₂(𝑡)𝑥^𝑐3₃(𝑡)𝑡≤𝑟₁𝑐₁−𝑟₂𝑐₂−𝑟₃𝑐₃<0; (67) then lim_𝑡→∞𝑥^𝑐1₁(𝑡)𝑥^𝑐2₂(𝑡)𝑥^𝑐3₃(𝑡)=0a.s. (68) Therefore, by the above arguments, we get the following conclusion.

Theorem 12. Let 𝑥(𝑡) be the solution of system (3) with any initial value 𝑥(0)∈𝑅³₊. Then
(1) if 𝑟₁<0, then lim_𝑡→∞𝑥_𝑖(𝑡)=0a.s.,𝑖=1,2,3, (69)
(2) if 𝑟₁>0, 𝑟₁−(𝑏₁₁/𝑏₂₁)𝑟₂<0, then lim_𝑡→∞𝑥_𝑖(𝑡)=0a.s.,𝑖=2,3, (70) and the distribution of 𝑥₁(𝑡) converges weekly to the probability measure with density: 𝑓^∗(𝜁)=𝐶₀𝜁^{2𝑟1/𝜎21−1}𝑒^{−2𝑏11𝜁/𝜎21}, (71) where 𝐶₀=(2𝑏₁₁/𝜎²₁)^{2𝑟1/𝜎21}/Γ(2𝑟₁/𝜎²₁) and lim_𝑡→∞1𝑡^𝑡₀𝑥₁(𝑠)𝑟𝑑𝑠=¹𝑏₁₁,a.s; (72)
(3) if 𝑟₁−(𝑏₁₁/𝑏₂₁)𝑟₂−((𝑏₁₁𝑏₂₂+𝑏₁₂𝑏₂₁)/𝑏₂₁𝑏₃₂)𝑟₃<0, then lim_𝑡→∞𝑥^𝑐1₁(𝑡)𝑥^𝑐2₂(𝑡)𝑥^𝑐3₃(𝑡)=0,a.s., (73) where 𝑐₁=𝑏₂₁, 𝑐₂=𝑏₁₁, and 𝑐₃=(𝑏₁₁𝑏₂₂+𝑏₁₂𝑏₂₁)/𝑏₃₂.

5. Numerical Simulation

In this section, we give out the numerical experiment to support our results. Consider 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁−𝑏₁₁𝑥₁(𝑡)−𝑏₁₂𝑥₂(𝑡)+𝜎₁𝑥₁(𝑡)𝐵̇₁(𝑡),𝑥̇₂(𝑡)=𝑥₂(𝑡)−𝑎₂+𝑏₂₁𝑥₁(𝑡)−𝑏₂₂𝑥₂(𝑡)−𝑏₂₃𝑥₃(𝑡)−𝜎₂𝑥₂(𝑡)𝐵̇₂(𝑡),𝑥̇₃(𝑡)=𝑥₃(𝑡)−𝑎₃+𝑏₃₂𝑥₂(𝑡)−𝑏₃₃𝑥₃(𝑡)−𝜎₃𝑥₃(𝑡)𝐵̇₃(𝑡). (74) By the Milstein method in [23], we have the difference equation: 𝑥_1,𝑘+1=𝑥_1,𝑘+𝑥_1,𝑘×𝑎₁−𝑏₁₁𝑥_1,𝑘−𝑏₁₂𝑥_2,𝑘Δ𝑡+𝜎₁𝜖_1,𝑘√𝜎Δ𝑡+²¹2𝜖²_1,𝑘,𝑥Δ𝑡−Δ𝑡_2,𝑘+1=𝑥_2,𝑘+𝑥_2,𝑘×−𝑎₂+𝑏₂₁𝑥_1,𝑘−𝑏₂₂𝑥_2,𝑘−𝑏₂₃𝑥_3,𝑘Δ𝑡−𝜎₂𝜖_2,𝑘√𝜎Δ𝑡+²²2𝜖²_2,𝑘,𝑥Δ𝑡−Δ𝑡_3,𝑘+1=𝑥_3,𝑘+𝑥_3,𝑘×−𝑎₃+𝑏₃₂𝑥_2,𝑘−𝑏₃₃𝑥_3,𝑘Δ𝑡−𝜎₃𝜖_3,𝑘√𝜎Δ𝑡+²³2𝜖²_3,𝑘Δ𝑡−Δ𝑡, (75) where 𝜖_1,𝑘, 𝜖_2,𝑘, and 𝜖_3,𝑘, 𝑖=1,2,3, are the Gaussian random variables 𝑁(0,1), 𝑟₁=𝑎₁−𝜎²₁/2>0, and 𝑟_𝑖=𝑎_𝑖+𝜎²_𝑖/2, 𝑖=2,3. Choosing (𝑥₁(0),𝑥₂(0),𝑥₃(0))∈𝑅³₊, and suitable parameters, by Matlab, we get Figures 1, 2, and 3.

