# Long-Time Behaviours of a Stochastic Predator-Prey System with Holling-Type II Functional Response and Regime Switching

Authors: Weili Kong; Yuanfu Shao
Journal: Journal of Mathematics
 (2021)

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## Abstract

Considering the impacts of white noise, Holling-type II functional response, and regime switching, we formulate a stochastic predator-prey model in this paper. By constructing some suitable functionals, we establish the sufficient criteria of the stationary distribution and stochastic permanence. By numerical simulations, we illustrate the results and analyze the influence of regime switching on the dynamics.


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## Body

## 1. Introduction

Functional responses are very important in the predator-prey system, which is the amount of prey catch per predator per unit of time and has significant effect on the dynamical properties. Usually there are two kinds of functional response: prey dependent (such as Holling II and Holling IV, see [1–3]) and predator dependent (such as Hassell–Varley, Beddington–DeAngelis, and Crowley–Martin, see [4, 5]). Recently, a number of researchers have devoted their efforts to the predator-prey system with functional response and obtained some nice results [1–7].For the ecological system, the growth rate of population is inevitably affected by environmental white noise, which almost exists everywhere in real world [8–10]. May reveals that due to stochastic fluctuations in environmental conditions, all the natural parameters exhibit a certain amount of random perturbations, and hence, random disturbance is introduced in many mathematical models to reveal the effect of white noise [10–15]. Besides the white noise, the growth of species also suffers from fluctuating environments such as hurricanes and earthquakes, which is described by colorful noise in mathematical modelling [16–18]. The colorful noise may take several values and switch among different regimes of environments. The switching is memoryless, and the waiting time for the next switching follows an exponential distribution. That is, in mathematical sense, it is a Markovian process. Actually, when the environments fluctuate frequently, colorful noise may bring great influence to population dynamics and even change the permanence and extinction of species, so the impacts of colorful noise on population dynamics have attracted many researchers, see, e.g., [19–22].Motivated by above discussion, in this article, we formulate a stochastic model with Holling-type II functional response and colorful noise. By stochastic analysis, we aim to study the stability in distribution and stochastic permanence of the system.The rest of this paper is structured as follows. Section2 begins with our model and some notations. Section 3 is devoted to the stability in distribution of the above system. Section 4 focuses on the stochastic permanence. Some examples are given to illustrate our main results in Section 5. Finally, a brief conclusion and discussion are given to end the paper in Section 6.


