# Stochastic Periodic Solution and Permanence of a Holling–Leslie Predator-Prey System with Impulsive Effects

Authors: Jinxing Zhao; Yuanfu Shao
Journal: Journal of Mathematics
 (2021)

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## Abstract

Considering the environmental effects, a Holling–Leslie predator-prey system with impulsive and stochastic disturbance is proposed in this paper. Firstly, we prove that existence of periodic solution, the mean time boundness of variables is found by integral inequality, and we establish some sufficient conditions assuring the existencle of periodic Markovian process. Secondly, for periodic impulsive differential equation and system, it is different from previous research methods, by defining three restrictive conditions, we study the extinction and permanence in the mean of all species. Thirdly, by stochastic analysis method, we investigate the stochastic permanence of the system. Finally, some numerical simulations are given to illustrate the main results.


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## Body

## 1. Introduction

Predator-prey phenomenon is very popular in natural world, and recently more and more researchers pay attention to investigate the complicated dynamical behaviors between predator and prey species, which has been and is long to be one of the most important hot topics in the future [1]. Hsu and Huang [2] proposed the following Holling–Tanner model:(1)dxt=xta1−b1xt−aytb+xtdt,dyt=yta2−b2ytxtdt,where xtandyt are the densities of prey and predator at time t, respectively; aii=1,2 is the intrinsic growth rate of prey or predator; and b1 is the density-dependent coefficient; ayt/b+xt represents Holling type II functional response, where a,b>0 denote the capturing rate and half capturing saturation constant, respectively. Function b2yt/xt is the Leslie–Gower term, which measures the loss in the predator population due to rarity of its favorite food, where a2x/b2 is the carrying capacity. For more biological meanings of this model, see [3–5].Moreover, in natural world, there are many kinds of functional responses. If the functional response is Holling type IV, then the model reads(2)dxt=xta1−b1xt−aytb+x2tdt,dyt=yta2−b2ytxtdt.Its dynamics has been sufficiently studied, and many better results have been obtained by Li and Xiao [6].On the contrary, in practice, the environmental white noise almost exists everywhere. For ecological system, the growth rate of population is inevitably affected by the white noise. In order to reveal the effect of white noise, random disturbance is introduced in many mathematical models [7–12]. Meanwhile, due to the individual life cycle and seasonal variation and so on, the birth rate, death rate, carrying capacity of species, and other parameters always exhibit cycle changes [13–15]. Therefore, Jiang et al. [16] proposed the following nonautonomous stochastic model:(3)dxt=xta1t−b1txt−atytbt+xntdt+σ1txtdB1t,dyt=yta2t−b2tytxtdt+σ2tytdB2t,where ait,bit,σiti=1,2,at,andbt are all positive T-periodic functions; B1t and B2t are independent standard Brownian motions defined on the probability space Ω,F,Ftt≥0,P with a filtration Ftt≥0 satisfying the usual conditions (i.e., it is right continuous, and F0 contains all p-null set); and σi2i=1,2 denotes the density of white noise. Parameter n=1 or n=2 means Holling type II or Holling type IV functional response, respectively.However, for population system, the effect from natural or man-made factors is very popular, and hence, the growth of species will have to suffer from some discrete changes of relatively short time interval at some periodic times, such as stocking and harvesting. These effects are often modeled by impulsive parameters. In the last decades, many impulsive dynamical systems have been proposed and many better results have been reported, see, [17–23] and references therein. For example, the authors Zuo and Jiang [23] investigated the periodic solution and boundary periodic solution of the following impulsive model:(4)dxt=xta1t−b1txt−atytbt+xtdt+σ1txtdB1tdyt=yta2t−b2tytxtdt+σ2tytdB2t,t≠tk,x1tk+=1+ξ1kx1tkx2tk+=1+ξ2kx2tk,t=tk.Inspired by the above discussion, considering some natural or man-made impulsive factors, we propose the following stochastic predator-prey model:(5)dxt=xta1t−b1txt−atytbt+xntdt+σ1txtdB1tdyt=yta2t−b2tytxtdt+σ2tytdB2t,t≠tk,x1tk+=1+ξ1kx1tkx2tk+=1+ξ2kx2tk,t=tk,where all coefficients a1t,a2t,b1t,b2t,at,bt,σ1t,andσ2t are bounded, continuous, and periodic with period T. The impulsive points satisfy 0<t1<t2<⋯, limk⟶∞tk=+∞, and there exists an integer p such that tk+p=tk+T,λi,k+p=λik,i=1,2. Furthermore, by the biological meanings, we assume ξik>−1 for i=1,2. For the biological meanings of all parameters, refer to [2, 6, 16, 23].Our main aim of this paper is as follows: firstly, for determinate system, the existence of equilibrium or periodic solution is an important topic for the dynamics of biological system [24–27]. Similarly, for stochastic system, it is very interesting to study whether there exists a periodic Markovian process or not.Secondly, for predator-prey system, the dynamical behaviors are another important topic [28–30]. By the comparison method, we establish some sufficient conditions assuring the extinction, permanence in the mean of all species, and the stochastic permanence of system (5).The rest of the work of this paper is organized as follows: Section2 begins with some notations, definitions, and important lemmas. Section 3 is devoted to the existence and uniqueness of the periodic Markovian process. Section 4 focuses on the extinction and permanence in the mean of species. Section 5 focuses on the stochastic permanence of system (5). Some numerical simulations are given to verify our main results in Section 6. Finally, we conclude this paper with a brief conclusion and discussion in Section 7.