Figure 1 

The solution of system (1) and system (3) with (𝑥1(0),𝑥2(0),𝑥3(0))=(1,0.8,0.5), 𝑎1=0.3, 𝑎2=0.4, 𝑎3=0.1, 𝑏11=0.1, 𝑏12=0.1, 𝑏21=0.6, 𝑏22=0.6, 𝑏23=0.6, 𝑏32=0.8, and 𝑏33=0.6. The red lines represent the solution of system (1), while the blue lines represent the solution of system (3) with 𝜎1=0.02, 𝜎2=0.01, and 𝜎3=0.01.

Figure 2 

Two of the species will die out in probability. The red lines represent the solution of system (1), while the blue lines represent the solution of system (3) with (𝑥1(0),𝑥2(0),𝑥3(0))=(1,0.8,0.5), 𝑎1=0.4, 𝑎2=0.4, 𝑎3=0.1, 𝑏11=0.1, 𝑏12=0.1, 𝑏21=0.6, 𝑏22=0.6, 𝑏23=0.6, 𝑏32=0.8, and 𝑏33=0.6. The red lines represent the solution of system (1), while the blue lines represent the solution of system (3) with 𝜎1=0.02, 𝜎2=3, and 𝜎3=0.01.

Figure 3 

One of the species or both species will die out in probability. The red lines represent the solution of system (1), while the blue lines represent the solution of system (3) with (𝑥1(0),𝑥2(0),𝑥3(0))=(1,0.8,0.5), 𝑎1=−0.1, 𝑎2=0.4, 𝑎3=0.1, 𝑏11=0.1, 𝑏12=0.1, 𝑏21=0.6, 𝑏22=0.6, 𝑏23=0.6, 𝑏32=0.8, and 𝑏33=0.6. The red lines represent the solution of system (1), while the blue lines represent the solution of system (3) with 𝜎1=0.02, 𝜎2=3, and 𝜎3=0.01.

In Figure 1, when the noise is small, choosing parameters satisfying the condition of Theorem 10, the solution of system (3) will persist in time average.

In Figure 2, we observe case (3) in Theorem 12 and choose parameters 𝑟₁>0, 𝑟₁−(𝑏₁₁/𝑏₂₁)𝑟₂<0. As Theorem 12 indicated that two predators will die out in probability. The prey solution of system (3) will persist in time average.

In Figure 3, we observe case (1) in Theorem 12 and choose parameters 𝑟₁<0. As Theorem 12 indicated that not only predators but also prey will die out in probability when the noise of the prey is large, and it does not happen in the deterministic system.

Acknowledgments

The work was supported by the Ministry of Education of China (no. 109051), the Ph.D. Programs Foundation of Ministry of China (no. 200918), and NSFC of China (no. 11371085), Natural Science Foundation of Jilin Province of China (no. 201115133).