## 2. The Model and Notations

Hsu and Huang [6] proposed the following predator-prey model with Holling-type II functional response:(1)dxt=xtr1−b1xt−c1yt1+xtdt+σ1xtdB1t,dyt=yt−r2−b2yt+c2xt1+xtdt+σ2ytdB2t,where r1>0 and −r2<0 represent the birth rate of prey and death rate of predator, respectively; b1 and b2 are intraspecific competition rate between species; c1>0 is the capture rate, and c2>0 is the conversion rate of food; σi2i=1,2 denotes the density of white noise; yt/1+xt is the Holling-type II functional response. 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). In view of the impact of regime switching (colorful noise) analyzed before, system (1) turns to the following:(2)dxt=xtr1αt−b1αtxt−c1αtyt1+xtdt+σ1αtxtdB1t,dyt=yt−r2αt−b2αtyt+c2αtxt1+xtdt+σ2αtytdB2t.The regime switchingαt is a Markovian chain in a finite state space S=1,2,…,N. The generator of αt is defined as χ=χijN×N with(3)Pαt+ϵ=j|αt=i=χijϵ+oϵ,i≠j,1+χiiϵ+oϵ,i=j,where ϵ>0,χij is the transition rate from the ith stage to the jth stage and χij≥0 if i≠j while χii=−∑i≠jχij. It is often assumed that every sample of αt is a right continuous step function and irreducible with a finite simple jumps in any finite subinterval of R+=0,∞. It obeys a unique stationary distribution π=π1,π2,…,πN satisfying πχ=0 and ∑k=1Nπk=1,πk>0,∀k∈S. The detailed switching mechanism of the hybrid system is referred to [19, 23].Letut=lnxt,vt=lnyt, then system (2) is equivalent to the following model:(4)dut=r1αt−σ12αt2−b1αteut−c1αtevt1+eutdt+σ1αtdB1t,dvt=−r2αt−σ22αt2−b2αtevt−c2αteut1+eutdt+σ2αtdB2t.For the later discuss, we introduce some notations about the Itô’s integral for stochastic differential equations with Markovian switching [19, 22]. Let(5)dxt=fxt,t,αtdt+gxt,t,αtdBt,where f:R2×R+×S⟶R2,g:R2×R+×S⟶R2×2 are measurable functions. Let V∈C2,1R2×R+×S,R2. Define the operator LV as follows:(6)LVx,t,k=Vtx,t,k+Vxx,t,kfx,t,k+12tracegTx,t,kVxxx,t,kgx,t,k+∑j=1NχkjVx,t,j,where Vtx,t,k=∂Vx,t,k/∂t, Vxx,t,k=∂Vx,t,k/∂x1,∂Vx,t,k/∂x2, and Vxxx,t,k=∂2Vx,t,k/∂xi∂xj2×2,i,j=1,2.The generalized Itô’s formula is defined as(7)dVx,t,k=LVx,t,kdt+Vxx,t,kgx,t,kdBt.Lemma 1 (see [21]).
If the following conditions hold.(i)
Fori≠j,χij>0.(ii)
For eachk∈S and any ς∈R2,ϱς2≤ςTgx,t,kgTx,t,kς≤ϱ−1ς2 holds with ϱ∈0,1 for all x∈R2.(iii)
There exists a bounded open subsetD⊂R2 with a regular boundary (i.e., smooth) such that, for any k∈S, there exists a nonnegative function V⋅,k:DC⟶R satisfying V⋅,k is twice continuously differentiable and for some ϵ>0,(8)LVx,k≤−ε,for anyx,k∈DC×S.Then, (5) is ergodic and positive recurrent; that is, there exists a unique stationary density μ⋅,⋅, for any Borel measurable function f⋅,⋅:R2×S⟶R with ∑k∈S∫R2fx,kμx,kdx<∞, we have(9)Plimt⟶∞1t∫0tfxs,αsds=∑k∈S∫R2fx,kμx,kdx=1.About the existence and uniqueness of positive solutions and the moment boundedness of (2), we have the following two lemmas. The proofs of them are very standard and are omitted here. Readers may refer to [3, 21].Lemma 2.
There is a unique positive solutionxt,yt,αt for system (2) on t≥0 with initial value x0,y0,α0∈R+2×S, and the solution will remain in R+2×S with probability 1.Lemma 3.
For any initial valuex0,y0,α0∈R+2×S and any p>0, there exists a constant Kp such that the solution xt,yt,αt for system (2) satisfying Ext+ytp≤Kp for all t≥0.For simplicity, we give some notations as follows:(10)Πik=rik−σi2k2,i=1,2,Πi=∑k∈SπkΠik,i=1,2,σ=maxk∈Sσ1k,σ2k,Πk=minΠ1k,Π2k,Π=∑k∈SπkΠk,f^=maxk∈Sfk,f⌣=maxk∈Sfk.