## 2. Preliminaries

For ann-dimensional stochastic differential equation [13],(6)dxt=fxt,tdt+gxt,tdBt,t≥t0,with initial value xt0=x0∈Rn, where Bt is an n-dimensional standard Brownian motion. The differential operator L associated (6) is defined by(7)L=∂∂t+∑k=1nfkx,t∂∂xk+12∑k,j=1ngTx,tgx,tkj∂2∂xk∂xj.For bounded and continuous functionft, set(8)fu=supt≥0ft,fl=inft≥0ft,f∗=limsupt⟶+∞ft,f∗=liminft⟶+∞ft,and if ft is integrable, then define(9)f=1t∫0tfsds,fT=1T∫0Tfsds.To investigate the dynamics of (5), we consider the following nonimpulsive system:(10)dXt=Xta1t+1T∑j=1pln1+ξ1k−b1tA1tXt−atA2tYtbt+A1tXtndt+σ1tXtdB1t,dYt=Yta2t+1T∑j=1pln1+ξ2k−b2tA2tYtA1tXtdt+σ2tYtdB2t,where(11)A1t=∏j=1p1+ξ1j−t/T∏0≤tk<t1+ξ1k,A2t=∏j=1p1+ξ2j−t/T∏0≤tk<t1+ξ2k.It is easy to show thatA1tandA2t both are periodic functions with period T (for details, see [27]). We assume the product equals unity if the number of factors is zero and ε stands for a sufficiently small positive constant whose value may be different at different places.Now we present the definitions of periodic Markovian process and the solution of impulsive stochastic differential equation, and some auxiliary results of the existence of periodic Markovian process.Definition 1.
(see [13, 27]). A stochastic process ξt=ξt,ω−∞<t<∞ is said to be periodic with periodic T if for every finite sequence of numbers t1,t2,…,tn, the joint distribution of random variables ξt1+h,…,ξtn+h is independent of h, where h=kT,k=±1,±2,….Definition 2.
(see [22, 27]). System (5) is stochastically permanent if for every ε∈0,1, there is a pair of constants α>0 and β>0 such that for any initial data x0,y0∈R+2, the solution xt,yt of (5) has the property that(12)liminft⟶+∞Pxt,yt≥α≥1−ε,limsupt⟶+∞Pxt,yt≤β≥1−ε,where P represents the probability of the events.Definition 3.
(see [20, 22]). For the following impulsive stochastic differential equation(13)dxt=ft,xtdt+gt,xtdBt,t≠tk,xtk+=1+λkxtk,k∈N,with initial value x0=x0, a stochastic process xt=x1t,x2t,…,xntT,t∈R+, is said to be a solution of the above system, if(i)
xt is ft adapted and is continuous on 0,t1 and each interval tk,tk+1⊂R+,k∈N, ft,xt∈LR+,Rn,andgt,xt∈L2R+,Rn, where LpR+,Rn is all Rn-valued measurable ft-adapted processes ft satisfying ∫0Tftpdt<+∞ almost surely for all T>0,p=1,2.(ii)
For everytk,k∈N, xtk+=limt⟶tk+xt and xtk−=limt⟶tk−xt exist, and xtk=xtk− with probability one.(iii)
For allt∈0,t1,xt obeys the integral equation(14)xt=x0+∫0tfs,xsds+∫0tgs,xsdBs,and for all t∈tk,tk+1,k∈N,xt obeys the following integral equation:(15)xt=xtk++∫tktfs,xsds+∫tktgs,xsdBs.Lemma 1.
(see [13, 27]). For the following Itô’s differential equation(16)dxt=bt,xtdt+σt,xtdBt,if all the coefficients are T−periodic in t and satisfy the linear growing condition and the Lipschitz condition in every cylinder Ul×R+ for l>0, where Ul=x:x≤l and there exists a function v=vt,x which is twice continuously differentiable with respect to x and once continuously differentiable with respect to t in Rn×R+, T is periodic in t and satisfies the following conditions:(17)infx>lvt,x⟶+∞,asl⟶+∞,Lvt,x≤−1outside some compact set,then there exists a solution of (16) which is T-periodic Markovian process.Lemma 2.
(see [20]). Suppose that Zt∈CΩ×0,+∞,R+ and limt⟶+∞Ft/t=0,a.s.(a)
If there exist two positive constantsT>0,λ0>0 such that, for all t>T,(18)lnZt≤λt−λ0∫0tzsds+Ft,a.s,then(19)Z∗≤λλ0,a.s.ifλ≥0,limt⟶+∞Zt=0,a.s.ifλ<0.(b)
If there exist some constantsT>0,λ0>0,λ such that, for all t>T,(20)lnZt≥λt−λ0∫0tzsds+Ft,a.s,then(21)Z∗≥λλ0,a.s.Lemma 3.
Letxt=A1tXt,yt=A2tYt; then(1)
ifxt,yt is the solution of (5), then Xt,Yt is the solution of (10).(2)
ifXt,Yt is the solution of (10), then xt,yt is the solution of (5).Remark 1.
The proof is similar with that of [20] and is omitted. Lemma 3 shows that the dynamics of (5) is equivalent to that of (10). Hence, in the later, we mainly consider (10) to reveal the dynamical properties of (5).
As to the existence of nontrivial positive solution of (5), we have Lemma 4.Lemma 4.
For any given initial valuex0,y0∈R+2, system (5) has a unique solution xt,yt on t≥0, and the solution remains in R+2 with probability one.Remark 2.
Lemma3 implies that the existence of solution xt,yt of (5) on t≥0 is equivalent to the existence of the solution Xt,Yt of (10). The proof of the existence of Xt,Yt of (10) is similar with that of [23] and is omitted here.


