# Persistence and Extinction of a Stochastic Modified Bazykin Predator-Prey System with Lévy Jumps

Authors: Zhangzhi Wei; Zheng Wu; Ling Hu; Lianglong Wang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

This paper is devoted to the dynamics of a stochastic modified Bazykin predator-prey system with Lévy jumps. First, we show that the system has a unique global positive solution and give some properties of solutions. Then, some sufficient conditions for persistence and extinction are derived by Itô formula and some inequalities on stochastic analysis. At last, some simulations are provided to check the main results.


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## Body

## 1. Introduction

In population biology, construction of models and relevant qualitative analysis are popular fields [1, 2]. In the last few decades, many predator-prey models with functional and numerical responses have been proposed and studied extensively. Particularly, ratio-dependent functional response is common in some classical literature [3–5].Alexeev and Bazykin firstly proposed a Bazykin system [1] (1)x˙t=xta-bxt-cyt1+Axt,y˙t=yt-g-hyt+fxt1+Axt,where x(t) and y(t) are the size of prey and predator at time t. a,b,c,f,g, and m are positive constants (some details refer to [1]).Considering the ratio effect in hunting process of predation, Haque built a modified Bazykin system [2] (2)x˙t=xta-bxt-cytyt+Axt,y˙t=yt-g-hyt+fxtyt+Axt,of which dynamical behavior near equilibrium point and bifurcation are observed. System (2) is more reasonable and many researchers began to pay more attention on it. Its generalized forms have been investigated and a lot of results were obtained [6–8].However, in the natural world, environmental noise is everywhere, and the growth rate of the populations is not constants. In this case, many systems are described by stochastic differential equations driven by Brownian motion of the actual research. For example, [9] studied the following stochastic modified Bazykin system: (3)dxt=xta-bxt-cytyt+Axtdt+αxtdw1t,dyt=yt-g-hyt+fxtyt+Axtdt+βytdw2t,where w1(t) and w2(t) are two independent Wiener processes defined on a complete probability space Ω,F,Ftt≥0,P and α and β stand for the level of the white noises. Some sufficient conditions for persistence and extinction are obtained.Recently, the stochastic differential equation driven by jump has drawn more and more researchers’ attention [10–17]. This is mainly according to sudden perturbation of environment, such as toxic contamination of water, torrential flood, and hurricane. Motivated by the above, this paper considers the following stochastic modified Bazykin system with Lévy jumps: (4)dxt=xt-a-bxt-cytyt+Axtdt+αxtdw1t+∫Zγuxt-N~dt,du,dyt=yt--g-hyt+fxtyt+Axtdt+βytdw2t+∫Zηuyt-N~dt,du,where x(t-), y(t-) represent the left limit of x(t), y(t). N~(dt,du)=N(dt,du)-λ(du)dt, N is a Poisson random measure, and λ is the characteristic measure of N on a measurable subset Z⊂R+=(0,+∞) with λ(Z)<+∞.The rest of this paper is organized as follows. In Section2, some properties of positive solutions to system (4) are discussed. In Section 3, the main results for persistence and extinction are given. Finally, the simulation results show the validity of our results.


