# Complex Dynamics of a Stochastic Two-Patch Predator-Prey Population Model with Ratio-Dependent Functional Responses

Authors: Rong Liu; Guirong Liu
Journal: Complexity
 (2021)

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## Abstract

This paper investigates a stochastic two-patch predator-prey model with ratio-dependent functional responses. First, the existence of a unique global positive solution is proved via the stochastic comparison theorem. Then, two different methods are used to discuss the long-time properties of the solutions pathwise. Next, sufficient conditions for extinction and persistence in mean are obtained. Moreover, stochastic persistence of the model is discussed. Furthermore, sufficient conditions for the existence of an ergodic stationary distribution are derived by a suitable Lyapunov function. Next, we apply the main results in some special models. Finally, some numerical simulations are introduced to support the main results obtained. The results in this paper generalize and improve the previous related results.


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## Body

## 1. Introduction

The dynamic relationship between predators and their preys has been universal in mathematical ecology. In the nature world, foraging behaviour is a common phenomenon. Ecological species have the ability to adapt through learning (see [1]). An individual will adjust its behaviour by learning in response to a change of the environment in order to survive and acquire the most food. In [1], the authors studied the two-patch predator-prey population model(1)dx1dt=x1r1−a1x1−s1x1vy1+h1s1x1,dx2dt=x2r2−a2x2−s2x21−vy1+h2s2x2,dydt=y−m1v−m21−v+s1x1e1v1+h1s1x1+s2x2e21−v1+h2s2x2,with nonnegative initial conditions. Here, xi denotes the density of prey in patch ii=1,2, and y represents the density of predators. v0≤v≤1 is the proportion of time that predators stay in patch 1 on average; rii=1,2 is the intrinsic growth rate of prey in patch i; ai is the intraspecific competition coefficient of the prey in patch i; si is the attacking rate of the predators in patch i; ei is the expected biomass of the prey converted to predators in patch i; mi is the per capita mortality rate of predators in patch i; and hi is the handling time of the predation in patch i, respectively.It is well known that the functional response between the predator and prey plays an important role in the population dynamics. In model (1), the authors assumed that an individual predator consumes the prey with functional response x/1+shx, which depends only on the prey. However, when predators have to search for food and, therefore, have to share or compete for food, a ratio-dependent functional response is more reasonable (see [2]). Based on the Holling-type II function, Arditi and Ginzburg [3] first proposed a ratio-dependent functional response of form αx/x+βy. Here, α is the encounter rate with prey by a searching predator, and β is the half saturation constant for the prey. Kuang and Beretta [4] investigated the predator-prey model with ratio-dependent functional response(2)dxdt=xr−ax−αyx+βy,dydt=y−d+eαxx+βy,with nonnegative initial conditions. Here, x and y represent population sizes of the prey and predator at time t, respectively. All parameters are positive constants. r and a, respectively, stand for the prey intrinsic growth rate and the intraspecific competition rate of the prey. d is the death rate of the predator population. α, β, and e, respectively, represent the encounter rate, half capturing saturation constant, and conversion rate that predator y preys on prey x.Note that population model (1) with the functional responses only depend on prey density. However, the ratio-dependent functional response depends not only on the prey but also on the predator. Thus, the ratio-dependent function of the prey and predator is more suitable to substitute for the model. Therefore, based on models (1) and (2), a two-patch predator-prey population model with ratio-dependent functional responses is expressed in the following form:(3)dx1dt=x1r1−a1x1−α1vx1yx1+β1y,dx2dt=x2r2−a2x2−α21−vx2yx2+β2y,dydt=y−m1v−m21−v+e1α1vx1x1+β1y+e2α21−vx2x2+β2y,where xi denotes the density of prey in patch ii=1,2 and y represents the density of predators. All meanings of the parameters are exact as or similar to those for model (1) except the following. Here, αi, βi, and eii=1,2 are the encounter rate, the half-saturation constant, and the conversion rate that y preys on xi, respectively.From [5], it can be seen that stochasticity or variability plays an important role in understanding the dynamics of predator-prey populations. Note that noise in models can lead to several interesting dynamical effects, which are not anticipated by their deterministic counterpart. Thus, in order to simulate population dynamics, environmental fluctuations should be considered in modeling. In general, environmental fluctuations can be simulated by a colored noise. From [6], it can be seen that if the colored noise is not strongly correlated, then one can approximate the colored noise by a white noise w˙t. In fact, the white noise w˙t is formally regarded as the derivative of a Brownian motion wt, i.e., w˙t=dwt/dt (see [7]). As a result, it is more objective to modeling stochastic population models with white noise in mathematical biology. Recently, many authors have paid their attention to stochastic prey-predator models with white noise, see [8–15] and the references therein. Reference [8] investigated the stability of a stochastic one-predator-two-prey population model with time delay, while [13] considered the stability of a stochastic two-predator one-prey population model with time delay. References [10, 11, 15] discussed the dynamic behaviors of stochastic population models with the Allee effect. Reference [12] is concerned with a stochastic three-species food web model with omnivory and ratio-dependent functional response.To the best of our knowledge, so far, there is no investigation on the dynamics of the stochastic two-patch prey-predator model with ratio-dependent functional responses. The purpose of this paper is to make some contribution in this direction. As in the work of Imhof and Walcher [16], assuming that the environmental noise is proportional to the variables, we obtain the following stochastic two-patch prey-predator model:(4)dx1t=x1tr1−a1x1t−α1vytx1t+β1ytdt+σ1x1tdw1t,dx2t=x2tr2−a2x2t−α21−vytx2t+β2ytdt+σ2x2tdw2t,dyt=yt−m1v−m21−v+e1α1vx1tx1t+β1yt+e2α21−vx2tx2t+β2ytdt+σ3ytdw3t,with initial value x10,x20,y0=x10,x20,=y0∈ℝ3+=x,y,z∈ℝ3:x>0,y>0,z>0. All meanings of the parameters are exact as or similar to those for model (3) except the following. Here, w=w1t,w2t,w3t:t≥0 represents the three-dimensional standard Brownian motion defined on a filtered complete probability space Ω,ℱ,ℱtt≥0,ℙ satisfying the usual conditions. σi2 represents the intensity of noise witi=1,2,3.Furthermore, if the intraspecific competition of the predator is considered in model (4), then one can obtain the following stochastic two-patch predator-prey model:(5)dx1t=x1tr1−a1x1t−α1vytx1t+β1ytdt+σ1x1tdw1t,dx2t=x2tr2−a2x2t−α21−vytx2t+β2ytdt+σ2x2tdw2t,dyt=yt−m1v−m21−v−byt+e1α1vx1tx1t+β1yt+e2α21−vx2tx2t+β2ytdt+σ3ytdw3t,with initial value x10,x20,y0=x10,x20,=y0∈ℝ3+. Here, b is the interspecific competition coefficient of the predator.In this paper, we first investigate the dynamics of the stochastic two-patch predator-prey population model (5). Then, we apply the main results in the stochastic predator-prey population model (4). The rest of this paper is organized as follows. In Section 2, we first prove that model (5) has a unique global positive solution by the stochastic comparison theorem. Then, we discuss the long-time properties of the solutions pathwise. Using the exponential martingale inequality and the Borel–Cantelli lemma, we show that the sample Lyapunov exponents of the solutions are nonpositive. Moreover, we prove that, under certain conditions, the sample Lyapunov exponents of the solutions are zero. In Section 3, we establish the sufficient conditions for the extinction and persistence in mean of model (5). In Section 4, we first prove the stochastic ultimate boundedness of model (5) by using two different methods. Then, we show that model (5) is stochastically permanent. Moreover, in section 5, by constructing a suitable Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution to model (5). Next, in Section 6, we apply the main results to two stochastic two-species predator-prey population models and stochastic two-patch predator-prey population model (4). Section 7 contains some numerical results, which are used to demonstrate the theoretical results in this paper. Moreover, through numerical calculation, we find other dynamic properties of the model. The paper ends with a conclusion.For simplicity, in the coming discussion, we introduce the notations(6)λ1=˙r1−α1vβ1,λ2=˙r2−α21−vβ2,λ3=˙e1α1v+e2α21−v−m1v−m21−v.