## 3. Stationary Distribution

In this section, we discuss the stationary distribution of (2).Theorem 1.
For any initial valuex0,y0,α0∈R+2×S and any k∈S, the solution xt,yt,αt of (2) is ergodic and has a unique stationary distribution in R+2×S if the following condition holds:(11)ζ=c⌣2r^1+b^1r⌣1−σ^122−r^2−σ^222>0.Proof.
According to the equivalent property of (2) and (4), we only need to prove it for (4). Define V1t,u,v=eu+pev2/2, where p=c⌣1/c^2, and then we have(12)LV1=eu+peveur1αt−σ12αt2−b1αteu−c1αtev1+eu+peu+pevev−r2αt−σ22αt2−b2αtev+c2αtev1+eu+σ12αt2e2u+eueu+pev+σ22αt2p2e2v+peveu+pev=eu+peveur1αt−σ12αt2−b1αte2u−c1αteu+v1+eu+eu+pev−pevr2αt+σ22αt2−pb2αte2v+pc2αteu+v1+eu+eu+pevσ12αt2eu+σ22αt2pev+σ12αt2e2u+σ22αt2p2e2v≤eu+peveur1αt−σ12αt2−b⌣1e2u−pevr2αt+σ22αt2−pb⌣2e2v+eu+pev2σ12αteu+pσ22αtev+σ12αt2e2u+σ22αt2p2e2v≤≤r1k−σ12k2eu+peveu−b⌣1e3u−p2b⌣2e3v+σ12αte2u+p2σ22αte2v+pσ12k+σ22k2eu+v≤−b⌣12e3u−p2b⌣22e3v+ϱ,where(13)ϱ=supu,v∈R2−b⌣12e3u−p2b⌣22e3v+r1k−σ12k2eu+peveu+σ12ke2u+p2σ22ke2v+pσ12k+σ22k2eu+v.
On the other hand,(14)L−v=r2αt+σ22αt2+b2αtev−c2αteu1+eu≤r2αt+σ22αt2+b^2ev−c⌣2eu1+eu≤r2k+σ22k2+b^2ev−c⌣2r1kr1k+b1k+c⌣2r1k+b1kr1k−b1keu1+eu.
Setq˜=c1^+b2^/r2ˇ, and similarly we have(15)Lln1+eu−u+q˜ev=eu1+eur1αt−σ12αt2−b1αteu−c1αtev1+eu+σ12αt2×eu1+eu2+−r1αt+σ12αt2+b1αteu+c1αtev1+eu+q˜ev−r2αt−σ22αt2−b2αtev+c2αteu1+eu+σ22αt2ev=eur1αt−b1αteu1+eu−c1αteu+v1+eu2−σ12αte2u21+eu2+−r1αt+σ12αt2+b1αteu+c1αtev1+eu+q˜−r2αtev−b2αte2v+c2αteu+v1+eu≤r1k−b1keu−r1k−b1keu1+eu−r1k+σ12k2+b1keu+c1kev+q˜−r2αtev+c2keu+v.
DefineV2=−v+c⌣2/r1αt+b1αtln1+eu−u+q˜ev, and then(16)LV2≤r2αt+σ22αt2+b^2ev−c⌣2r1αtr1αt+b1αt+c⌣2r1αt+b1αtr1αt−b1αteu1+eu+c⌣2r1αt+b1αtr1αt−b1αteu−r1αt−b1αteu1+eu−r1αt+σ12αt2+b1αteu+c1αtev+q˜−r2αtev+c2αteu+v≤r2k+σ22⋅k2−c⌣2r1kr1k+b1k+c⌣2r1k+b1kσ12k2+q˜c^2eu+v≤−c⌣2r1k+b1kr1k−σ12k2+r2k+σ22k2+c⌣2r^1+b^1q˜c^2eu+v≤−ζ+qeu+v,where q=c⌣2c^2/r^1+b^1q˜. Let V=V1+MV2, where M=2/ζmax2,supu,v∈R+2−b⌣1/4e3u−p2b⌣2/4e3v+ϱ. It is easy to observe that(17)LV=LV1+MLV2≤−Mζ+Mqeu+v−b⌣12e3u−p2b⌣22e3v+ϱ,and(18)Mζ4≥1.
Define a bounded closed set as follows:(19)U=u,v:u≤lnε−1,v≤lnε−1,u,v∈R2,where ε is a sufficiently small number, and then the set UC=R+2/U contains the following four domains:(20)Uε1=u,v∈R2:−∞≤u≤lnε,Uε2=u,v∈R2:−∞≤v≤lnε,Uε3=u,v∈R2:u≥lnε−1,Uε4=u,v∈R2:v≥lnε−1.