## 2. Properties of Positive Solutions

Throughout this paper, we require thatw1(t), w2(t), and N are independent and(5)H1max⁡∫Zγupλdu,∫Zηupλdu<+∞,p>0H2max⁡∫Zln⁡1+γu2λdu,∫Zln⁡1+ηu2λdu<+∞with min⁡1+γu,1+ηu>0,u∈Z.First, we present the global existence of positive solutions.Lemma 1.
For any given value(x0,y0)∈R+2, system (4) has a unique positive solution X(t)=(x(t),y(t)) on t≥0 and the solution will be in R+2 a.s. (almost surely).Proof.
Letut=ln⁡xt,vt=ln⁡yt, then we consider the following system: (6)dut=a-beut-cevtevt+Aeut-α22-∫Zγu-ln⁡1+γuλdudt+αdw1t+∫Zln⁡1+γuN~dt,du,dvt=-g-hevt+feutevt+Aeut-β22-∫Zηu-ln⁡1+ηuλdudt+βtdw2t+∫Zln⁡1+ηuN~dt,duwith initial date u(0)=ln⁡x0,v(0)=lny0 on t≥0. Since the coefficients of system (6) are locally Lipschitz continuous, then there is a unique local solution on [0,T] a.s., where T is blow-up time. Then x=eu(t),y=ev(t) is the unique positive local solution to system (4) with initial data x0>0,y0>0. We will show that T=+∞, and this mean the solution is global.
Consider the following equations:(7)dX1t=X1ta-c-bX1tdt+αX1tdw1t+∫ZX1tγuN~dt,du,X10=x0,dX2t=X2ta-bX2tdt+αX2tdw1t+∫ZX2tγuN~dt,du,X20=x0,dY1t=Y1t-g+fA-h+fA2X1tY1tdt+βY1tdw2t+∫ZY1tηuN~dt,du,Y10=y0,dY2t=Y2t-g+fA-hY2tdt-βY2tdw2t+∫ZY2tηuN~dt,du,Y20=y0.
By the comparison theorem of stochastic differential equation, we conclude that(8)X1t≤xt≤X2t,Y1t≤yt≤Y2ta.s.
According to [10], we can give(9)X1t=exp⁡a-c-ρ1t+αw1t+κ1tx0-1+b∫0texp⁡a-c-ρ1s+αw1s+κ1sds,X2t=exp⁡a-ρ1t+αw1t+κ1tx0-1+b∫0texp⁡a-ρ1s+αw1s+κ1sds,Y1t=exp⁡-g+f/A-ρ2t+βw2t+κ2ty0-1+∫0th+f/A2X1texp⁡f/A-g-ρ2s+βw2s+κ2sds,Y2t=exp⁡-g+f/A-ρ2t+βw2t+κ2ty0-1+h∫0texp⁡f/A-g-ρ2s+βw2s+κ2sds,where (10)ρ1=0.5α2+∫Zγu-ln⁡1+γuλdu,ρ2=0.5β2+∫Zηu-ln⁡1+ηuλdu,κ1t=∫0t∫Zln⁡1+γuN~dt,du,κ2t=∫0t∫Zln⁡1+ηuN~dt,du.
Becauset≥0 is the existence range of solutions X1(t),X2(t),Y1(t),Y2(t), that means T=+∞.Next, we will show the asymptotic property of the solution to system (4).Theorem 2.
The solutions of system (4) are bounded in mean.Proof.
LetV1≔V1(t,x)=etxp(p>0). Direct computation, by the formula EV1(t,x(t))-EV1(0,x0)=E∫0tLV(s)ds [18], now leads to(11)Eetxp=xp0+E∫0tLV1ds,where (12)LV1=etxp+etpxpa-bx+0.5p-1α2-cyy+Ax+∫Z1+γup-1λdu≤etxp+etpxpa-bx+0.5p-1α2+∫Z1+γup-1λdu.From actual meanings of parameters and assumption H1, we get that(13)xp+pxpa-bx+0.5p-1α2+∫Z1+γup-1λduis bounded. There exists a constant K1>0, such that (14)Eetxpt≤x0p+∫0tK1esdswhen 1+xp+pxp[a-bx+0.5(p-1)α2∫Z[(1+γ(u))p-1]λ(du)]>0; otherwise Eetxp(t)≤x0p. Let K2=max{K1,0}; we have Eetxp(t)≤x0p+K2et. Denote K=K2+x0p, then Exp(t)≤K<+∞. Similarly, Eyp(t)≤K.