## 2. Global Positive Solution and Pathwise Estimation

In this section, we first show that model (5) has a unique positive global solution by the stochastic comparison theorem. Then, we discuss the long-time properties of the solutions pathwise.Theorem 1.
For any given initial valuex10,x20,y0∈ℝ+3, model (5) has a unique global solution x1t on t≥0 and the solution will remain in ℝ+3 with probability one.Proof.
We consider the following system:(7)dX1t=r1−a1eX1t−α1veYteX1t+β1eYt−σ122dt+σ1dw1t,dX2t=r2−a2eX2t−α21−veYteX2t+β2eYt−σ222dt+σ2dw2t,dYt=−m1v−m21−v−beYt+e1α1veX1teX1t+β1eX1t+e2α21−veX2teX2t+β2eYt−σ322dt+σ3dw3t,where X10,X20,Y0=lnx10,lnx20,lny0. It is clear that the coefficients of system (7) are locally Lipschitz continuous. Hence, system (7) has a unique maximal local solution X1t,X2t,Yt on 0,τe, where τe is the explosion time. Let xit=eXiti=1,2 and yt=eYt. From Itô formula, it follows that x1t,x2t,yt=eX1t,eX2t,eYt is the unique positive local solution of model (5) with initial value x10,x20,y0 on 0,τe.
If we can verify thatτe=∞ a.s., then X1t,X2t,Yt is a global solution to system (7). Now, using the stochastic comparison theorem, we show that τe=∞ a.s. We consider the following two stochastic differential systems:(8)dΦ1t=Φ1tr1−a1Φ1tdt+σ1Φ1tdw1t,dΦ2t=Φ2tr2−a2Φ2tdt+σ2Φ2tdw2t,dΨt=Ψte1α1v+e2α21−v−bΨtdt+σ3Ψtdw3t,with initial value Φ10,Φ20,Ψ0=x10,x20,y0∈ℝ+3 and(9)dϕ1t=ϕ1tλ1−a1ϕ1tdt+σ1ϕ1tdw1t,dϕ2t=ϕ2tλ2−a2ϕ2tdt+σ2ϕ2tdw2t,dψt=ψtλ3−b+e1α1vβ1ϕ1t+e2α21−vβ2ϕ2tψtdt+σ3ψtdw3t,with initial value ϕ10,ϕ20,ψ0=x10,x20,y0∈ℝ+3.
Thanks to Lemma 4.2 in [17], systems (8) and (9) can be explicitly solved as follows:(10)Φ1t=expr1−σ12/2t+σ1w1t1/x10+a1∫0texpr1−σ12/2z+σ1w1zdz,Φ2t=expr2−σ22/2t+σ2w2t1/x20+a2∫0texpr2−σ22/2z+σ2w2zdz,Ψt=expe1α1v+e2α21−v−σ32/2t+σ3w3t1/y0+b∫0texpe1α1v+e2α21−v−σ32/2z+σ3w3zdz,ϕ1t=expλ1−σ12/2t+σ1w1t1/x10+a1∫0texpλ1−σ12/2z+σ1w1zdz,ϕ2t=expλ2−σ22/2t+σ2w2t1/x20+a2∫0texpλ2−σ22/2z+σ2w2zdz,ψt=expλ3−σ32/2t+σ3w3t1/y0+∫0tb+e1α1vβ1/ϕ1s+e2α21−vβ2/ϕ2sexpλ3−σ32/2z+σ3w3zdz.It is clear that the local solution x1t,x2t,yt is positive on 0,τe. Thus, from the stochastic comparison theorem (see Theorem 3.1 in [18]), it follows that 0<ϕit≤xit≤Φit (i=1,2) and 0<ψt≤yt≤Ψt almost surely for t∈0,τe. Thus, for t∈0,τe,(11)lnϕit≤Xit≤lnΦit,lnψt≤Yt≤lnΨt,a.s.,i=1,2.
Note thatlnϕit, lnΦit, lnψt, and lnΨt (i=1,2) exist on 0,∞. Thus, τe=∞ a.s. This means that, for any X10,X20,Y0=lnx10,lnx20,lny0∈ℝ3, system (7) has a unique global solution X1t,X2t,Yt on 0,∞ a.s. Thus, for any initial value x10,x20,y0∈ℝ+3, model (5) has a unique global positive solution x1t,x2t,yt=eX1t,eX2t,eYt on 0,∞ a.s.
Now, we discuss the long-time properties of the solutions pathwise. We denoteut=1/t∫0tusds.Theorem 2.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. Then,(12)limsupt⟶∞Inxitt≤0,limsupt⟶∞Inytt≤0a.s.,i=1,2.Proof.
We consider the stochastic processΦ1t in system (8). Applying Itô’s formula to etlnΦ1 leads to(13)etlnΦ1t=lnx10+∫0teslnΦ1s+r1−a1Φ1s−σ122ds+M1t,where M1t=∫0tσ1esdw1s is a continuous local martingale with M10=0 and M1,M1t=∫0tσ12e2sds. Let n=1,2,…, γ>0, θ>1 and 0<ε<1. We choose T=nγ, α=εe−nγ, and β=θenγlnn/ε. By the exponential martingale inequality (see Theorem 1.7.4 in [7]), one can get(14)ℙsup0≤t≤TM1t−α2M1,M1t>β≤e−αβ=1nθ.
Since∑n=0∞1/nθ<∞ for θ>1, the Borel–Cantelli lemma (see Lemma 1.2.4 in [7]) implies that there exists a set Ω0∈ℱ with ℙΩ0=1 and an integer-valued random variable n0=n0ω such that, for every ω∈Ω0, M1t≤θenγlnn/ε+εe−nγ/2M1,M1t holds for all 0≤t≤nγ, n≥n0. Substituting the abovementioned inequality into (13), we have(15)etlnΦ1t≤lnx10+∫0teslnΦ1s+r1−a1Φ1sds−12∫0tσ12esds+εe−nγ2∫0tσ12e2sds+θenγlnnε,which holds for all 0≤t≤nγ, n≥n0. Note that, for 0≤s≤t≤nγ,(16)12εe−nγσ12e2s−12σ12es=12σ12esεes−nγ−1≤12σ12esε−1<0.
Thus, it follows from (15) that(17)etlnΦ1t≤lnx10+∫0teslnΦ1s+r1−a1Φ1sds+θenγlnnεholds for all 0≤t≤nγ, n≥n0. Consider function q1x=lnx+r1−a1x on 0,∞. Obviously, q1 has maximum value for x=1/a1>0 and q1,max=ln1/a1+r1−1. We denote K1=˙ln1/a1+r1−1∨1. Then,(18)etlnΦ1t≤lnx10+K1et+θenγlnnε,holds for all 0≤t≤nγ, n≥n0. Thus, for all 0≤n−1γ≤t≤nγ, n≥n0, we have(19)lnΦ1tlnt≤lnx10etlnt+K1lnt+θeγlnnεlnn−1γ.
Lettingn⟶∞ (and so t⟶∞), we obtain limsupt⟶∞lnΦ1t/lnt≤θeγ/ε a.s. Moreover, letting θ↓1, γ↓0 and ε↑1, one can get limsupt⟶∞lnΦ1t/lnt≤1 a.s. This, together with limt⟶∞lnt/t=0, yields limsupt⟶∞lnΦ1t/t≤0 a.s. Note that 0<x1t≤Φ1t a.s. for any t∈0,∞. Then, we have(20)limsupt⟶∞Inx1tt≤limsupt⟶∞lnΦ1tt≤0,a.s.
By a similar discussion as that mentioned above forx1, we also have(21)limsupt⟶∞Inx2tt≤0,limsupt⟶∞Inytt≤0,a.s.
The proof is, therefore, complete.Lemma 1.
(See [12]). We consider the one-dimensional stochastic differential equation(22)dxt=xta−bxtdt+σxtdwt,where a, b, and σ are positive constants and and wt is the standard Brownian motion. For any x0>0, let xt be the solution of equation (22) with x0=x0. If a>σ2/2, then(23)limt⟶∞lnxtt=0,limt⟶∞xt=a−σ2/2b,a.s.Theorem 3.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. If λi−σi2/2>0 (i=1,2,3), then(24)limt⟶∞lnxitt=0,limt⟶∞lnytt=0,a.s.i=1,2.
Moreover, the solution obeys(25)0<λ1−σ12/2a1≤liminft⟶∞x2t≤limsupt⟶∞x1t≤r1−σ12/2a1,a.s,0<λ1−σ22/2a2≤liminft⟶∞x2t≤limsupt⟶∞x1t≤r1−σ22/2a2,a.s.Proof.
From Theorem1, it follows that, for any t∈0,∞,(26)0<ϕit≤xit≤Φit,0<ψt≤yt≤Ψt,a.s.i=1,2.
Hereϕ1t and Φ1t are the solutions of stochastic equations, respectively.(27)dϕ1t=ϕ1tλ1−a1ϕ1tdt+σ1ϕ1tdw1t,dΦ1t=Φ1tr1−a1Φ1tdt+σ1Φ1tdw1t,where ϕ10=Φ10=x10>0. From Lemma 1, if λ1−σ12/2>0, then(28)limt⟶∞lnϕ1tt=0,limt⟶∞ϕ1t=λ1−σ12/2a1,a.s,limt⟶∞lnΦ1tt=0,limt⟶∞Φ1t=r1−σ12/2a1,a.s.
This, together with (12), yields(29)limt⟶∞lnx1tt=00<λ1−σ12/2a1≤liminft⟶∞x1t≤limsupt⟶∞x1t≤r1−σ12/2a1,a.s.
Similarly, ifλ2−σ22/2>0, then(30)limt⟶∞lnx2tt=00<λ2−σ22/2a2≤liminft⟶∞x2t≤limsupt⟶∞x2t≤r2−σ22/2a2,a.s.
Now, we showlimt⟶∞lnyt/t=0 a.s. Note that Ψt is the solution of the stochastic equation(31)dΨt=Ψte1α1v+e2α21−v−bΨtdt+σ3Ψtdw3t,with Ψ0=y0>0. Note that λ3−σ32/2>0. Thus, from Lemma 1, limt⟶∞lnΨt/t=0 a.s. From limt⟶∞lnϕ1t/t=0 and limt⟶∞lnϕ2t/t=0 a.s., it follows that, for any ε>0, there exists T1>0 such that(32)e−εt≤ϕit≤eεtfort≥T1,i=1,2.
Moreover, from the strong law of large numbers, it follows thatlimt⟶∞σ3w3t/t=0 a.s. Thus, for the abovementioned ε>0, there exists T2>0 such that(33)−εt≤σ3w3t≤εtfort≥T2.
Letκ=λ3−σ32/2. From the expression of ψt, it follows that, for any t>s≥T=T1∨T2,(34)1ψt=1yTe−κt−T−σ3w3t−w3T+b∫Tte−κt−s−σ3w3t−w3sds+∫Tte1α1vβ1ϕ1s+e2α21−vβ2ϕ2se−κt−s−σ3w3t−w3sds≤1yTe−κt−T+εt+T+b∫Tte−κt−s+εt+sds+e1α1vβ1+e2α21−vβ2∫Tteεse−κt−s+εt+sds.
Hence, fromκ>0, ε>0 and t>T, we have(35)e−3εt+Tψt≤1yTe−κt−T−2εt+T+bκ+εe−εte−3εT1−e−κ+εt−T+e1α1vβ1+e2α21−vβ2κ+2εe−3εT1−e−κ+2εt−T≤1yTe−κt−T−2εt+T+bκ+e1α1vβ1+e2α21−vβ2κ≤1yT+bκ+e1α1vβ1+e2α21−vβ2κ=˙K.
Thus,1/ψt≤Ke3εt+T a.s., which implies −lnψt≤lnK+3εt+T a.s. Then, from the arbitrariness of ε, it follows that liminft⟶∞lnψt/t≥0 a.s. Consequently,(36)0≤liminft⟶∞Inψtt≤liminft⟶∞Inytt≤limsupt⟶∞Inytt≤limt⟶∞InΨtt=0,a.s.
This implies(37)limx⟶∞lnytt=0,a.s
The proof is, therefore, complete.From proof of Theorem3, we can get the following result with the proof being omitted.Corollary 1.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. For i=1,2, if λi−σi2/2>0, then(38)limt⟶∞Inxitt=0,0<λi−σi2/2ai≤liminft⟶∞xitlimsupt⟶∞xit≤ri−σi2/2aia.s.