Takeε sufficiently small enough such that(21)0<ε<ζ4q,0<ε<p2b⌣24Mq,0<ε<b⌣14Mq,(22)−Mζ−b⌣14ε+η1≤−1,−ζ−p2b⌣24ε+η2≤−1,where η1,η2 are defined later. Next, we verify LVu,v≤−1 for all u,v∈UC=Uε1∪Uε2∪Uε3∪Uε4.Case 1.
Ifu,v∈Uε1, namely, −∞≤u≤lnε, then eu+v≤εev≤ε1+e3v. By (17), (18), and (21), we have(23)LV≤−Mζ+Mqεev−b⌣12e3u−p2b⌣22e3v+ϱ≤−Mζ+Mqε+Mqεe3v−b⌣12e3u−p2b⌣22e3v+ϱ=−Mζ4+−Mζ4+Mqε+−p2b⌣24+Mqεe3v−b⌣14e3u+−Mζ2−p2b⌣24e3v−b⌣14e3u+ϱ≤−Mζ4+−Mζ4+Mqε+−p2b⌣24+Mqεe3v−b⌣14e3u+−Mζ2+supu,v∈R+2−p2b⌣24e3v−b⌣14e3u+ϱ≤−1.Case 2.
Ifu,v∈Uε2, namely, −∞≤v≤lnε, then eu+v≤εeu≤ε1+e3u, and similarly we have(24)LV≤−Mζ+Mqεeu−b⌣12e3u−p2b⌣22e3v+ϱ≤−Mζ4+−Mζ4+Mqε+−b⌣14+Mqεe3u−p2b⌣24e3v+−Mζ2+supu,v∈R+2−b⌣14e3u−p2b⌣24e3v+ϱ≤−1.Case 3.
Ifu,v∈Uε3, then we derive from (17) and (22) that(25)LV≤−Mζ+Mqεeu−b⌣12e3u−p2b⌣22e3v+ϱ≤−Mζ−b⌣14ε+η1≤−1,where η1=supu,v∈R+2−b⌣1/4e3u−p2b⌣2/2e3v+Mqeu+v+ϱ.Case 4.
Ifu,v∈Uε4, similarly, from (17) and (22) we have(26)LV≤−Mζ+Mqεeu−b⌣12e3u−p2b⌣22e3v+ϱ≤−Mζ−p2b⌣24ε+η2≤−1,where η2=supu,v∈R+2−b⌣1/2e3u−p2b⌣2/4e3v+Mqeu+v+ϱ.
Consequently, we deduce thatLVu,v≤−1 on all u,v∈UC. Obviously, the other condition of Lemma 1 holds too, so we conclude from Lemma 1 that system (4) is ergodic and has a unique stationary distribution in R+2×S; that is, system (2) is ergodic and has a unique stationary distribution in R+2×S. This completes the proof.
For (2), if the state Markovian chain αt takes value in space S=1, namely, there is no switching, then (2) turns to the following subsystem:(27)dxt=xtr1−b1xt−c1yt1+xtdt+σ1xtdB1t,dyt=yt−r2−b2yt+c2xt1+xtdt+σ2ytdB2t.
For (27), from Theorem 1, we can easily obtain the following conclusion.Corollary 1.
For any initial valuex0,y0∈R+2, the solution xt,yt of (27) is ergodic and has a unique stationary distribution in R+2 if the following condition holds:(28)ζ˜=c2r1+b1r1−σ122−r2−σ222>0.Remark 1.
It is clear that, for any positive integerk,c2k/r1k+b1kr1k−σ12k/2−r2k−σ22k/2>c⌣2/r^1+b^1r⌣1−σ^12/2−r^2−σ^22/2. That is, Theorem 1 shows that switching system (2) has stationary distribution only under the condition that every subsystem of (2) has stationary distribution. If there exists no switching, Corollary 1 gives the sufficient condition of stationary distribution of (27), which is accordant with Theorem 1 of [3].