## 3. Persistence and Extinction

In this section, some properties of the solutions of system (4) are investigated. Some sufficient conditions for persistence and extinction are shown. To proceed, some definitions and lemma are as follows.Definition 3 (see [19]).
(1) The populationx is called to be extinct if x(t)→0(t→∞)a.s.
(2) The populationx is called to be persistent if liminft→∞⁡1/t∫0tx(s)ds>0 a.s.
(3) System (4) is called to be persistent if populations x and y are all persistent a.s.Lemma 4 (see [20]).
UnderH1 and H2, suppose Y(t)∈C(Ω×[0,+∞),R+).
(1) If there exist three positiveT,k,k0 such that (15)ln⁡Yt≤kt-k0∫0tYsds+∑i=1nkiwit+∑i=1nci∫0t∫Zln⁡1+γuN~ds,dua.s.for all t>T, where ki,ci are constants, then(16)limsupt→∞⁡1t∫0tYsds≤kk0a.s.If k<0 and other conditions are the same, then limt→∞⁡Y(t)=0 a.s.
(2) If there exist three positiveT,k,k0 such that (17)ln⁡Yt≥kt-k0∫0tYsds+∑i=1nkiwit+∑i=1nci∫0t∫Zln⁡1+γuN~ds,dua.s.for all t>T, where ki,ci are constants, then (18)liminft→∞⁡1t∫0tYsds≥kk0a.s.
Now, the main results about persistence and extinction of system (4) are as follows.Theorem 5.
(1) The populationsx and y are extinct if a-ρ1<0 a.s.
(2) The populationx is persistent and y is extinct if a-c-ρ1>0,f/A-g-ρ2<0 a.s.
(3) The populationsx and y are persistent if a-c-ρ1>0,f/A-g-ρ2>0 a.s.Proof.
(1) By using the Itô formula, the following formulas are hold:(19)dln⁡xt=a-bxt-cytyt+Axt-ρ1dt+αdw1t+∫Zln⁡1+γuN~dt,du,dln⁡yt=-g-hyt+fxtyt+Axt-ρ2dt+βdw2t+∫Zln⁡1+ηuN~dt,du.Calculated by integral, we have the following form: (20)ln⁡xt-ln⁡x0=a-ρ1t+αw1t-∫0tbxsds-∫0tcysys+Axsds+κ1t≤a-ρ1t-b∫0txsds+αw1t+κ1t.
Then by Lemma4, note that a<ρ1, and therefore x(t)→0(t→∞) (population x is extinct) a.s. Then, for arbitrarily small ε>0, there exist a sufficiently large T, when t>T, and we have x(t)<ε and (21)dln⁡yt≤-g-hyt+ε-ρ2dt+βdw2t+∫Zln⁡1+ηuN~dt,du,where -g-ε-ρ2<0. It is clear that y(t)→0(t→∞) (population y is extinct) a.s. Thus (1) is correct.
(2) From (1), the following forms are clear.(22)ln⁡xt-ln⁡x0=a-ρ1t+αw1t-∫0tbxsds-∫0tcysys+Axsds+κ1≤a-ρ1t-b∫0txsds+αw1t+κ1,ln⁡xt-ln⁡x0=a-ρ1t+αw1t-∫0tbxsds-∫0tcysys+Axsds+κ1≥a-c-ρ1t-b∫0txsds+αw1t+κ1.Then by Lemma 4, we have (23)a-c-ρ1b≤liminft→∞⁡1t∫0txsds≤limsupt→∞⁡1t∫0txsds≤a-ρ1ba.s.Therefore the population x is persistent in mean. For population y, we have (24)dln⁡yt≤-g-hyt+fA-ρ2dt+βdw2t+∫Zln⁡1+ηuN~dt,du.
From Lemma4 and condition f/A-g-ρ2<0, we know that y(t)→0(t→∞) (i.e., population y is extinct) a.s.
(3) In view of above, we can conclude the following whena-c-ρ1>0,g/A-g-ρ2>0, and(25)ln⁡yt-ln⁡y0≥-g+fA-ρ2t-h+fA2X1∗∫0tysds+βw2t+κ2t.Then, we have (26)f/A-g-ρ2h+f/A2X1∗≤liminft→∞⁡1t∫0tysds≤limsupt→∞⁡1t∫0tysds≤f/A-g-ρ2h,a.s.,where X1∗ is the minimum of X1(t)>0.
It is clear that populationy is persistent.Remark 6.
For (1) in Theorem5, it implies that the population x of extinction a.s. leads to the extinction of population y a.s. As shown in Figure 2, the simulations also affirm this point. In Figure 2, we can see that population x becomes extinct, and after a while, population y becomes extinct.