## 3. Persistence in Mean and Extinction

In this section, we show that, under some conditions, model (5) is persistent in mean and extinct.Lemma 2.
(See [19]). Assume that u∈CΩ×0,+∞,ℝ+, G∈CΩ×0,+∞,ℝ, and limt⟶∞Gt/t=0 a.s.(i)
If there areϱ≥0, ϱ0>0 and T>0 satisfying(39)lnut≤ϱt−ϱ0∫0tusds+Gt,a.s.,t≥T,thenlimsupt⟶∞ut≤ϱ/ϱ0 a.s. Furthermore, if ϱ=0, then limsupt⟶∞ut=0 a.s.(ii)
If there existϱ>0, ϱ0>0 and T>0 satisfying(40)lnut≥ϱt−ϱ0∫0tusds+Gt,a.s.,t≥T,thenliminft⟶∞ut≥ϱ/ϱ0 a.s.Theorem 4.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. If λi−σi2/2>0,i=1,2,3, then(41)liminft⟶∞xit≥λi−σi2/2ai>0,a.s.,i=1,2;liminft⟶∞yt+M1ytx1t+M2ytx2t≥λ3−σ32/2b>0,a.s.
Here,M1=e1α1vβ1/b and M2=e2α21−vβ2/b. This means that model (5) is persistent in mean.Proof.
From Theorem3, it follows that if λi−σi2/2>0,i=1,2, then(42)liminft⟶∞x1t≥λ1−σ12/2a1>0,liminft⟶∞x2t≥λ2−σ22/2a2>0,a.s.
For the predatoryt, using the Itô formula, we obtain(43)lnyt=∫0t−m1v−m21−v−σ322−bys+e1α1vx1sx1s+β1ys+e2α21−vx2sx2s+β2ysds+σ3w3t+lny0,which implies(44)lnyt≥λ3−σ322t−b∫0tys+M1ysx1s+M2ysx2sds+σ3w3t+lny0.
Hence,(45)1t∫0tbys+M1ysx1s+M2ysx2sds≥λ3−σ322+σ3w3tt+lny0t−lnytt.
Lettingt⟶∞ and by the strong law of numbers and Theorem 3, we have(46)liminft⟶∞yt+M1ytx1t+M2ytx2t≥λ3−σ32/2b>0,a.s.
The proof is complete.Theorem 5.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. If r1−σ12/2<0, r2−σ22/2<0, and λ3−σ32/2<0, then(47)limt⟶∞x1t=0,limt⟶∞x2t=0,limt⟶∞yt=0,a.s.
That is, for any initial valuex10,x20,y0∈ℝ+3, model (5) is extinct with probability one.Proof.
From the Itô formula, it follows that(48)lnx1t=∫0tr1−a1x1s−α1vysx1s+β1ys−σ122ds+σ1w1t+lnx10≤r1−σ122t+σ1w1t+lnx10.
This, together withlimt⟶∞σ1w1t/t+lnx10/t=0, yields limsupt⟶∞lnxit/t≤r1−σ12/2<0 a.s. Thus,(49)limt⟶∞x1t=0,a.s.
Similarly, fromr2−σ22/2<0, it follows that(50)limt⟶∞x2t=0,a.s.
Moreover, from (43), we have(51)lnyt≤∫0te1α1v+e2α21−v−m1v−m21−v−σ322ds+σ3w3t+lny0=λ3−σ322t+σ3w3t+lny0.
Fromlimt⟶∞σ3w3t/t+lny0/t=0, it follows that limsupt⟶∞lnyt/t≤λ3−σ32/2<0 a.s. This implies(52)limt⟶∞yt=0,a.s.
Therefore, model (5) is extinct exponentially. The proof is complete.Theorem 6.
(I) If the predator is absent, i.e.,yt=0 a.s. for all t≥0, then the quantities of prey x1t and prey x2t satisfy the following:(i)
Ifr1−σ12/2>0 and r2−σ22/2>0, then(53)limt⟶∞x1t=r1−σ12/2a1,limt⟶∞x2t=r2−σ22/2a2,a.s(ii)
Ifr1−σ12/2>0 and r2−σ22/2<0, then(54)limt⟶∞x1t=r1−σ12/2a1,limt⟶∞x2t=0,a.s(iii)
Ifr1−σ12/2<0 and r2−σ22/2>0, then(55)limt⟶∞x1t=0,limt⟶∞x2t=r2−σ22/2a2,a.s(iv)
Ifr1−σ12/2<0 and r2−σ22/2<0, then(56)limt⟶∞x1t=0,limt⟶∞x2t=0,a.s(II) If the prey in patch 2 is absent, i.e.,x2t=0 a.s. for all t≥0, then the quantities of prey x1t and predator yt satisfy the following:(i)
Ifλ1−σ12/2>0 and e1α1v−m1v−m21−v−σ32/2>0, then(57)liminft⟶∞x1t≥λ1−σ12/2a1,a.s;liminft⟶∞yt+M1ytx1t≥e1α1v−m1v−m21−v−σ32/2b,a.s(ii)
Ifr1−σ12/2<0 and e1α1v−m1v−m21−v−σ32/2<0, then(58)limt⟶∞x1t=0,limt⟶∞yt=0,a.s(III) If the prey in patch 1 is absent, i.e.,x1t=0 a.s. for all t≥0, then the quantities of prey x2t and predator yt satisfy the following:(i)
Ifλ2−σ22/2>0 and e2α21−v−m1v−m21−v−σ32/2>0, then(59)liminft⟶∞x2t≥λ2−σ22/2a1,a.s.;liminft⟶∞yt+M2ytx2t≥e2α21−v−m1v−m21−v−σ32/2b,a.s(ii)
Ifr2−σ22/2<0 and e2α21−v−m1v−m21−v−σ32/2<0, then(60)limt⟶∞x2t=0limt⟶∞yt=0,a.s(IV) If the prey is absent, i.e.,x1t=x2t=0 a.s. for all t≥0, then the predator dies with probability oneProof.
(I) In the absence of the predator, from the Itô formula, it follows that(61)lnxit=ri−σi22t−ai∫0txisds+σiwit+lnxi0,a.s.,i=1,2.
Note thatlimt⟶∞σiwit/t+lnxi0/t=0 a.s., i=1,2. Thus, from Lemma 2, it follows that if ri−σi2/2>0,i=1,2, then(62)limt⟶∞xit=ri−σi2/2ai,a.s.,i=1,2.
Moreover, from Lemma2, it follows that if ri−σi2/2<0,i=1,2, then(63)limt⟶∞xit=0,a.s.,i=1,2.
Thus, (I) holds.
Next, we prove (II). From Theorem3, it follows that if λ1−σ12/2>0, then(64)limt⟶∞x1t≥λ1−σ12/2a1,a.s.
Moreover, in the absence of the prey in patch 2, from the Itô formula, it follows that(65)lnyt=∫0t−m1v−m21−v−σ322−bys+e1α1vx1sx1s+β1ysds+σ3w3t+lny0≥e1α1v−m1v−m21−v−σ322t−b∫0tys+M1ysx1sds+σ3w3t+lny0.
Hence,(66)bt∫0tys+M1ysx1sds≥e1α1v−m1v−m21−v−σ322+σ3w3tt+lny0t−lnytt.
Lettingt⟶∞ and by the strong law of numbers and Theorem 2, we have(67)bliminft⟶∞yt+M1ytx1t≥e1α1v−m1v−m21−v−σ322−limsupt⟶∞lnytt≥e1α1v−m1v−m21−v−σ322a.s.
Thus, we have(68)liminft⟶∞yt+M1ytx1t≥e1α1v−m1v−m21−v−σ32/2b,a.s.
Furthermore, from the proof of (I), ifr1−σ12/2<0, then(69)limt⟶∞x1t=0,a.s.
Moreover, in the absence of the prey in patch 2, from the Itô formula, it follows that(70)lnyt≤e1α1v−m1v−m21−v−σ322t−b∫0tysds+σ3w3t+lny0.
Note thatlimt⟶∞σ3w3t/t+lny0/t=0 a.s. Thus, from Lemma 2, if e1α1v−m1v−m21−v−σ32/2<0, then(71)limt⟶∞yt=0,a.s.
Hence, (II) holds. The proof of (III) is similar to (II) and, hence, is omitted.
At last, we prove (IV). In the absence of the prey, from the Itô formula, it follows that(72)lnyt=∫0t−m1v−m21−v−σ322−bysds+σ3w3t+lny0=−m1v−m21−v−σ322t−b∫0tysds+σ3w3t+lny0.
Note thatlimt⟶∞σ3w3t/t+lny0/t=0 a.s. Thus, from Lemma 2, it follows that(73)limt⟶∞yt=0,a.s.
The proof is, therefore, complete.


## 4. Stochastic Permanence

In this section, we investigate the stochastic permanence of model (5).


### 4.1. Stochastically Ultimate Boundedness

In this subsection, we first use two different ways to prove the boundedness of model (5) and then show that model (5) is stochastically ultimately bounded by Chebyshev’s inequality. The definition of stochastically ultimate boundedness of model (5) was introduced in the literature [20, 21] as follows.Definition 1.
(See [20, 21]). Model (5) is called stochastically ultimately bounded if, for any ε∈0,1, there exist positive constants Hi=Hiε,i=1,2,3 such that the solution x1t,x2t,yt of model (5) with any initial value x10,x20,y0∈ℝ+3 has the property that(74)limsupt⟶∞ℙxit>Hi<ε,limsupt⟶∞ℙyt>H3<ε,i=1,2.Theorem 7.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. Then, for any p>0, the solution x1t,x2t,yt obeys(75)limsupt⟶∞Exipt≤pp+1p+11/p+ri+p/2σi2p+1aip=˙Kip,i=1,2,limsupt⟶∞Eypt≤pp+1p+11/p+e1α1v+e2α21−v+p/2σ32p+1bp=˙K3p.Proof.
Applying the Itô formula toetΦip leads to(76)EetΦ1pt=x10p+pE∫0tesΦ1ps1p+r1+p−12σ12−a1Φ1sds≤x10p+pE∫0tesΦ1ps1p+r1+p2σ12−a1Φ1sds.
Clearly, functionfx=xp1/p1/p+r1+p/2σ12−a1x reaches its maximum value at x=p1/p+r1+p/2σ12/a1p+1>0 and fmax=p/a1p1/p+r1+p/2σ12/p+1p+1. Thus,(77)EetΦ1pt≤x10p+ppa1p1/p+r1+p/2σ12p+1p+1et−1,which implies(78)limsupt⟶∞EΦ1pt≤pp+1p+11/p+r1+p/2σ12p+1a1p.
By a similar discussion as inΦ1t, we also have(79)limsupt⟶∞EΦ2pt≤pp+1p+11/p+r2+p/2σ22p+1a2p,limsupt⟶∞EΨpt≤pp+1p+11/p+e1α1v+e2α21−v+p/2σ32p+1bp.
From (12), it follows that 0<xit≤Φit and 0<yt≤Ψt a.s. on t∈0,+∞,i=1,2. Then, for any p>0, we have(80)0<Exipt≤EΦipt,0<Eypt≤EΨpt,i=1,2.
Thus, the desired results can be obtained immediately. The proof is complete.Theorem 8.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. Then,(81)limsupt⟶∞Exit≤K2eiλ,limsupt⟶∞Eyt≤K2λ,i=1,2.where K2=e1r1+λ2/4a1+e2r2+λ2/4a2 and λ=m1v+m21−v.Proof.
We denoteλ=m1v+m21−v and Nt=e1x1t+e2x2t+yt. From model (5) and the Itô formula, it follows that(82)ENt=N0+E∫0t−λNs+e1r1+λx1s−e1a1x12s+e2r2+λx2s−e2a2x22s−by2sds.
This, together witheiri+λxit−eiaixi2t≤eiri+λ2/4ai,i=1,2, yields(83)dENtdt=E−λNt+e1r1+λx1t−e1a1x12t++e2r2+λx2t−e2a2x22t−by2t≤−λENt+Ee1r1+λx1t−e1a1x12t+Ee2r2+λx2t−e2a2x22t≤−λENt+e1r1+λ24a1+e2r2+λ24a2=˙K2−λENt,where K2=e1r1+λ2/4a1+e2r2+λ2/4a2. Thus, by the comparison theorem, we have(84)0≤limsupt⟶∞ENt≤K2λ.
From the solution of model (5) which is positive, it follows that(85)limsupt⟶∞Exit≤K2eiλ,limsupt⟶∞Eyt≤K2λ,i=1,2.
The proof is complete.Theorem 9.
Model (5) is stochastically ultimately bounded.Proof.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. From Theorem 7, it follows that(86)limsupt⟶∞Exit≤Ki1,limsupt⟶∞Eyt≤K31,i=1,2.
For anyε∈0,1, let Hi=Ki1/ε+1, i=1,2,3. From Chebyshev’s inequality, it follows that(87)ℙxit>Hi≤ExitHi,ℙyt>H3≤EytH3,i=1,2.
Hence,(88)limsupt⟶∞ℙxit>Hi≤limsupt⟶∞ExitHi<ε,i=1,2,limsupt⟶∞ℙyt>H3≤limsupt⟶∞EytH3<ε.
The proof is, therefore, complete.
Similarly, from Theorem8, together with Chebyshev’s inequality, one can say that model (5) is also stochastically ultimately bounded.


### 4.2. Stochastic Permanence

In this section, we show that the model (5) is stochastically permanent. The definition of stochastic permanence of model (5) is introduced as follows.Definition 2.
(See [20, 21]). Model (5) is called stochastically permanent if, for any ε∈0,1, there exist positive constants δi=δiε, Hi=Hiε, and δi<Hi, i=1,2,3, such that solution x1t,x2t,yt of model (5) with any initial value x10,x20,y0∈ℝ+3 has the property that(89)liminft⟶∞ℙxit≤Hi≥1−εlimsupt⟶∞ℙyt≤H3≥1−ε,i=1,2.liminft⟶∞ℙxit≥δi≥1−εandlimsupt⟶∞ℙyt≥δ3≥1−ε,i=1,2.
For simplicity, we denoteκi=λi−σi2,i=1,2,3. To prove that model (5) is stochastically permanent, we define(90)u1=1x1,u2=1x2,u3=1y.
From the Itô formula, it follows that(91)du1t=a1−r1u1t+σ12u1t+α1vu3t+β1u1tu12tdt−σ1u1tdw1t,du2t=a2−r2u2t+σ22u2t+α21−vu3t+β2u2tu22tdt−σ2u2tdw2t,du3t=u3tbu3t+m1v+m21−v+σ32−e1α1vu3tu3t+β1u1t−e2α21−vu3tu3t+β2u2tdt−σ3u3tdw3t,with initial value u10,u20,u30=1/x10,1/x20,1/y0. Furthermore, we consider the following auxiliary system:(92)dθ1t=a1−κ1θ1tdt−σ1θ1tdw1t,dθ2t=a2−κ2θ2tdt−σ2θ2tdw2t,dθ3t=b−κ3θ3t+e1α1vβ1θ1t+e2α21−vβ2θ2tdt−σ3θ3tdw3t,with initial value θ10,θ20,θ30=1/x10,1/x20,1/y0.
Thanks to Lemma 4.2 in [17], system (92) has the exact solution. Moreover, from the stochastic comparison theorem, it follows that, for t∈0,∞,(93)0<uit≤θit,a.s.,i=1,2,3.Theorem 10.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. If κi>0,i=1,2,3, then(94)limsupt⟶∞E1xit≤aiκi=˙Mi,i=1,2,limsupt⟶∞E1yt≤bκ3+a1e1α1vβ1κ1κ3+a2e2α21−vβ2κ2κ3=˙M3.Proof.
Integrating the both sides of the first equation of (92) and taking the expectation yields(95)Eθ1t=1x10+E∫0ta1−κ1θ1sds.
Then, we have the differential equationdEθ1t/dt=a1−κ1Eθ1t with Eθ10=1/x10. Thus, one can get(96)Eθ1t=1x10e−κ1t+a1κ11−e−κ1t.
This, together withκ1>0, it yields limt⟶∞Eθ1t=a1/κ1. From 1/x1t=u1t and (93), it follows that(97)limsupt⟶∞E1x1t=limsupt⟶∞Eu1t≤limsupt⟶∞Eθ1t=a1κ1=˙M1.
Similarly, one can get(98)limsupt⟶∞E1x2t=limsupt⟶∞Eu2t≤limsupt⟶∞Eθ2t=a2κ2=˙M2.
Integrating the both sides of the third equation of system (92) and taking the expectation yields(99)Eθ3t=1y0+E∫0tb−κ3θ3s+e1α1vβ1θ1s+e2α21−vβ2θ2sds.
Thus, we can getdEθ3t/dt=b−κ3Eθ3t+e1α1vβ1Eθ1t+e2α21−vβ2Eθ2t with Eθ30=1/y0. Therefore, we have(100)Eθ3t=1y0e−κ3t+bκ31−e−κ3t+e1α1vβ1∫0te−κ3t−sEθ1sds+e2α21−vβ2∫0te−κ3t−sEθ2sds.
From L’Hospital’s rule and (97), it follows that(101)limt⟶∞∫0te−κ3t−sEθ1sds=limt⟶∞∫0teκ3sEθ1sdseκ3t=limt⟶∞Eθ1tκ3=a1κ1κ3.
Similarly, we also have(102)limt⟶∞∫0te−κ3t−sEθ2sds=limt⟶∞∫0teκ3sEθ2sdseκ3t=limt⟶∞Eθ2tκ3=a1κ2κ3.
This, together with (100) and κ3>0, yields(103)limsupt⟶∞Eθ3t≤bκ3+a1e1α1vβ1κ1κ3+a2e2α21−vβ2κ2κ3=˙M3.
Thus, from1/yt=u3t and (93), it follows that(104)limsupt⟶∞E1yt=limsupt⟶∞Eu3t≤limsupt⟶∞Eθ3t≤M3.
The proof is, therefore, complete.Theorem 11.
Ifκi>0,i=1,2,3, then model (5) is stochastically permanent.
Proof. For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. From Theorem 10, it follows that(105)limsupt⟶∞E1xit≤Mi,limsupt⟶∞E1yt≤M3,1=1,2.
For anyε∈0,1, let δi=ε/Mi,i=1,2,3; then,(106)ℙxit<δi=ℙ1xit>1δi≤E1/xit1/δi=δiE1xit,i=1,2,ℙyt<δ3=ℙ1yt>1δ3≤E1/yt1/δ3=δ3E1yt.
Thus,(107)limsupt⟶∞ℙxit<δi≤limsupt⟶∞δiE1xit=ε,i=1,2,limsupt⟶∞ℙyt<δ3≤limsupt⟶∞δ3E1yt=ε.
This implies(108)liminft⟶∞ℙxit≥δi≥1−ε,liminft⟶∞ℙyt≥δ3≥1−ε,i=1,2.
Letε∈0,1 be sufficiently small such that δi<Hi,i=1,2,3. Then, from Theorem 9 and Definition 2, one can say that model (5) is stochastically permanent. The proof is, therefore, complete.