## 3. Existence of Periodic Solution

In this section, we focus on the existence of periodic Markovian process of (5). Above all, we give the following assumption.Theorem 1.
Suppose the following condition holds,(22)H1ζ≔a1t−σ12tT+1T∑j=1pln1+ξ1k>0,η≔a2t−σ22tT+1T∑j=1pln1+ξ2k>0,then there exists a solution for system (5), which is a T-periodic Markovian process.Proof.
According to the equivalent property and existence of solutions (Lemmas3 and 4), we only need to prove that, under H1, the solution of system (10) is a periodic Markovian process. Lemma 1 shows that it suffices to find a C2-function Vt,X,Y and a closed set U∈R+2 such that all conditions of Lemma 1 hold. Define(23)V1t,X=eμ1tX+lnX,V2t,Y=qeμ2tY+clnY,where qandc are two constants defined later and μ1t and μ2t are the positive continuous function such that(24)μ1′t=a1t−σ12t−a1t−σ12tT,μ2′t=a2t−σ22t−a2t−σ22tT.
It is not difficult to verify thatμiti=1,2 is T-periodic on 0,∞, and(25)liminfX,Y∈R+2/UkVt,X,Y⟶∞,ask⟶∞,where Uk=X,Y:X,Y∈1/k,k×1/k,k. Hence Vt,X,Y is T-periodic and satisfies the first condition of Lemma 1. Applying the Itô’s formula to V1t,X and V2t,Y, then(26)LV1t,X=eμ1tXμ1′t−a1t−1T∑j=1pln1+ξ1k+σ12t+a1t+1T∑j=1pln1+ξ1k−b1tA1tX−atA2tYbt+A1tXn+eμ1tb1tA1t+eμ1tXatA2tYbt+A1tXn−σ12t2=a1t+1T∑j=1pln1+ξ1k−σ12t2+eμ1tb1tA1t−b1tA1tX+eμ1tXμ1′t−a1t−1T∑j=1pln1+ξ1k+σ12t+eμ1tXatA2tYbt+A1tXn−atA2tYbt+A1tXn≤a1u+1T∑j=1pln1+ξ1k−σ12l2+eμ1ub1uA1u−b1lA1lX−eμ1tXa1t−σ12tT+1T∑j=1pln1+ξ1k+auA2ueμ1uYblX−alA2lYbu+A1unXn,LV2=qeμ2tYμ2′t−a2t−1T∑j=1pln1+ξ2k+σ22t+ca2t+1T∑j=1pln1+ξ2k−cb2tA2tYA1tX+qb2teμ2tA2tA1tX−cσ22t2≤−qeμ2tYa2t−σ22tT+1T∑j=1pln1+ξ2k+qb2ueμ2uA2uA1lX−cb2lA2lYA1uX+ca2u+1T∑j=1pln1+ξ2k−σ22l2.
Letc=auA1uA2ueμ1u/blb2lA2l, then we have(27)LVt,X,Y≤ϑ−b1lA1lX−A1leμ1lζ−qeμ2ub2uA2uA1lX−qeμ2lYη−alA2lYb1u+A1unXn,where ζandη are defined as above, and(28)ϑ≔eμ1ub1uA1u+a1u+1T∑j=1pln1+ξ1k−σ12tl2+auA1uA2ueμ1ublb2lA2la2u+1T∑j=1pln1+ξ2k−σ22tl2.
Takeq=A1leμ1lζ/2eμ2ub2uA2u, then(29)LVt,X,Y≤ϑ−b1lA1lX−eμ1lζ2X−A1leμ1l+μ2l−μ2u2b2uA2uYζη−alA2lYb1u+A1unXn≔ΦX,Y.
Choose any small positiveε1 and ε2 such that(30)0<ε1<minb1lA1lϑ+1,eμ1lζ2ϑ+1,0<ε2<A1leμ1l+μ2l−μ2u2b2uA2uϑ+1,ϑ+1<alA2lA1uε1nε2b1uA1lε1n+1.
Define an open subset as follows:(31)Dε1,2=X,Y:X,Y∈ε1,1ε1×ε2,1ε2.
Obviously,Dε1,2 is compact and its component DεC=R+2/Dε=∪i=14Dεi, where(32)Dε1=X,Y|0<X<ε1,Dε2=X,Y|0<Y<ε2,Dε3=X,Y|X>1ε1,Dε4=X,Y|Y>1ε2.
It is easy to verify thatΦX,Y⟶−∞ when X⟶0+, or Y⟶0+, or X⟶∞, or Y⟶∞. Therefore, LVX,Y≤−1 holds for any X,Y∈R+2/Dε, which means the second condition of Lemma 1 holds. Using Lemma 1, then the existence of periodic solution of (10) is obtained. This completes the proof.Remark 3.
For system (5), if there is no impulsive effect, i.e., ξik=0i=1,2, by Theorem 1, we can obtain the sufficient conditions assuring the existence of T-periodic solution, which is in accordance with Theorem 3 in Reference [16]. And if n=1, i.e., the case of Holling type II functional response, Theorem 1 yields the same result as Theorem 3.3 in reference [23]. It is in this sense that we improved or generalized the main results in [16, 23].