## 4. Stochastic Permanence

For (2), if we consider the birth rate instead of the death rate of predator, then (2) turns to the following model:(29)dxt=xtr1αt−b1αtxt−c1αtyt1+xtdt+σ1αtxtdB1t,dyt=ytr2αt−b2αtyt+c2αtxt1+xtdt+σ2αtytdB2t,where r2⋅>0 is the birth rate of species yt and other parameters are the same as before. Now, we consider the stochastic permanence of (29).Definition 1.
(see [16])System (29) is stochastically permanent if for every ε∈0,1 and any k∈S, there is a pair of constants ℳ>0 and N>0 such that for any initial data x0,y0,α0∈R+2×S, the solution Xt=xt,yt of (29) has the property that(30)liminft⟶∞PXt≥ℳ≥1−ε,limsupt⟶∞PXt≤N≥1−ε,where P represents the probability of events.Assumption 1.
For somek∈S,χkj>0,k≠j.Lemma 4.
Under Assumption1, if Π>0, then there exists δ0>0 such that for any 0<δ<δ0,Gδ=diagυ1δ,υ2δ,…,υNδ−χ is a nonsingular M-matrix, where υkδ=δΠk−1/2δ2σ2.Remark 2.
The proof is rather standard. Readers may refer to the details in [24] or [21, 23].Theorem 2.
For any initial valuex0,y0,α0∈R+2×S, system (29) is stochastically permanent under conditions of Lemma 3.Proof.
The proof is motivated by [22]. Let G be a matrix or vector, and denote by G≫0 all the elements of G are positive. Under the hypotheses, Lemma 2 shows Gδ is a nonsingular M-matrix, and then by M-matrix theory (see Theorem 2.1 [22]), there exists ρ=ρ1,ρ2,…,ρNT≫0 such that Gδρ≫0, that is, ρkυkδ−∑j=1Nχkjρj>0,k∈S. So, there exists a constant τ>0 such that(31)ρkυkδ−∑j=1Nχkjρj−τρk>0,k∈S.
Define functionalV=ρk1+V˜δ, where V˜=1/x+y, then for above τ>0, we compute τV+LV as follows:(32)τV+LV=τρk1+V˜δ+∑j=1Nχkjρj1+V˜δ+ρkδ1+V˜δ−1−V˜2×r1αtx+r2αty−b1αtx2−b2αty2+c2αt−c1αt1+xxy+ρkδ1+V˜δ−1V˜σ1αtx+σ2αtyx+y2+12ρkδδ−11+V˜δ−2−V˜2σ1αtx−σ2αty2=τρk1+V˜δ+∑j=1Nχkjρj1+V˜δ−ρkδ1+V˜δ−1V˜r1αtx+r2αtyx+y+ρkδ1+V˜δ−1b1αtx2+b2αty2−c2αt−c1αt/1+xxyx+y2+ρkδ1+V˜δ−1V˜σ1αtx+σ2αtyx+y2+12ρkδδ−11+V˜δ−2V˜2σ1αtx+σ2αtyx+y2=V˜δfV˜δ+gV˜,where limV⟶∞gV˜/V˜δ=0, and(33)fV˜δ=τρk+∑j=1Nχkjρj−ρkδr1αtx+r2αtyx+y+ρkδσ1αtx+σ2αtyx+y2+12ρkδδ−1σ1αtx+σ2αtyx+y2≤τρk+∑j=1Nχkjρj−ρkδΠ1kxx+y−ρkδΠ2kyx+y+ρkδ2σ22≤τρk+∑j=1Nχkjρj−ρkΠkδ+ρkδ2σ22=τρk+∑j=1Nχkjρj−ρkυkδ<0.
By Ito’s formula, we have(34)LeτtVt=eτtτV+LV≤eτtτρk−ρkυkδ+∑j=1NχkjρjV˜δ+oV˜δ≤Yδeτt,where Yδ=maxk∈SsupV˜∈R+2τρk−ρkυkδ+∑j=1NχkjρjV˜δ+oV˜δ,1. Integrating deτtVt from 0 to t and taking expectation give(35)Eρkeτt1+V˜δ−ρk1+V˜0δ≤Yδkeτt−1.
Hence,(36)E1+V˜δ≤Yδkmink∈Sρk+e−τt1+1x0+y0δ.
LetHδ=Yδ/kmink∈Sρk, then(37)limsupt⟶∞Ext+yt−δ≤Hδ.
SinceX=x2+y21/2, then we deduce that EX−δ≤2δ/2Hδ, and hence,(38)PX<22εHδ1/δ=ℳ≤EX−δℳ−δ≤2δ/2Hδ2δ/2ε/Hδ−1=ε.
Therefore,PX≥ℳ≥1−ε holds. By Lemma 3, using Chebyshev’s inequality again, it is clear that PX≤N≥1−ε for some constant N. Therefore, (29) is stochastically permanent by Definition 1. The proof is completed.
Obviously, if there is no switching, we can similarly obtain the following corollary.Corollary 2.
For any initial valuex0,y0∈R+2, the subsystem of (29) is stochastically permanent if ri−σi2/2>0,i=1,2.Remark 3.
Theorem2 reveals that when some subsystems of (2) are no stochastic permanent, if we give a suitable switching, then switching system (2) may be stochastic permanent, which implies the switching has very important influence to the dynamics of (2). By simulation, we can verify it directly, see Figure 1.Figure 1
The effect of switchingπ on the dynamics of (39) and (40). (a) The nonstochastic permanence of (39) and (40) with π=0.1,0.9. (b) The stochastic permanence of (39) and (40) with π=0.1,0.9.
(a)(b)