## 4.1. Stochastically Ultimate Boundedness

In this subsection, we first use two different ways to prove the boundedness of model (5) and then show that model (5) is stochastically ultimately bounded by Chebyshev’s inequality. The definition of stochastically ultimate boundedness of model (5) was introduced in the literature [20, 21] as follows.Definition 1.
(See [20, 21]). Model (5) is called stochastically ultimately bounded if, for any ε∈0,1, there exist positive constants Hi=Hiε,i=1,2,3 such that the solution x1t,x2t,yt of model (5) with any initial value x10,x20,y0∈ℝ+3 has the property that(74)limsupt⟶∞ℙxit>Hi<ε,limsupt⟶∞ℙyt>H3<ε,i=1,2.Theorem 7.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. Then, for any p>0, the solution x1t,x2t,yt obeys(75)limsupt⟶∞Exipt≤pp+1p+11/p+ri+p/2σi2p+1aip=˙Kip,i=1,2,limsupt⟶∞Eypt≤pp+1p+11/p+e1α1v+e2α21−v+p/2σ32p+1bp=˙K3p.Proof.
Applying the Itô formula toetΦip leads to(76)EetΦ1pt=x10p+pE∫0tesΦ1ps1p+r1+p−12σ12−a1Φ1sds≤x10p+pE∫0tesΦ1ps1p+r1+p2σ12−a1Φ1sds.
Clearly, functionfx=xp1/p1/p+r1+p/2σ12−a1x reaches its maximum value at x=p1/p+r1+p/2σ12/a1p+1>0 and fmax=p/a1p1/p+r1+p/2σ12/p+1p+1. Thus,(77)EetΦ1pt≤x10p+ppa1p1/p+r1+p/2σ12p+1p+1et−1,which implies(78)limsupt⟶∞EΦ1pt≤pp+1p+11/p+r1+p/2σ12p+1a1p.
By a similar discussion as inΦ1t, we also have(79)limsupt⟶∞EΦ2pt≤pp+1p+11/p+r2+p/2σ22p+1a2p,limsupt⟶∞EΨpt≤pp+1p+11/p+e1α1v+e2α21−v+p/2σ32p+1bp.
From (12), it follows that 0<xit≤Φit and 0<yt≤Ψt a.s. on t∈0,+∞,i=1,2. Then, for any p>0, we have(80)0<Exipt≤EΦipt,0<Eypt≤EΨpt,i=1,2.
Thus, the desired results can be obtained immediately. The proof is complete.Theorem 8.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. Then,(81)limsupt⟶∞Exit≤K2eiλ,limsupt⟶∞Eyt≤K2λ,i=1,2.where K2=e1r1+λ2/4a1+e2r2+λ2/4a2 and λ=m1v+m21−v.Proof.
We denoteλ=m1v+m21−v and Nt=e1x1t+e2x2t+yt. From model (5) and the Itô formula, it follows that(82)ENt=N0+E∫0t−λNs+e1r1+λx1s−e1a1x12s+e2r2+λx2s−e2a2x22s−by2sds.
This, together witheiri+λxit−eiaixi2t≤eiri+λ2/4ai,i=1,2, yields(83)dENtdt=E−λNt+e1r1+λx1t−e1a1x12t++e2r2+λx2t−e2a2x22t−by2t≤−λENt+Ee1r1+λx1t−e1a1x12t+Ee2r2+λx2t−e2a2x22t≤−λENt+e1r1+λ24a1+e2r2+λ24a2=˙K2−λENt,where K2=e1r1+λ2/4a1+e2r2+λ2/4a2. Thus, by the comparison theorem, we have(84)0≤limsupt⟶∞ENt≤K2λ.
From the solution of model (5) which is positive, it follows that(85)limsupt⟶∞Exit≤K2eiλ,limsupt⟶∞Eyt≤K2λ,i=1,2.
The proof is complete.Theorem 9.
Model (5) is stochastically ultimately bounded.Proof.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. From Theorem 7, it follows that(86)limsupt⟶∞Exit≤Ki1,limsupt⟶∞Eyt≤K31,i=1,2.
For anyε∈0,1, let Hi=Ki1/ε+1, i=1,2,3. From Chebyshev’s inequality, it follows that(87)ℙxit>Hi≤ExitHi,ℙyt>H3≤EytH3,i=1,2.
Hence,(88)limsupt⟶∞ℙxit>Hi≤limsupt⟶∞ExitHi<ε,i=1,2,limsupt⟶∞ℙyt>H3≤limsupt⟶∞EytH3<ε.
The proof is, therefore, complete.
Similarly, from Theorem8, together with Chebyshev’s inequality, one can say that model (5) is also stochastically ultimately bounded.


## 4.2. Stochastic Permanence

In this section, we show that the model (5) is stochastically permanent. The definition of stochastic permanence of model (5) is introduced as follows.Definition 2.
(See [20, 21]). Model (5) is called stochastically permanent if, for any ε∈0,1, there exist positive constants δi=δiε, Hi=Hiε, and δi<Hi, i=1,2,3, such that solution x1t,x2t,yt of model (5) with any initial value x10,x20,y0∈ℝ+3 has the property that(89)liminft⟶∞ℙxit≤Hi≥1−εlimsupt⟶∞ℙyt≤H3≥1−ε,i=1,2.liminft⟶∞ℙxit≥δi≥1−εandlimsupt⟶∞ℙyt≥δ3≥1−ε,i=1,2.
For simplicity, we denoteκi=λi−σi2,i=1,2,3. To prove that model (5) is stochastically permanent, we define(90)u1=1x1,u2=1x2,u3=1y.
From the Itô formula, it follows that(91)du1t=a1−r1u1t+σ12u1t+α1vu3t+β1u1tu12tdt−σ1u1tdw1t,du2t=a2−r2u2t+σ22u2t+α21−vu3t+β2u2tu22tdt−σ2u2tdw2t,du3t=u3tbu3t+m1v+m21−v+σ32−e1α1vu3tu3t+β1u1t−e2α21−vu3tu3t+β2u2tdt−σ3u3tdw3t,with initial value u10,u20,u30=1/x10,1/x20,1/y0. Furthermore, we consider the following auxiliary system:(92)dθ1t=a1−κ1θ1tdt−σ1θ1tdw1t,dθ2t=a2−κ2θ2tdt−σ2θ2tdw2t,dθ3t=b−κ3θ3t+e1α1vβ1θ1t+e2α21−vβ2θ2tdt−σ3θ3tdw3t,with initial value θ10,θ20,θ30=1/x10,1/x20,1/y0.
Thanks to Lemma 4.2 in [17], system (92) has the exact solution. Moreover, from the stochastic comparison theorem, it follows that, for t∈0,∞,(93)0<uit≤θit,a.s.,i=1,2,3.Theorem 10.
For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. If κi>0,i=1,2,3, then(94)limsupt⟶∞E1xit≤aiκi=˙Mi,i=1,2,limsupt⟶∞E1yt≤bκ3+a1e1α1vβ1κ1κ3+a2e2α21−vβ2κ2κ3=˙M3.Proof.
Integrating the both sides of the first equation of (92) and taking the expectation yields(95)Eθ1t=1x10+E∫0ta1−κ1θ1sds.
Then, we have the differential equationdEθ1t/dt=a1−κ1Eθ1t with Eθ10=1/x10. Thus, one can get(96)Eθ1t=1x10e−κ1t+a1κ11−e−κ1t.
This, together withκ1>0, it yields limt⟶∞Eθ1t=a1/κ1. From 1/x1t=u1t and (93), it follows that(97)limsupt⟶∞E1x1t=limsupt⟶∞Eu1t≤limsupt⟶∞Eθ1t=a1κ1=˙M1.
Similarly, one can get(98)limsupt⟶∞E1x2t=limsupt⟶∞Eu2t≤limsupt⟶∞Eθ2t=a2κ2=˙M2.
Integrating the both sides of the third equation of system (92) and taking the expectation yields(99)Eθ3t=1y0+E∫0tb−κ3θ3s+e1α1vβ1θ1s+e2α21−vβ2θ2sds.
Thus, we can getdEθ3t/dt=b−κ3Eθ3t+e1α1vβ1Eθ1t+e2α21−vβ2Eθ2t with Eθ30=1/y0. Therefore, we have(100)Eθ3t=1y0e−κ3t+bκ31−e−κ3t+e1α1vβ1∫0te−κ3t−sEθ1sds+e2α21−vβ2∫0te−κ3t−sEθ2sds.
From L’Hospital’s rule and (97), it follows that(101)limt⟶∞∫0te−κ3t−sEθ1sds=limt⟶∞∫0teκ3sEθ1sdseκ3t=limt⟶∞Eθ1tκ3=a1κ1κ3.
Similarly, we also have(102)limt⟶∞∫0te−κ3t−sEθ2sds=limt⟶∞∫0teκ3sEθ2sdseκ3t=limt⟶∞Eθ2tκ3=a1κ2κ3.
This, together with (100) and κ3>0, yields(103)limsupt⟶∞Eθ3t≤bκ3+a1e1α1vβ1κ1κ3+a2e2α21−vβ2κ2κ3=˙M3.
Thus, from1/yt=u3t and (93), it follows that(104)limsupt⟶∞E1yt=limsupt⟶∞Eu3t≤limsupt⟶∞Eθ3t≤M3.
The proof is, therefore, complete.Theorem 11.
Ifκi>0,i=1,2,3, then model (5) is stochastically permanent.
Proof. For anyx10,x20,y0∈ℝ+3, let x1t,x2t,yt be the solution of model (5) with initial value x10,x20,y0. From Theorem 10, it follows that(105)limsupt⟶∞E1xit≤Mi,limsupt⟶∞E1yt≤M3,1=1,2.
For anyε∈0,1, let δi=ε/Mi,i=1,2,3; then,(106)ℙxit<δi=ℙ1xit>1δi≤E1/xit1/δi=δiE1xit,i=1,2,ℙyt<δ3=ℙ1yt>1δ3≤E1/yt1/δ3=δ3E1yt.
Thus,(107)limsupt⟶∞ℙxit<δi≤limsupt⟶∞δiE1xit=ε,i=1,2,limsupt⟶∞ℙyt<δ3≤limsupt⟶∞δ3E1yt=ε.
This implies(108)liminft⟶∞ℙxit≥δi≥1−ε,liminft⟶∞ℙyt≥δ3≥1−ε,i=1,2.
Letε∈0,1 be sufficiently small such that δi<Hi,i=1,2,3. Then, from Theorem 9 and Definition 2, one can say that model (5) is stochastically permanent. The proof is, therefore, complete.