## 4. Extinction and Permanence in the Mean

In this section, we discuss the extinction and permanence in the mean of system (5). Firstly, we provide a lemma on the presentation of the solution for an impulsive stochastic differential equation.Lemma 5.
For the following periodic impulsive differential equation(33)dxt=xta1t−b1txtdt+σ1txtdBt,t≠tk,xtk+=1+ξ1kxtk,let xt be a solution with any given initial data x0>0. Then x˜t is a unique positive T-periodic solution such that limt⟶+∞xt−x˜t=0,a.s, where(34)xt=∏0<tk<t1+ξ1kexp∫0ta1s−σ12s/2ds+∫0tσ1sdBs1/x0+∫0t∏0<tk<s1+ξ1kb1sexp∫0sa1θ−σ12θ/2dθ+∫0sσ1θdBθds.Remark 4.
The existence ofT-periodic solution can be obtained by Theorem 1. The presentation of xt and the global attractivity are referred to [21].Lemma 6.
For the solutionxt,yt of system (5), we have(35)limsupt⟶+∞Pxt<β1>1−ε,limsupt⟶+∞Pyt<β2>1−ε,that is, the solution of (5) is stochastically ultimately bounded.Proof.
From the first equation of (5), we have(36)dxt≤xta1t−b1txtdt+σ1txtdB1t.
Consider the following comparison system:(37)dut=uta1t−b1utdt+σ1tutdB1t,t≠tk,utk+=1+ξ1kutk.
By Lemma5, the solution ut of (37) is positive and T-periodic, right continuous, and globally attractive. Therefore, ut has maximum value and minimum value. Define u∗=maxt⟶+∞ut, then by comparison theorem for stochastic equation, for any sufficiently small positive ε>0, we have(38)xt≤ut≤u∗+ε≔β1.
Therefore,limsupt⟶+∞Pxt<β1>1−ε is established.
On the contrary, usingxt≤β1, we can obtain from (5) that(39)dyt≤yta2t−b2tytβ1dt+σ2tytdB2t.
In the same manner, we have(40)yt≤vt≤v∗+ε≔β2,
wherevt is the solution of the following stochastic comparison system:(41)dvt=vta2t−b2tytβ1dt+σ2tvtdB2t,t≠tk,vtk+=1+ξ2kvtk,and v∗=maxt⟶+∞vt. Therefore, limsupt⟶+∞Pyt<β2>1−ε. This completes the proof.
For convenience, denote(42)ζ^≔a1t−σ12t2T+1T∑j=1pln1+ξ1k,η^≔a2t−σ22t2T+1T∑j=1pln1+ξ2k,ζ˜≔a1t−σ12t2−atA2tβ2btT+1T∑j=1pln1+ξ1k.Theorem 2.
For system (5), the following results hold:(i)
Ifζ^>0, then x∗≤M1, that is, species xt is bounded in the mean.(ii)
Ifη^>0, then y∗≤M2 and y∗≥m2, that is, species yt is permanent in the mean.(iii)
Ifζ^<0, then limt⟶+∞xt=0, that is, species xt is extinct, and if η^<0, then limt⟶+∞yt=0, that is, species yt is extinct.(iv)
Ifζ^>0,η^<0, then x∗≤M1 and x∗≥m˜1. If ζ˜>0,η^>0, then x∗≤M1 and x∗≥m1, that is, species xt is permanent in the mean, where m˜1,mi,Mii=1,2 are constants defined later in the proof.Proof.