## 4. Numerical Simulations

We will demonstrate our results with the help of numerical simulations by using the Euler-Maruyama scheme [21, 22]. In numerical simulation, randomly selected parameters are as follows a=0.55,b=0.21,c=0.15,f=0.93,h=0.11,A=1.23,α=0.75,β=0.73,Z=(0,+∞), and λ(Z)=1, with simulation time span T=50 and step size Δt=T/N, where N=212. The initial data is (x0,y0)=(0.3,0.11) in Figure 2, and others are (3,1).(1) As illustrated in Figure1, we choose γ(u)=0.99,η(u)=0.85, then ρ1=a-0.5α2-∫Z(γ(u)-ln(1+γ(u)))λ(du)=0.58312, ρ2=a-0.5β2-∫Z(η(u)-ln(1+η(u)))λ(du)=0.50126, and a-ρ1=-0.03312<0,f/A-g-ρ2=-0.09517<0. By Theorem 5, the populations x and y are extinct a.s. Numerical experiments verify the correctness of (1) in Theorem 5.Figure 1Figure 2(2) As illustrated in Figure2, we choose γ(u)=0.99,η(u)=0.05, then ρ1=a-0.5α2-∫Z(γ(u)-ln(1+γ(u)))λ(du)=0.58312, ρ2=a-0.5β2-∫Z(η(u)-ln(1+η(u)))λ(du)=0.26766, and a-ρ1=-0.03312<0,f/A-g-ρ2=0.13844>0. By Theorem 5, the populations x and y are extinct a.s. Numerical experiments also verify the correctness of (1) in Theorem 5.(3) As shown in Figure3, we choose γ(u)=0.15,η(u)=0.91, then ρ1=a-0.5α2-∫Z(γ(u)-ln(1+γ(u)))λ(du)=0.29149, ρ2=a-0.5β2-∫Z(η(u)-ln(1+η(u)))λ(du)=0.52935, and a-c-ρ1=0.048512>0,f/A-g-ρ2=-0.12325<0. By Theorem 5, the population x is persistent a.s. and population y is extinct a.s. The correctness of (2) in Theorem 5 is verified.Figure 3(4) As shown in Figure4, we choose γ(u)=0.15,η(u)=0.05, then ρ1=a-0.5α2-∫Z(γ(u)-ln(1+γ(u)))λ(du)=0.29149, ρ2=a-0.5β2-∫Z(η(u)-ln(1+η(u)))λ(du)=0.52935, and a-c-ρ1=0.048512>0,f/A-g-ρ2=0.13844>0. By Theorem 5, the populations x and y are persistent a.s. The validity of correctness of (3) in Theorem 5 is verified.Figure 4


## 5. Conclusions

In this paper, a stochastic modified Bazykin system with Lévy jumps is discussed. Firstly, the existence of global positive solutions is investigated, and boundedness in mean is also given. Further, under related assumption, we obtain the sufficient conditions for the stochastic permanence and extinction of system (4). There are some interesting topics which deserve further discussion. For example, many authors consider the stationary distribution for stochastic predator-prey model with harvesting and delays [23–25], epidemic model with regime switching [26], impulsive stochastic model [27], and budworm growth model with Markovian switching [28]. The above investigations are left for future work.



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