## 5. Stationary Distribution and Ergodicity

This section will show that there is an ergodic stationary distribution for the solution of model (5). Let ℬℝ+3 be the Borel σ-algebra on ℝ+3 and ℝ+3,ℬℝ+3,ℱtt≥0,ℙ be a filtered complete probability space satisfying usual hypotheses. Consider a Markov process Xt in the state space ℝ+3,ℬℝ+3,ℙ that satisfies the following stochastic differential equation:(109)dXt=bXtdt+gXtdWt,X0=X0.Here,Wt is a standard 3-dimensional standard Brownian motion, b:ℝ+3⟶ℝ3, and g:ℝ+3⟶ℝ3×3 are all locally Lipschitz functions. The diffusion matrix of Xt is defined as JX=gXgTX=aijX. Let X=x1,x2,y,(110)bX=x1r1−a1x1−α1vyx1+β1yx2r2−a2x2−α21−vyx2+β2yy−m1v−m21−v−by+e1α1vyx1+β1y+e2α21−vyx2+β2y,gX=diagσ1x1,σ2x2,σ3y, and W=w1,w2,w3T. Then, model (5) reduces to the abstract form (110) with diffusion matrix JX=diagσ12x12,σ22x22,σ32y2. The norm X is given by X=x12+x22+y2. Moreover, we denote PtX0,A the transition probability(111)PtX0,A=ℙXt∈A|X0=X0,∀t∈ℝ+,∀X0∈ℝ+3,∀X0∈ℬℝ+3.Definition 3.
(See [22]). Let ℙt,X,· be the probability measure induced by Xt in (110) with X0=X0. That is, ℙt,X0,A=ℙXt∈A|X0=X0, for any Borel set A∈ℬℝ+3. If there exists a probability measure μ· such that limt⟶∞ℙt,X0,A=μA for all X0∈ℝ+3 and A∈ℬℝ+3, then equation (110) has a stationary distribution μ·.Lemma 3.
(See [23, 24]). The Markov process Xt in (110) has a unique ergodic stationary distribution μ· if there exists a bounded open domain D⊂E3 (E3 denotes 3-dimensional Euclidean space) with regular boundary Γ, and(A1): there is a positive numberM such that ∑i,j=13aijXξiξj≥M|ξ|2, X∈D, ξ∈ℝ3(A2): there exists a nonnegativeC2 function V such that LV is negative for any X∈E3/D
Then,(112)ℙlimT⟶∞1T∫0TfXtdt=∫Edfxμdx=1,for all x∈E3, where The caption of Figure 5 is unclear. Please rephrase the caption for clarity and correctness. is a function integrable with respect to the measure μ.Theorem 12.
Ifκ1−e1α1vβ1>0, κ2−e2α21−vβ2>0, and κ3>0, then for any initial value x10,x20,y0∈ℝ+3, model (5) has a stationary distribution μ· and the solutions have ergodic property.Proof.
In what follows, for the simplification, we denotex1t, x2t, and yt as x1, x2, and y, respectively. We define C2 function V1: ℝ+3⟶ℝ+ by(113)V1X=x1+x2+y,for X=x1,x2,y∈ℝ+3. From the Itô formula, it follows that(114)LV1X=x1r1−a1x1−α1vyx1+β1y+x2r2−a2x2−α21−vyx2+β2y+y−m1v−m21−v−by+e1α1vx1x1+β1y+e2α21−vx2x2+β2y≤x1r1−a1x1+x2r2−a2x2+ye1α1v+e2α21−v−by,=−a1x12+r1x1−a2x22+r2x2−by2+e1α1v+e2α21−vy.
We defineV2: ℝ+3⟶ℝ+ by(115)V2X=x1−1+x2−1+y−1,for X=x1,x2,y∈ℝ+3. Applying the Itô formula, we have(116)LV2X=−x1−1r1−a1x1−α1vyx1+β1y−σ12−x2−1r2−a2x2−α21−vyx2+β2y−σ22−y−1−m1v−m21−v−by+e1α1vx1x1+β1y+e2α21−vx2x2+β2y−σ32≤−x1−1κ1−a1x1−x2−1κ2−a2x2−y−1κ3−by−e1α1vβ1yx1+β1y−e2α21−vβ2yx2+β2y≤−x1−1κ1−a1x1−x2−1κ2−a2x2−y−1κ3−by−e1α1vβ1yx1−e2α21−vβ2yx2,=−κ1−e1α1vβ1x1−1+a1−κ2−−e2α21−vβ2x2−1+a2−κ3y−1+b.
LetVX=V1X+V2X=x1+x2+y+x1−1+x2−1+y−1. Then,(117)LVX=LV1X+LV2X≤−a1x12+r1x1−κ1−e1α1vβ1x1−1+a1−a2x22+r2x2−κ2−e2α21−vβ2x2−1+a2−by2+e1α1v+e2α21−vy−κ3y−1+b,=˙fx1+gx2+hy,where(118)fx1=−a1x12+r1x1−κ1−e1α1vβ1x1−1+a1,gx2=−a2x22+r2x2−κ2−e2α21−vβ2x2−1+a2,hy=−by2+e1α1v+e2α21−vy−κ3y−1+b.
Obviously,fx1, gx2, and hy are all functions with an upper bound on ℝ+. Thus, we denote(119)fu=supx1∈ℝ+fx1,gu=supx2∈ℝ+gx2,hu=supy∈ℝ+hy.
Letρ be a sufficiently small positive number. We define a bounded open set as follows:(120)D=x1,x2,y∈ℝ+3|ρ<x1<1ρ,ρ<x2<1ρ,ρ<y<1ρ⊂ℝ+3.
Now, we prove thatLVX≤−1 on ℝ+3/D. From κ1−e1α1vβ1>0, it follows that(121)LVX≤fx1+gx2+hy≤fx1+gu+hu⟶−∞,a.s.,x1⟶0+orx1⟶+∞.
Similarly, fromκ2−e2α21−vβ2>0 and κ3>0, we also have(122)LVX≤fx1+gx2+hy≤fu+gx2+hu⟶−∞,a.s.,x2⟶0+orx2⟶+∞.LVX≤fx1+gx2+hy≤fu+gu+hy⟶−∞,a.s.,y⟶0+ory⟶+∞.
Consequently, for sufficiently smallρ, one can see(123)LVX≤−1,for allx1,x2,y∈ℝ+3D.
Hence, condition (A2) of Lemma3 holds.
Letσ2=σ12∧σ22∧σ32 and M=ρ2σ2. Then, for any X=x1,x2,y∈D and ξ=ξ1,ξ2,ξ2∈ℝ3, we have(124)∑i,j=13aijXξiξj=σ12x12ξ12+σ22x22ξ22+σ32y2ξ32≥ρ2σ2ξ12+ξ22+ξ32=˙Mξ2.
Thus, (A1) in Lemma3 is satisfied. From Lemma 3, we can say that model (5) has a stationary distribution μ· and the solutions of model (5) have ergodic property.


## 6. Application of Main Results

In this section, we first apply the main results to two stochastic two-species predator-prey models. Then, we present the application of the main results to stochastic two-patch predator-prey model (4).


### 6.1. Two-Species Predator-Prey Model

If the predator only stays in one patch, then one can obtain the following stochastic predator-prey model (obtained by takingv=0 or v=1 in model (5)).(125)dxt=xtr−axt−αytxt+βytdt+σ1xtdw1t,dyt=yt−m−byt+eαxtxt+βytdt+σ2ytdw2t,where x0,y0=x0,y0∈ℝ+2. Model (125) was discussed in [25]. Furthermore, Linh and Ton [26] considered the corresponding nonautonomous model of (125). By a similar discussion as in Theorem 1, for any x0,y0∈ℝ+2, model (21) has a unique global positive solution xt,yt. Moreover, for model (125), we have the following results. For simplicity, we denote δ1=r−α/β and δ2=eα−m.Corollary 2.
For anyx0,y0∈ℝ+2, let xt,yt be the solution of model (125) with initial value x0,y0.(I)
By a similar discussion as in Theorems2–6, we can obtain the following results:(i)
The solutionxt,yt of model (126) obeys(126)limsupt⟶∞lnxtt≤0,limsupt⟶∞lnytt≤0,a.s.Moreover, ifδ1−σ12/2>0 and δ2−σ22/2>0; then,(127)limt⟶∞lnxtt=0,limt⟶∞lnytt=0,a.s.(ii)
Ifδ1−σ12/2>0 and δ2−σ22/2>0, then(128)liminft⟶∞xt≥δ1−σ12/2a>0,liminft⟶∞yt+Mytxt≥δ2−σ22/2b>0,a.s.Here,M=eαβ/b. This means that model (125) is persistent in mean.(iii)
Ifr−σ12/2<0 and δ2−σ22/2<0, then(129)limt⟶∞xt=0,limt⟶∞yt=0,a.s.(iv)
If the predator is absent, i.e.,yt=0 a.s. for all t≥0, then the quantity of prey in model (125) satisfies the following:(130)limt⟶∞xt=r−σ12/2a1,a.s.,ifr−σ12/2>0;limt⟶∞xt=0,a.s.,ifr−σ12/2<0.(v)
If the prey is absent, i.e.,xt=0 a.s. for all t≥0, then the predator in model (125) dies with probability one.(II)
By a similar discussion as in Theorems7, 10 and 11, one can get the following results:(i)
The solutionxt,yt of model (125) obeys(131)limsupt⟶∞Expt≤pp+1p+11/p+r+p/2σ12p+1aplimsupt⟶∞Eypt≤pp+1p+11/p+eα+p/2σ22p+1bp.Moreover, from Chebyshev’s inequality, model (125) is stochastically ultimately bounded.(ii)
Ifςi=˙δi−σi2>0i=1,2, then the solution xt,yt of model (125) obeys(132)limsupt⟶∞E1xt≤aς1,limsupt⟶∞E1yt≤bς2+aeαβς1ς2.Furthermore, ifςi>0,i=1,2, then model (125) is stochastically permanent.By a similar discussion as in Theorem12, if ς1−eαβ>0 and ς2>0, then any x0,y0∈ℝ+2, model (125) has a stationary distribution μ· and the solutions have an ergodic property.Remark 1.
In [25], the authors show that model (125) has a unique global positive solution by using stopping times and contradiction. In this paper, the stochastic comparison theorem is used to prove that the model has a unique global positive solution. Reference [25] only shows that if δ1−σ12/2>0 and δ2−σ22/2>0, then limt⟶∞lnxt/t=0 and limt⟶∞lnyt/t=0 a.s. However, we also show that the sample Lyapunov exponents of the solutions are nonpositive in the absence of conditions. In [25], the authors only show that the solutions are uniformly bounded in the p-th moment. However, we give the concrete upper bound for the p-th moment. It is clear that the results of (ii) and (iii) in Corollary 2 (I) are consistent with Theorems 7 and 8 in [25]. However, the result of (III) in Corollary 2 is not reflected in [25]. Thus, our work can be seen as the extension of [25].Remark 2.
For the deterministic version of model (125), from [26], we can see that limt⟶∞yt=0 holds under some special conditions, i.e., the predator dies out, but it never gets limt⟶∞xt=0 (if limt⟶∞yt=0, then liminft⟶∞xt≥r/a>0). However, the result of (iii) in Corollary 2 (I) shows that great noise intensities σ12 and σ22 can make both the prey and predator in model (125) die out. This means that a relatively large stochastic perturbation can cause the extinction of the population.
However, Linh and Ton [26] only consider the asymptotic estimations of moments, the upper-growth rates, and exponential death rates of species in the corresponding nonautonomous stochastic model of model (125). Moreover, the results of (iv) and (v) in Corollary 2 (I) are consistent with Theorems 4.3 and 4.4 in [26]. Thus, our paper can be regarded as the extension and supplement of [26].
Furthermore, if we do not consider the intraspecific competition of the predator, i.e.,b=0 in model (125), then one can obtain the following stochastic model:(133)dxt=xtr−axt−αytxt+βytdt+σ1xtdw1t,dyt=yt−m+eαxtxt+βytdt+σ2ytdw2t,with initial value x0,y0∈ℝ+2. This is a stochastic predator-prey model discussed in [27]. Wu et al. [10] considered the corresponding nonautonomous model of stochastic model (133). By a similar discussion as in Theorem 1, for any x0,y0∈ℝ+2, model (133) has a unique global positive solution xt,yt.Corollary 3.
For anyx0,y0∈ℝ+2, let xt,yt be the solution of model (133) with initial value x0,y0.(I)
By a similar discussion as in Theorems5 and 6, one can get the following results:(i)
Ifr−σ12/2>0 and δ2−σ22/2<0, then(134)limt⟶∞xt=0,limt⟶∞yt=0,a.s.(ii)
If the predator is absent, i.e.,yt=0 a.s. for all t≥0, then the quantity of prey in model (133) satisfies the following:(135)limt⟶∞xt=r−σ12/2a1,a.s.,ifr−σ12/2>0;limt⟶∞xt=0,a.s.,ifr−σ12/2<0(iii)
If the prey is absent, i.e.,xt=0 a.s. for all t≥0, then the predator in model (133) dies with probability one(II)
By a similar discussion as in Theorems8 and 10, one can get the following results:(i)
The solutionxt,yt of model (133) obeys(136)limsupt⟶∞Ext≤K3em,limsupt⟶∞Eyt≤K3m.whereK3=er+m2/4a. Then, from Chebyshev’s inequality, model (133) is stochastically ultimately bounded.(ii)
Ifςi=˙δi−σi2>0,i=1,2, then the solution xt,yt of model (125) obeysFurthermore, by a similar discussion as in Theorem11, if ςi>0,i=1,2, then model (125) is stochastically permanent.Remark 3.
Ifςi=˙δi−σi2>0,i=1,2, then by Theorem 3.3 in [27], model (133) is persistent in mean. However, from (II) in Corollary 3, model (133) is stochastically permanent. This means that Theorem 11 generalizes and improves Theorem 3.3 in [27].Remark 4.
From Theorem 4.11 in [14], it ca be seen that if δi−3/2σi2>0,i=1,2, then model (133) is stochastically permanent. However, the results in Corollary 3 show that if δi−σi2>0,i=1,2, then model (133) is stochastically permanent. Obviously, if δi−3/2σi2>0,i=1,2 holds, then δi−σi2>0,i=1,2 holds. On the contrary, it is not set up. Thus, we can say that Corollary 3 generalizes and improves Theorem 4.11 in [14].