Make use of Lemma3, we only need to prove these conclusions hold for (10) for some constants m˜1′,mi′,Mi′i=1,2. For system (10), by applying the Itô’s formula to lnX and lnY, we have(43)dlnX=a1t+1T∑j=1pln1+ξ2k−b1tA1tX−atA2tYtbt+A1tXtn−σ12t2dt+σ1tdB1t,(44)dlnY=a2t+1T∑j=1pln1+ξ2k−b2tA2tYtA1tXt−σ22t2dt+σ2tdB2t.(i)
Ifζ^>0, then integrating both sides of (43) from 0 to t yields(45)lnXtX0≤1t∫0ta1s−σ12s2+1T∑j=1pln1+ξ1kds−b1uA1u∫0tXsds+1t∫0tσ1sdB1s.Sincelimt⟶+∞1/T∫0tσ1sdB1s=0, then we obtain from Lemma 2 that(46)X∗≤a1t−σ12/2T+1/T∑j=1pln1+ξ1kb1uA1u≔M1′.(ii)
Lemma4 implies the solution of (10) satisfies X>0, and hence there exists a positive constant ε>0 such that xt>ε. From (44), we have(47)dlnY≥a2t−σ22t2+1T∑j=1pln1+ξ2k−b2tA2tA1tεYdt+σ2tdB2t,which leads to(48)lnYtY0≥∫0ta2s−σ22s2+1T∑j=1pln1+ξ1kds−b2uA2uA1lε∫0tYsds+1t∫0tσ2sdB2s.Under the conditionη^>0, we obtain from Lemma 2 that(49)Y∗≥A1lεa2t−σ22t/2T+1/T∑j=1pln1+ξ1kb2uA2u≔m2′.On the contrary, combining Lemma6, we have(50)dlnY≤a2t−σ22t2+1T∑j=1pln1+ξ1k−b2lA2lA1uβ1Ydt+σ2tdB2t.Integrating both sides of above inequality yields(51)lnYtY0≤1t∫0ta2s−σ22s2+1T∑j=1pln1+ξ1kds−b2lA2lA1uβ1∫0tYsds+1t∫0tσ2tdB2t.Ifη^>0, using Lemma 2 again, then(52)Y∗≤A1uβ1a2t−σ22t/2T+1/T∑j=1pln1+ξ1kb2lA2l≔M2′.Therefore, speciesYt is permanent in the mean.(iii)
Ifζ^<0, by Case (i), obviously limt⟶+∞Xt=0. If η^<0, similarly, by the proof of Case (ii), we have limt⟶+∞Yt=0.(iv)
Ifζ^>0, then Case (i) indicates X∗≤M1′. If η^<0, by use of Case (ii), we have limt⟶+∞Yt=0, then (43) reads(53)dlnX=a1t+1T∑j=1pln1+ξ1k−b1tA1tX−σ12t2dt+σ1tdB1t,and hence,(54)dlnX≥a1t+1T∑j=1pln1+ξ1k−b1uA1uX−σ12t2dt+σ1tdB1t.
According to Lemma2, then(55)X∗≥a1t−σ12t/2T+1/T∑j=1pln1+ξ1kb1uA1u≔m˜1′.
Ifζ˜>0, then obviously ζ^>0, and hence X∗≤M1′. Therefore,(56)m˜1′≤X≤M1′.
Furthermore, ifη^>0, then from (33) and Lemma 6, we have(57)dlnX≥a1t+1T∑j=1pln1+ξ1k−b1uA1uX−atA2tβ2bt−σ12t2dt+σ1tdB1t,and hence,(58)lnXtX0≥1t∫0ta1s−σ12s2+1T∑j=1pln1+ξ1k−asA2sβ2bsds−b1uA1u∫0tXsds+1t∫0tσ1tdB1t.
Then(59)X∗≥a1t−σ12t/2−atA2tβ2/btT+1/T∑j=1pln1+ξ1kb1uA1u≔m1′.
CombiningX∗≤M1′, then species Xt is permanent in the mean.
To summarize, the above conclusions hold for system (10). Using Lemma 3, the required assertion is directly obtained. The proof is completed.Remark 5.
For system (5), if the impulsive is absent, then Theorem 2 implies the sufficient conditions of the extinction of species xt or yt, which is accordance with Theorem 2 in [16].


## 5. Stochastic Permanence

In this section, we consider the stochastic permanence of (5).Theorem 3.
Ifζ^>0,η^>0, then system (5) is stochastically permanent.Proof.
This proof is motivated by Reference [21]. Let Ωtε=ω∈Ω,Xt<β1,Yt<β2, Ω∞ε=ω∈Ω,limsupt⟶+∞PΩtε>1−ε, then Lemma 6 implies that PΩ∞ε>1−ε. Hence, for any ω∈Ω∞ε, there exists k1>0 such that Xt<β1,Yt<β2 for any t≥k1T.
On the contrary, by the assumptionζ^>0, we can choose a positive constant ε>0 such that(60)ζ^−b1uA1uε−auA1uA2uεblb2lA2l>0.
Firstly, we prove that there existsς1>0, for any ω∈Ω∞ε, we have(61)liminft⟶+∞PXt>ς1>1−ε.
Suppose it is not true, i.e., there exist a solutionX,Y with initial data X0>0,Y0>0, a positive integer k2>k1, and a nonempty set Ω1⊂Ωtε with PΩ1>0 such that Xt,ω<ς1 for any ω∈Ω1 and t≥k2T. From (44), we have(62)dlnY=a2t+1T∑j=1pln1+ξ2k−b2tA2tYtA1tXt−σ22t2dt+σ2tdB2t≤a2t−σ22t2+1T∑j=1pln1+ξ2k−b2lA2lYtA1uς1+σ2tdB2t.
Then Lemma2 implies(63)Y∗≤A1ua2t−σ22/2+1/T∑j=1pln1+ξ2kς1b2lA2l≔κ.
On the contrary,(64)dlnX=a1t+1T∑j=1pln1+ξ2k−2b1tA1tX−atA2tYtbt+A1tXtn−σ12t2dt+σ1tdB1t.
Integrating (64) from k2T to t yields(65)lnXt=lnXk2T+∫k2Tta1s+1T∑j=1pln1+ξ2k−σ12s2ds−∫k2Tt2b1sA1sXds−∫k2TtasA2sYsbs+A1sXsnds+∫k2Ttσ1sdB1s.
Then we derive from (65) that(66)liminft⟶+∞lnXt≥liminft⟶+∞a1t−σ12t2T+1T∑j=1pln1+ξ1k−b1uA1uς1−auA2uκbl>0.
Then we concludeXt⟶+∞ as t⟶+∞, which contradicts with the boudedness of Xt. Hence we have liminft⟶+∞PXt>ς1>1−ε for any ω∈Ω∞ε.
Secondly, we prove there exists a constantα1ε>0 such that(67)liminft⟶+∞Pω∈Ω∞ε:α1ε≤Xt,ω≤β1ε≥1−ε.
We claim thatliminft⟶+∞Pxt>α1>1−ε holds for all ω∈Ω∞ε. If not, then there exists a sequence νn⊂R+ and a nonempty set Ω2⊂Ωtε with PΩ2>0, and a positive integer k3k3>k1 such that supt>k3TXt,νn,ω<α1ε/n+1 for all ω∈Ω2, where Xt,νn,Yt,νn is the solution of (10) with initial data νn. Using (64), for all t≥k3T, we have(68)dlnXt,νnIΩ2dt≥a1t+1T∑j=1pln1+ξ1k−σ12t2−2b1tA1tβ1ε−atA2tβ2εbtPΩ2.
According to the previous proof, there is a nonempty setΩ3⊆Ω2 where PΩ3>0, and two positive integers k4 and k5 with k5>k4>k3 such that(69)Xk4T,νn≥α1ε,Xk5T,νn≤α1εn+1hold for all ω∈Ω3. Define(70)τ∗=inft:Xt,νn>α1εn+1,ω∈Ω3,t>k5T,τ∗=supt<τ∗:Xt,νn<α1ε,ω∈Ω3,t>k4T.
ThenXτ∗,νn=α1ε/n+1,Xτ∗,νn=α1ε, and for any t∈τ∗,τ∗ and ω∈Ω3, we have(71)α1εn+1<Xt,νn<α1ε.
Integrating both sides of (65) from τ∗ to τ∗ yields(72)∫τ∗τ∗σ12t2−a1t−1T∑j=1pln1+ξ1k+2b1tAtβ1ε+atBtβ2εbtdt≥−lnXτ∗,νnIΩ3+lnXτ∗,νnIΩ3=lnn+1PΩ3,which leads to τ∗−τ∗⟶+∞ as n⟶+∞.On the contrary, by the given condition, there exist two constantsτ0 and γ such that for all t≥τ0 and l>0, we have(73)η1l+t−∫ll+tb1uA1uε+auA1uA2uεblb2lA2lds>γ,a.s.Proof.
whereη1t=1/T∫0ta1s−σ12s/2ds+1/T∑j=1pln1+ξ2k.
For anyt∈τ∗,τ∗ and ω∈Ω3, we have(74)dlnY≤a2t−σ22t2+1T∑j=1pln1+ξ2k−b2lA2lYtA1uα1+σ2tdB2t.
The previous proof shows that there exists a positive integerk6>t0/T such that Y∗≤κ for any t>k6T+τ∗ and ω∈Ω3. Due to T∗−T∗⟶∞, then there exists a positive integer n0>0 such that τ∗>k6T+τ∗ for any n≥n0, and hence(75)Y∗≤A1ua2t−σ22t/2T+1/T∑j=1pln1+ξ2kα1b2lA2l≔κ˜,for any t∈k6T+τ∗,τ∗ and ω∈Ω3. Using (65) again, we have(76)lnXtn,νnIΩ3≥lnXk6T+τ∗,νn∗∗∗IΩ3+η1τ∗−∫k6T+τ∗τ∗b1uA1uα1+κ˜dsIΩ3,which leads to(77)α1εn+1≥α1εn+1exp1PΩ3η1τ∗−∫k6T+τ∗τ∗b1uA1uα1+κ˜dsIΩ3≥α1εn+1eγ.
Obviously it implies a contradiction, and our claim is obtained. Therefore(78)liminft⟶+∞Pω∈Ω∞ε:α1ε≤Xt,ω≤β1ε≥1−εholds.
In a similar way, we can derive thatliminft⟶+∞Pω∈Ω∞ε:α2ε≤Yt,ω≤β2ε≥1−ε. Finally, making use of Lemma 3 yielding the required assertion. The proof is completed.