### 6.2. Two-Patch Predator-Prey Model (4)

By a similar discussion as in Theorem1, for any x10,x20,y0∈ℝ+3, model (4) has a unique global positive solution x1t,x2t,yt. Moreover, for model (4), we have the following results.Corollary 4.
(I)
By a similar discussion as in Theorems5 and 6, we can obtain the following results:(i)
Ifr1−σ12/2<0, r2−σ22/2<0 and λ3−σ23/2<0, then(138)limt⟶∞x1t=0,limt⟶∞x2t=0,limt⟶∞yt=0,a.s.(ii)
If the predator is absent, i.e.,yt=0 a.s. for all t≥0, then the prey xii=1,2 in model (4) satisfies the following:(139)limt⟶∞xit=ri−σi2/2ai,a.s.,ifri−σi2/2>0;limt⟶∞xit=0,a.s.,ifri−σi2/2<0(iii)
If the prey is absent, i.e.,x1t=x2t=0 a.s. for all t≥0, then the predator in model (4) dies with probability one(II)
By a similar discussion as in Theorems8 and 10, one can get the following results:(i)
The solutionx1t,x2t,yt of model (4) obeys(140)limsupt⟶∞Exit≤K2eiλ,limsupt⟶∞Eyt≤K2λ,i=1,2,whereK2=e1r1+λ2/4a1+e2r2+λ2/4a2. Then, from Chebyshev’s inequality, model (4) is stochastically ultimately bounded.(ii)
Ifκi>0,i=1,2,3, then the solution x1t,x2t,yt of model (4) obeys(141)limsupt⟶∞E1xit≤aiκi,i=1,2,limsupt⟶∞E1yt≤a1e1α1vβ1κ1κ3+a2e2α21−vβ2κ2κ3.Moreover, ifκi>0,i=1,2,3, then model (4) is stochastically permanent.


## 6.1. Two-Species Predator-Prey Model

If the predator only stays in one patch, then one can obtain the following stochastic predator-prey model (obtained by takingv=0 or v=1 in model (5)).(125)dxt=xtr−axt−αytxt+βytdt+σ1xtdw1t,dyt=yt−m−byt+eαxtxt+βytdt+σ2ytdw2t,where x0,y0=x0,y0∈ℝ+2. Model (125) was discussed in [25]. Furthermore, Linh and Ton [26] considered the corresponding nonautonomous model of (125). By a similar discussion as in Theorem 1, for any x0,y0∈ℝ+2, model (21) has a unique global positive solution xt,yt. Moreover, for model (125), we have the following results. For simplicity, we denote δ1=r−α/β and δ2=eα−m.Corollary 2.
For anyx0,y0∈ℝ+2, let xt,yt be the solution of model (125) with initial value x0,y0.(I)
By a similar discussion as in Theorems2–6, we can obtain the following results:(i)
The solutionxt,yt of model (126) obeys(126)limsupt⟶∞lnxtt≤0,limsupt⟶∞lnytt≤0,a.s.Moreover, ifδ1−σ12/2>0 and δ2−σ22/2>0; then,(127)limt⟶∞lnxtt=0,limt⟶∞lnytt=0,a.s.(ii)
Ifδ1−σ12/2>0 and δ2−σ22/2>0, then(128)liminft⟶∞xt≥δ1−σ12/2a>0,liminft⟶∞yt+Mytxt≥δ2−σ22/2b>0,a.s.Here,M=eαβ/b. This means that model (125) is persistent in mean.(iii)
Ifr−σ12/2<0 and δ2−σ22/2<0, then(129)limt⟶∞xt=0,limt⟶∞yt=0,a.s.(iv)
If the predator is absent, i.e.,yt=0 a.s. for all t≥0, then the quantity of prey in model (125) satisfies the following:(130)limt⟶∞xt=r−σ12/2a1,a.s.,ifr−σ12/2>0;limt⟶∞xt=0,a.s.,ifr−σ12/2<0.(v)
If the prey is absent, i.e.,xt=0 a.s. for all t≥0, then the predator in model (125) dies with probability one.(II)
By a similar discussion as in Theorems7, 10 and 11, one can get the following results:(i)
The solutionxt,yt of model (125) obeys(131)limsupt⟶∞Expt≤pp+1p+11/p+r+p/2σ12p+1aplimsupt⟶∞Eypt≤pp+1p+11/p+eα+p/2σ22p+1bp.Moreover, from Chebyshev’s inequality, model (125) is stochastically ultimately bounded.(ii)
Ifςi=˙δi−σi2>0i=1,2, then the solution xt,yt of model (125) obeys(132)limsupt⟶∞E1xt≤aς1,limsupt⟶∞E1yt≤bς2+aeαβς1ς2.Furthermore, ifςi>0,i=1,2, then model (125) is stochastically permanent.By a similar discussion as in Theorem12, if ς1−eαβ>0 and ς2>0, then any x0,y0∈ℝ+2, model (125) has a stationary distribution μ· and the solutions have an ergodic property.Remark 1.
In [25], the authors show that model (125) has a unique global positive solution by using stopping times and contradiction. In this paper, the stochastic comparison theorem is used to prove that the model has a unique global positive solution. Reference [25] only shows that if δ1−σ12/2>0 and δ2−σ22/2>0, then limt⟶∞lnxt/t=0 and limt⟶∞lnyt/t=0 a.s. However, we also show that the sample Lyapunov exponents of the solutions are nonpositive in the absence of conditions. In [25], the authors only show that the solutions are uniformly bounded in the p-th moment. However, we give the concrete upper bound for the p-th moment. It is clear that the results of (ii) and (iii) in Corollary 2 (I) are consistent with Theorems 7 and 8 in [25]. However, the result of (III) in Corollary 2 is not reflected in [25]. Thus, our work can be seen as the extension of [25].Remark 2.
For the deterministic version of model (125), from [26], we can see that limt⟶∞yt=0 holds under some special conditions, i.e., the predator dies out, but it never gets limt⟶∞xt=0 (if limt⟶∞yt=0, then liminft⟶∞xt≥r/a>0). However, the result of (iii) in Corollary 2 (I) shows that great noise intensities σ12 and σ22 can make both the prey and predator in model (125) die out. This means that a relatively large stochastic perturbation can cause the extinction of the population.
However, Linh and Ton [26] only consider the asymptotic estimations of moments, the upper-growth rates, and exponential death rates of species in the corresponding nonautonomous stochastic model of model (125). Moreover, the results of (iv) and (v) in Corollary 2 (I) are consistent with Theorems 4.3 and 4.4 in [26]. Thus, our paper can be regarded as the extension and supplement of [26].
Furthermore, if we do not consider the intraspecific competition of the predator, i.e.,b=0 in model (125), then one can obtain the following stochastic model:(133)dxt=xtr−axt−αytxt+βytdt+σ1xtdw1t,dyt=yt−m+eαxtxt+βytdt+σ2ytdw2t,with initial value x0,y0∈ℝ+2. This is a stochastic predator-prey model discussed in [27]. Wu et al. [10] considered the corresponding nonautonomous model of stochastic model (133). By a similar discussion as in Theorem 1, for any x0,y0∈ℝ+2, model (133) has a unique global positive solution xt,yt.Corollary 3.
For anyx0,y0∈ℝ+2, let xt,yt be the solution of model (133) with initial value x0,y0.(I)
By a similar discussion as in Theorems5 and 6, one can get the following results:(i)
Ifr−σ12/2>0 and δ2−σ22/2<0, then(134)limt⟶∞xt=0,limt⟶∞yt=0,a.s.(ii)
If the predator is absent, i.e.,yt=0 a.s. for all t≥0, then the quantity of prey in model (133) satisfies the following:(135)limt⟶∞xt=r−σ12/2a1,a.s.,ifr−σ12/2>0;limt⟶∞xt=0,a.s.,ifr−σ12/2<0(iii)
If the prey is absent, i.e.,xt=0 a.s. for all t≥0, then the predator in model (133) dies with probability one(II)
By a similar discussion as in Theorems8 and 10, one can get the following results:(i)
The solutionxt,yt of model (133) obeys(136)limsupt⟶∞Ext≤K3em,limsupt⟶∞Eyt≤K3m.whereK3=er+m2/4a. Then, from Chebyshev’s inequality, model (133) is stochastically ultimately bounded.(ii)
Ifςi=˙δi−σi2>0,i=1,2, then the solution xt,yt of model (125) obeysFurthermore, by a similar discussion as in Theorem11, if ςi>0,i=1,2, then model (125) is stochastically permanent.Remark 3.
Ifςi=˙δi−σi2>0,i=1,2, then by Theorem 3.3 in [27], model (133) is persistent in mean. However, from (II) in Corollary 3, model (133) is stochastically permanent. This means that Theorem 11 generalizes and improves Theorem 3.3 in [27].Remark 4.
From Theorem 4.11 in [14], it ca be seen that if δi−3/2σi2>0,i=1,2, then model (133) is stochastically permanent. However, the results in Corollary 3 show that if δi−σi2>0,i=1,2, then model (133) is stochastically permanent. Obviously, if δi−3/2σi2>0,i=1,2 holds, then δi−σi2>0,i=1,2 holds. On the contrary, it is not set up. Thus, we can say that Corollary 3 generalizes and improves Theorem 4.11 in [14].