## 6. Examples and Simulations

In this section, we give some examples and apply the Melstein method [31] to illustrate our theoretical results and reveal the effects of random disturbance and impulsive factors. For system (5), except some special mentions later, we always take(79)a1t=1+0.2sint2,bt=0.8+0.5cost2,σ1t=0.02+0.02sint2,ξ2k=0.01+0.01sint2,b1t=0.8+0.1cost2,a2t=0.6+0.1sint2,σ2t=0.01+0.01sint2,tk=0.1kT,k∈N,p=10.at=1+0.3cost2,b2t=0.3+0.05sint2,ξ1k=0.01+0.01sint2.For system (5), if n=1, then we get the stochastic Holling–Tanner system. By computation,(80)ζ≔a1t−σ12tT+1T∑j=1pln1+ξ1k=0.9994>0,η≔a2t−σ22tT+1T∑j=1pln1+ξ2k=0.5998>0.Then conditionH1 holds, and Theorem 1 implies that the solution of the stochastic Holling–Tanner system is a 4π-periodic Markovian process (Figure 1). Figures 1(a) and 1(b) are the time series graphs of x and y, respectively. Under the condition H1 holding, Figures 1(c)–1(f) are the phase graphs of periodic solution of deterministic system σit=0,ξik=0,i=1,2, impulsive system σit=0,ξik≠0,i=1,2, stochastic system σit≠0,ξik=0,i=1,2, and stochastic impulsive system σit≠0,ξik≠0,i=1,2, respectively. Similarly, the conditions of Case (iv) of Theorems 2 and 3 hold, and the system is permanent in the mean and stochastically permanent, see Figures 2(a), 2(b), and 3, respectively. On the contrary, let all parameters are as before except σ1t=1.5+0.02sint/2,σ2t=1.2+0.01sint/2 or ξ1k=−0.1+0.01sintk/2,ξ2k=−0.1+0.01sintk/2,tk=0.1k,k∈N, then an easy computing yields ζ^<0,η^<0; therefore, x and y are both extinct, illustrated in Figures 4(a) and 4(b), respectively. If σ1t=1+0.02sint/2,σ2t=0.01+0.01sint/2, then Theorem 2 indicates y is always permanent (Figure 5(a)). If σ1t=0.02+0.02sint/2,σ2t=1+0.01sint/2, then Theorem 2 implies that x is permanent, but y is extinct (Figure 5(b)). For system (5), if n=2, then we get a stochastic periodic Holling type IV predator-prey system. It is clear that Theorem 1 implies that the solution is a periodic Markovian process (Figure 6). Figures 6(a)–6(d) are the phase graphs of periodic solution of deterministic system σit=0,ξik=0,i=1,2, impulsive system σit=0,ξik≠0,i=1,2, stochastic system σit≠0,ξik=0,i=1,2, and stochastic impulsive system σit≠0,ξik≠0,i=1,2, respectively. Similarly, the conditions of Case (iv) of Theorems 2 and 3 hold, and the system is permanent in the mean and stochastically permanent, see Figures 7(a), 7(b), and 8, respectively.Figure 1
Periodic Markov process of (5) with n=1 and x0=y0=0.02. (a) The time series graph of x. (b) The time series graph of y. (c) The phase graph of deterministic system ξik=0,σit=0,i=1,2. (d) The phase graph of impulsive system σit=0,i=1,2. (e) The phase graph of stochastic system σit=0,i=1,2. (f) The phase graph of stochastic impulsive system with σ1t=0.02+0.02sint/2,σ2t=0.01+0.01sint/2,ξ1k=0.01+0.01sint/2,ξ2k=0.01+0.01sint/2,tk=0.1k,k∈N.