## 6.2. Two-Patch Predator-Prey Model (4)

By a similar discussion as in Theorem1, for any x10,x20,y0∈ℝ+3, model (4) has a unique global positive solution x1t,x2t,yt. Moreover, for model (4), we have the following results.Corollary 4.
(I)
By a similar discussion as in Theorems5 and 6, we can obtain the following results:(i)
Ifr1−σ12/2<0, r2−σ22/2<0 and λ3−σ23/2<0, then(138)limt⟶∞x1t=0,limt⟶∞x2t=0,limt⟶∞yt=0,a.s.(ii)
If the predator is absent, i.e.,yt=0 a.s. for all t≥0, then the prey xii=1,2 in model (4) satisfies the following:(139)limt⟶∞xit=ri−σi2/2ai,a.s.,ifri−σi2/2>0;limt⟶∞xit=0,a.s.,ifri−σi2/2<0(iii)
If the prey is absent, i.e.,x1t=x2t=0 a.s. for all t≥0, then the predator in model (4) dies with probability one(II)
By a similar discussion as in Theorems8 and 10, one can get the following results:(i)
The solutionx1t,x2t,yt of model (4) obeys(140)limsupt⟶∞Exit≤K2eiλ,limsupt⟶∞Eyt≤K2λ,i=1,2,whereK2=e1r1+λ2/4a1+e2r2+λ2/4a2. Then, from Chebyshev’s inequality, model (4) is stochastically ultimately bounded.(ii)
Ifκi>0,i=1,2,3, then the solution x1t,x2t,yt of model (4) obeys(141)limsupt⟶∞E1xit≤aiκi,i=1,2,limsupt⟶∞E1yt≤a1e1α1vβ1κ1κ3+a2e2α21−vβ2κ2κ3.Moreover, ifκi>0,i=1,2,3, then model (4) is stochastically permanent.


## 7. Numerical Simulations

In this section, we use the Milstein method (see [28]) to substantiate the main results. The parameters are given in the following table.Example 1.
denotek1v=r1−α1v/β1−σ12/2, k2v=r2−α21−v/β2−σ22/2, and k3v=e1α1−m1v+e2α2−m21−v−σ32/2. By Theorem 4, if kiv>0,i=1,2,3, then model (5) is persistent in mean. If we take the parameter values as in Table 1, σ1=0.05, σ2=0.05, and σ3=0.02, then one can get Figure 1. It can be seen from Figure 1 that when 0.042<v<0.44875, model (5) will be persistent in mean.Table 1
Physical interpretation of the parameters.
ParametersDescriptionValuesriIntrinsic growth rate of the prey in patchir1=0.45,r2=0.6eiExpected biomass of the prey converted to predators in patchie1=0.11,e2=0.21miPer capita mortality rate of predators in patchim1=0.03,m2=0.01aiIntraspecific competition coefficient of the prey in patchia1=0.0002,a2=0.0004bIntraspecific competition coefficient of the predatorb=0.0003αiEncounter rate with the prey in patchiα1=0.6,α2=0.5βiHalf saturation constant for the prey in patchiβ1=0.6,β2=0.8Figure 1
The trajectories ofk1v, k2v, and k3v with σ1=0.05, σ2=0.05, and σ3=0.02 (color figure online).Example 2.
We denotek1v,σ12=r1−α1v/β1−σ12/2, k2v,σ22=r2−α21−v/β2−σ22/2 and k3v,σ32=e1α1−m1v+e2α2−m21−v−σ32/2. If we take the parameter values as in Table 1, then we can obtain figures about kiv,σi2 (see Figure 2).
To illustrate the results, we take the parameter values as in Table1 and v=0.4. Here we give numerical simulations of model (5) with x10,x20,y0=1200,1000,500 and different noise intensities. In Figure 3, we choose σ12=σ22=σ32=0 and get the solution of deterministic model (3).Figure 2
kiv,σi2 and its phase plan i=1,2,3 (color figure online): (a) k1v,σ12, (b) k2v,σ22, (c) k3v,σ32, (d) phase plan of k1v,σ12, (e) phase plan of k2v,σ22, and (f) phase plan of k3v,σ32.
(a)(b)(c)(d)(e)(f)Figure 3
The trajectories of model (5) with σ12=σ22=σ32=0 (color figure online).Example 3.
We assume thatσ12=0.04, σ22=0.04, and σ32=0.01. By a simple computation, λ1−σ12/2=0.03>0, λ2−σ22/2=0.205>0, and λ3−σ32/2=0.0664>0. Thus, the conditions of Theorem 4 hold. In view of Theorem 4, all the populations in model (5) will be persistent in mean (see Figure 4).Figure 4
The trajectories of stochastic model (5) with σ12=0.04, σ22=0.04, and σ32=0.01 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Example 4.
We assume thatσ12=1, σ22=1.24, and σ32=0.2. Thus, r1−σ12/2=−0.05<0, r2−σ22/2=−0.02<0, and λ3−σ32/2=−0.0286<0. Then, from Theorem 5, it follows that limt⟶∞x1t=0, limt⟶∞x2t=0, and limt⟶∞yt=0 a.s. This means that all the population in model (5) will go to extinction (see Figure 5).Figure 5
The trajectories of stochastic model (5) with σ12=1, σ22=1.24, and σ32=0.2 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Example 5.
We assume thatσ1=0.05, σ2=0.05, and σ3=0.02. By a simple computation, κ1=0.0475>0, κ2=0.2225>0, and κ3=0.071>0. Thus, from Theorem 11, it follows that model (5) is stochastically permanent (see Figure 6).Figure 6
The trajectories of stochastic model (5) with σ1=0.05, σ2=0.05, and σ3=0.02 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Example 6.
We assume thatσ1=0.05, σ2=0.05, and σ3=0.02. By a simple computation, κ1−e1α1vβ1=0.0317>0, κ2−e2α21−vβ2=0.1721>0, and κ3=0.071>0. Thus, the conditions of Theorem 12 hold. According to Theorem 12, model (5) has a stationary distribution (see Figure 7–11).Figure 7
The density based on 10 stochastic simulations for each population. (a)–(c) The density att=80,000 with different initial value; (d)–(f) the density with initial value 1500,100,500 at different time periods (color figure online). (a) PDF of prey x1t, (b) PDF of prey x2t, (c) PDF of prey yt, (d) PDF of prey x1t, (e) PDF of prey x2t, and (f) PDF of prey yt.
(a)(b)(c)(d)(e)(f)Figure 8
The trajectories of stochastic model (5) with σ12=0.04 and σ22=0.04, in the absence of the predator (color figure online): (a) prey x1t and (b) prey x2t.
(a)(b)Figure 9
t The trajectories of stochastic model (5) with σ12=0.04 and σ22=1.24, in the absence of the predator (color figure online): (a) prey x1t and (b) prey x2t.
(a)(b)Figure 10
The trajectories of stochastic model (5) with σ12=1 and σ22=0.04, in the absence of the predator (color figure online): (a) prey x1t and (b) prey x2t.
(a)(b)Figure 11
The trajectories of stochastic model (5) with σ12=1 and σ22=1.24, in the absence of the predator (color figure online): (a) prey x1t and (b) prey x2t.
(a)(b)Example 7.
The predator is absent, i.e.,yt=0 a.s. for all t≥0.(i)
We assume thatσ12=0.04 and σ22=0.04. By a simple computation, r1−σ12/2=0.43>0 and r2−σ22/2=0.58>0. Thus, from Theorem 6, limt⟶∞x1t=2150 and limt⟶∞x2t=1450 a.s. This means that, in the absence of the predator, the prey xi,i=1,2 will be persistent in mean in the absence of the predator y (see Figure 8).(ii)
We assume thatσ12=0.04 and σ22=1.24. Thus, r1−σ12/2=0.43>0 and r2−σ22/2=−0.02<0. From Theorem 6, it follows that limt⟶∞x1t=2150 and limt⟶∞x2t=0 a.s. This means that, in the absence of the predator y, the prey x1 will be persistent in mean, while the prey x2 will go to extinction (see Figure 9).(iii)
We assume thatσ12=1 and σ22=0.04. By a simple computation, r1−σ12/2=−0.05<0 and r2−σ22/2=0.58>0. Thus, from Theorem 6, limt⟶∞x1t=0 and limt⟶∞x2t=1450 a.s. This means that, in the absence of the predator y, the prey x2 will be persistent in mean, while the prey x1 will go to extinction (see Figure 10).(iv)
We assume thatσ12=1 and σ22=1.24. Thus, r1−σ12/2=−0.05<0 and r2−σ22/2=−0.02<0. According to Theorem 6, we have limt⟶∞x1t=0 and limt⟶∞x2t=0 a.s. Thus, the prey x1 and x2 will go to extinction in the absence of the predator y (see Figure 11).Example 8.
The prey in patch 2 is absent, i.e.,x2t=0 a.s. for all t≥0 (Figures 12–15).(i)
We assume thatσ12=0.04 and σ32=0.002. By a simple computation, λ1−σ12/2=0.03>0 and e1α1v−m1v−m21−v−σ32/2=0.0074>0. Thus, from Theorem 6, it follows that(142)liminft⟶∞x1t≥150,liminft⟶∞yt+e1α1vβ1bytx1t≥24.67,a.s.This means that the preyx1 and the predator y will be persistent in mean in the absence of the prey x2 (see Figure 12).(ii)
We assume thatσ12=1 and σ32=0.04. By a simple computation, r1−σ12/2=−0.05<0 and e1α1v−m1v−m21−v−σ32/2=−0.0116<0. Thus, from Theorem 6, limt⟶∞x1t=0 and limt⟶∞yt=0 a.s. This means that, in the absence of the prey x2, the prey x1 and predator population y will go to extinction (see Figure 13).Figure 12
The trajectories of (5) with σ12=0.04 and σ32=0.002, in the absence of the prey in patch 2 (color figure online): (a) prey x1t and (b) prey x2t.
(a)(b)Figure 13
The trajectories of (5) with σ12=1 and σ32=0.04, in the absence of the prey in patch 2 (color figure online): (a) prey x1t and (b) prey x2t.
(a)(b)Figure 14
The trajectories of (5) with σ22=0.04 and σ32=0.02, in the absence of the prey in patch 1 (color figure online): (a) prey x2t and (b) predator yt.
(a)(b)Figure 15
The trajectories of (5) with σ22=1.24 and σ32=0.2, in the absence of the prey in patch 1 (color figure online): (a) prey x2t and (b) predator yt.
(a)(b)Example 9.
The prey in patch 1 is absent, i.e.,x1t=0 a.s. for all t≥0.(i)
We assume thatσ22=0.04 and σ32=0.02. By a simple computation, λ2−σ22/2=0.205>0 and e2α21−v−m1v−m21−v−σ32/2=0.035>0. From Theorem 6, it follows that(143)liminft⟶∞x2t≥512.5,liminft⟶∞yt+e2α21−vβ2bytx1t≥116.67,a.s.This means that, in the absence of the preyx1, the prey x2 and the predator y will be persistent in mean (see Figure 14).(ii)
We assume thatσ22=1.24 and σ32=0.2. Thus, r2−σ22/2=−0.02<0 and e2α21−v−m1v−m21−v−σ32/2=−0.055<0. From Theorem 6, limt⟶∞x2t=0 and limt⟶∞yt=0 a.s. This means that the prey x2 and the predator yt will go to extinction in the absence of the prey x1 (see Figure 15).Example 10.
The prey is absent, i.e.,x1t=x2t=0 a.s. for all t≥0. From Theorem 6, it follows that the predator yt will go to extinction (see Figure 16–22).Figure 16
The trajectories of (5) with σ32=0.02, in the absence of the prey (color figure online).Figure 17
The trajectories of stochastic model (5) with σ12=0.04, σ22=0.04, and σ32=0.2 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Figure 18
The trajectories of stochastic model (5) with σ12=0.04, σ22=1.24, and σ32=0.03 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Figure 19
The trajectories of stochastic model (5) with σ12=0.04, σ22=1.24, and σ32=0.002 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Figure 20
The trajectories of stochastic model (5) with σ12=1, σ22=0.04, and σ32=0.1 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Figure 21
The trajectories of stochastic model (5) with σ12=1, σ22=0.04, and σ32=0.02 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Figure 22
The trajectories of stochastic model (5) with σ12=1, σ22=1.24, and σ32=0.002 (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)Example 11.
(i)
We assume thatσ12=0.04, σ22=0.04, and σ32=0.2. Thus, r1−σ12/2=0.43>0, r2−σ22/2=0.58>0, and λ3−σ32/2=−0.0286<0. From Theorem 5, we have limt⟶∞yt=0 a.s. Moreover, from Figure 17, we can see that limt⟶∞x1t=2150 and limt⟶∞x2t=1450 a.s.(ii)
We assume thatσ12=0.04, σ22=1.24, and σ32=0.03. By a simple computation, r1−σ12/2=0.43>0, r2−σ22/2=−0.02<0, and e1α1v−m1v−m21−v−σ32/2=−0.0066<0. From Theorem 5, it follows that limt⟶∞x2t=0 a.s. Moreover, from Figure 18, we can see that limt⟶∞x1t=2150 and limt⟶∞yt=0 a.s.(iii)
We assume thatσ12=0.04, σ22=1.24, and σ32=0.002. Thus, λ1−σ12/2=0.03>0, r2−σ22/2=−0.02<0, and e1α1v−m1v−m21−v−σ32/2=0.0074>0. From Theorem 5, it follows that limt⟶∞x2t=0 a.s. Moreover, from Figure 19, we can see that prey x1 and predator y will be persistent in mean.(iv)
We assume thatσ12=1, σ22=0.04, and σ32=0.1. By a simple computation, r1−σ12/2=−0.05<0, r2−σ22/2=0.58>0, and e2α21−v−m1v−m21−v−σ32/2=−0.005<0. Thus, from Theorem 5, it follows that limt⟶∞x1t=0 a.s. Moreover, from Figure 20, we can see that limt⟶∞x2t=1450 and limt⟶∞yt=0 a.s.(v)
We assume thatσ12=1, σ22=0.04, and σ32=0.02. Thus, r1−σ12/2=−0.05<0, λ2−σ22/2=0.205>0, and e2α21−v−m1v−m21−v−σ32/2=0.035>0. From Theorem 5, it follows that limt⟶∞x1t=0 a.s. Moreover, from Figure 21, we can see that prey x2 and predator y will be persistent in mean.(vi)
We assume thatσ12=1, σ22=1.24, and σ32=0.002. Thus, r1−σ12/2=−0.05<0, r2−σ22/2=−0.02<0, and λ3−σ32/2=0.0704>0. Then, from Theorem 5, it follows that limt⟶∞x1t=0 and limt⟶∞x2t=0 a.s. Moreover, from Figure 22, we can see that predator y will go to extinction.Example 12.
We assume thatr1=0.9, e2=0.1, m1=0.03, m2=0.03, α2=0.2, σ12=0.02, σ22=0.02, and σ32=0.002, and the values of other parameters are shown in Table 1. Moreover, if we take v=0.8, then κ1=λ1−σ12=0.08>0, κ2=λ2−σ22=0.53>0, and κ3=λ3−σ32=0.0248>0. Thus, from Theorem 11, model (5) is stochastically permanent (see Figures 23(a)–23(c)). However, if we take v=0, then r1−σ12/2=0.89>0, r2−σ22/2=0.59>0, and λ3−σ32/2=−0.011<0. From Theorem 5, it follows that limt⟶∞yt=0 a.s. Moreover, from Figures 23(d) and 23(e), one can see that limt⟶∞x1t=4450 and limt⟶∞x2t=1475 a.s. This means that the prey population xiti=1,2 will be persistent in mean and the predator population yt will go to extinction. Furthermore, comparing Figures 23(c) and 23(f), we conclude that the patch structure is conducive to the survival of the predator population.Figure 23
The trajectories of stochastic model (5) with different v (color figure online): (a) prey x1twith v=0.8, (b) prey x2twith v=0.8, (c) predator ytwith v=0.8, (d) prey x1twith v=0, (e) prey x2twith v=0, and (f) predator ytwith v=0.
(a)(b)(c)(d)(e)(f)Example 13.
We assume thatσ1=0.05, σ2=0.05, and σ3=0.02. If we take v=0.1 (v=0.2, v=0.3, or v=0.4); then, all the conditions of Theorem 11 hold. Thus, from Theorem 11, it follows that model (5) is stochastically permanent. Here, we give the numerical simulations of model (5) with different v (see Figure 24). As can be seen from Figure 24, with the increase of v, that is, the proportion of time that predators stay in patch 1 increases, the number of prey in patch 1 decreases, while the number of prey in patch 2 increases. This has a reasonable biological significance. However, the mortality, the encounter rate with the prey, the half-saturation constant, and the conversion rate of the predators are different in different patches, and it is impossible to determine the number of predators with the change of v.Figure 24
The trajectories of stochastic model (5) with different v (color figure online): (a) prey x1t, (b) prey x2t, and (c) predator yt.
(a)(b)(c)