(a)(b)(c)(d)(e)(f)Figure 2
Permanence in the mean of (5) with n=1 and x0=y0=0.02. (a) The permanence in the mean of x. (b) The permanence in the mean of y.
(a)(b)Figure 3
Stochastic persistence of (5) with n=1 and x0=y0=0.02.Figure 4
The extinction of all species of system (5) with n=1 and x0=y0=0.02. (a) The extinction of x and y with σ1t=1.5+0.02sint/2,σ2t=1.2+0.01sint/2. (b) The extinction of x and y with ξ1k=−0.1+0.01sintk/2,ξ2k=−0.1+0.01sintk/2,tk=0.1k,k∈N.
(a)(b)Figure 5
The dynamics ofx and y of (5) with n=1 and x0=y0=0.02. (a) The permanence in the mean of y with σ1t=1+0.02sint/2,σ2t=0.01+0.01sint/2. (b) The permanence in the mean of x and extinction of y with σ1t=0.02+0.02sint/2,σ2t=1+0.01sint/2.
(a)(b)Figure 6
Periodic Markovian process of (5) with n=2 and x0=y0=0.02. (a) The phase graph of deterministic system ξik=0,σit=0,i=1,2. (b) The phase graph of impulsive system σit=0,i=1,2. (c) The phase graph of stochastic system σit=0,i=1,2. (d) The phase graph of stochastic impulsive system with σ1t=0.02+0.02sint/2,σ2t=0.01+0.01sint/2,ξ1k=0.01+0.01sint/2,ξ2k=0.01+0.01sint/2,tk=0.1k,k∈N.
(a)(b)(c)(d)Figure 7
Permanence in the mean of (5) with n=2 and x0=y0=0.02. (a) The permanence in the mean of x. (b) The permanence in the mean of y.
(a)(b)Figure 8
Stochastic persistence of (5) with n=2 and x0=y0=0.02.


## 7. Conclusions and Discussion

In this paper, we study a stochastic predator-prey system with impulsive effects and Holling type II or Holling type IV functional responses, which contains many models such as those in [16, 23]. Theorem 1 gives the sufficient conditions of the existence of periodic Markovian process. Theorem 2 represents the extinction and permanence in the mean of predator and prey species. Theorem 3 shows the stochastic permanence of this system. Finally, by writing Matlab codes, some simulations (Figures 1 and 8) are provided to verify the main results. Our numerical examples reveal that impulsive and stochastic factors bring much influence to the dynamics of this system.By comparison analysis, we give Remarks3 and 5 to show that our main results improve or generalize the corresponding results in [16, 23]. We apply stochastic analysis techniques instead of constructing some suitable functionals to study the stochastic permanence, which is less applied and relatively new in some sense. In the process of our analysis, Holling-type functional responses bring some difficulties, and we apply inequality techniques to overcome them. In view of too many kinds of functional responses, then how to deal with other functional response such as Beddington-DeAngelis type? Further, time delays often appear in biological models, and how to discuss the effect of time delays? All these are necessary and very interesting for us to study in the future.



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