## 8. Conclusions and Discussion

This paper is concerned with a stochastic two-patch predator-prey model with ratio-dependent functional responses. First, by using the comparison theorem of stochastic differential equations, we show that the model has a unique global positive solution. Then, the long-time properties of the solutions are discussed pathwise. Using the exponential martingale inequality and Borel–Cantelli lemma, we show that the sample Lyapunov exponents are nonpositive. Moreover, under certain conditions, we show that the sample Lyapunov exponents are zero. Next, the sufficient conditions for the extinction and persistence in mean of the model are given. Then, we investigate the stochastically ultimate boundedness and stochastic persistence of the model. Moreover, by constructing a suitable Lyapunov function, we show that the model has an ergodic stationary distribution. Next, we apply the main results to two special stochastic population models. Finally, some numerical simulations are introduced to support the main results. Furthermore, other dynamic properties of the model are found through numerical simulations.In Section2, by using the stochastic comparison theorem, we show that the model has a unique global positive solution. Then, the long-time properties of the solutions are discussed pathwise. Using the exponential martingale inequality and Borel–Cantelli lemma, we show that the sample Lyapunov exponents are nonpositive. Moreover, we show that if the noise intensities σi2i=1,2,3 are small compared to the other parameters, then the sample Lyapunov exponents are zero.Section3 reveals the effects of stochastic perturbations on the persistence and extinction of prey x1, prey x2, and predator y. From Theorem 4, if the noise intensities σi2i=1,2,3 are small such that λi−σi2/2>0,i=1,2,3, then all populations in model (5) will be persistent in mean. Furthermore, from Theorem 5, if r1−σ12/2<0, r2−σ22/2<0, and λ3−σ32/2<0, then the solution of model (5) tends to zero almost surely. This means that great noise intensities σi2i=1,2,3 can make all populations in model (5) will become extinct.Furthermore, Theorem6 discusses the effects of noise on the dynamics of other species in the absence of the predator y, prey in patch 1, and prey in patch 2, respectively. In the absence of the predator, if ri−σi2/2>0,i=1,2, then limt⟶∞xit=ri−σi2/2/ai a.s., i=1,2; if ri−σi2/2>0 and rj−σj2/2<0,i≠j,i,j=1,2, then limt⟶∞xit=ri−σi2/2/ai and limt⟶∞xjt=0 a.s. i≠j,i,j=1,2; moreover, if ri−σi2/2<0,i=1,2, then limt⟶∞xit=0 a.s., i=1,2. Hence, in the absence of the predator, with the increase of noise intensity σi2, the prey in patch i will go to extinction, while the environment noise w˙it has no effect on the extinction of the prey in patch ji≠j,i,j=1,2. Moreover, in the absence of the predator, with the decrease of noise intensity σi2, the prey in patch i can be persistent better, while the environment noise w˙it has no effect on the persistent level of the prey in patch j,i≠j,i,j=1,2. In the absence of the prey in patch 2, if λ1−σ12/2>0 and e1α1v−m1v−m21−v−σ32/2>0, then the prey in patch 1 and the predator y will be persistent in mean, while if r1−σ12/2<0 and e1α1v−m1v−m21−v−σ32/2<0, then the prey in patch 1 and the predator y will go to extinction exponentially. This means that, in the absence of the prey in patch 2, with the increase of noise intensity σi2,i=1,3, the prey in patch 1 and the predator y will become extinct. In the absence of the prey in patch 1, if λ2−σ22/2>0 and e2α21−v−m1v−m21−v−σ32/2>0, then the prey in patch 2 and the predator y will be persistent in mean, while if r2−σ22/2<0 and e2α21−v−m1v−m21−v−σ32/2<0, then the prey in patch 2 and the predator y will go to extinction exponentially. Hence, in the absence of the prey in patch 1, with the increase of noise intensity σi2,i=2,3, the prey in patch 2 and the predator y will become extinct. Moreover, from Theorem 6, in the absence of the prey, the predator dies with probability one.Theorem6 shows the effects of noise on the dynamics of other species in the absence of the predator y, prey in patch 1, and prey in patch 2, respectively. However, through the numerical simulation of Example 11, we can conclude the following results: (i) In the case of the predator extinction, the prey has the same dynamic behavior same as that in the absence of the predator. (ii) In the case of the prey x1 extinction, the dynamic behaviors of the prey in patch 2 and the predator y are the same as those in the absence of the prey x1. (iii) In the case of the prey x2 extinction, the dynamic behaviors of the prey in patch 1 and the predator y are the same as those in the absence of the prey x2. (iv) If the prey extinction, then the predator will go to extinction. This is consistent with the results in the absence of the prey.In Section4, we investigate the stochastically ultimate boundedness and stochastic persistence of the model. First, we use two different ways to prove the boundedness of the model and then show that the model is stochastically ultimately bounded by Chebyshev’s inequality. Next, we investigate the stochastic persistence of the model. The results show that if the noise intensities σi2 (i = 1, 2, 3) are small such that κi>0,i=1,2,3, then model (5) is stochastically permanent. This means that the species in model (5) will survive forever at low noise levels.In Section5, by constructing a suitable Lyapunov function, we show that the solution of model (5) has an ergodic stationary distribution. From Theorem 12, if κ1−e1α1vβ1>0, κ2−e2α21−vβ2>0, and κ3>0, then for any initial value x10,x20,y0∈ℝ+3, model (5) has a stationary distribution μ· and the solutions have an ergodic property. Hence, the small noise intensities σi2 (i = 1, 2, 3) can ensure that the solution of the model has an ergodic stationary distribution.In Section6, we first apply the main results to two stochastic two-species predator-prey models. Then, we present the application of the main results to stochastic two-patch predator-prey model (4). Moreover, we compare the results with the known closely related models.In [25], the authors discussed the stochastic model (125). It is clear that the results of (ii) and (iii) in Corollary 2 (I) are consistent with Theorems 7 and 8 in [25]. Moreover, [25] only shows that if δ1−σ12/2>0 and δ2−σ22/2>0, then limt⟶∞lnxt/t=0 and limt⟶∞lnyt/t=0 a.s. However, we also show that the sample Lyapunov exponents of the solutions are nonpositive in the absence of conditions. Furthermore, the ergodic stationary distribution of model (125) is not reflected in [25]. Thus, our work can be seen as the extension of [25].Linh and Ton [26] considered the corresponding nonautonomous model of (125). Moreover, the results of (iv) and (v) in Corollary 2 (I) are consistent with Theorems 4.3 and 4.4 in [26]. Furthermore, for the deterministic version of model (125), from [26], if limt⟶∞yt=0, then liminft⟶∞xt≥r/a>0. This means that when the predator dies out, the prey must survive forever. However, the result of (iii) in Corollary 2 (I) shows that great noise intensities σ12 and σ22 can make both the prey and predator in model (125) go to extinction.Ji et al. [27] considered the stochastic model (133), while Wu, Huang, and Wang [14] discussed the corresponding nonautonomous model of (133). On the one hand, from Theorem 3.3 in [27], if ςi=˙δi−σi2>0,i=1,2, then model (133) is persistent in mean. However, from (II) in Corollary 3, model (133) is stochastically permanent. Thus, Theorem 11 generalizes and improves Theorem 3.3 in [27]. On the other hand, from Theorem 4.11 in [14], if δi−3/2σi2>0,i=1,2, then model (133) is stochastically permanent. However, the results in Corollary 3 show that if δi−σi2>0,i=1,2, then model (133) is stochastically permanent. Obviously, if δi−3/2σi2>0,i=1,2 holds, then δi−σi2>0,i=1,2 holds. On the contrary, it is not set up. Thus, we can say that Corollary 3 generalizes and improves Theorem 4.11 in [14].



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*Source: 6671499-2021-01-16.xml*