## 5. Examples and Simulations

In this section, some examples are given to illustrate our theoretical results and reveal the effects of regime switching and stochastic factors [25]. For simplicity, we assume that the continuous-time discrete state Markovian chain αt takes value in the space S=1,2, then system (2) reduces to the following subsystems:(39)dxt=xtr11−b11xt−c11yt1+xtdt+σ11xtdB1t,dyt=yt−r21−b21yt+c21xt1+xtdt+σ21ytdB2t,(40)dxt=xtr12−b12xt−c12yt1+xtdt+σ12xtdB1t,dyt=yt−r22−b22yt+c22xt1+xtdt+σ22ytdB2t.We letr11=0.8,b11=0.2,c11=0.8,σ11=0.3;r21=0.5,b21=0.8,c21=1,σ21=0.4; r12=0.7,b12=0.3,c12=0.8,σ12=0.2;andr22=0.4,b22=0.3,c22=1,σ21=0.4. By Corollary 1, we know that (39) and (40) both have stationary distribution, see Figure 2.Figure 2
(a) The stationary distribution of system (39). (b) The stationary distribution of system (40).
(a)(b)Suppose the distribution ofαt is π=0.4,0.6 (see Figure 3). It is easy to verify that ζ>0. Theorem 1 implies that (2) has stationary distribution, see Figure 4.Figure 3
Distribution of the switchingαt. (a) The time series of αt. (b) The histogram of αt.
(a)(b)Figure 4
The stationary distribution of switching systems (39) and (40) with π=0.4,0.6. (a) The time series graph of (xt and yt). (b) The density graph of (xt and yt).
(a)(b)Ifr21=−0.2,r22=−0.3,andπ=0.4,0.6, then Theorem 2 shows that (39) and (40) are stochastic permanence, see Figure 5.Figure 5
The stochastic permanence of (39) and (40) with π=0.4,0.6.Ifσ12=1.2,σ22=1, and π=0.1,0.9, then the switching system is not stochastic permanence, but if we take π=0.9,0.1, then the system is stochastic permanence, which implies the switching is very important to make (39) and (40) be permanent, see Figure 1.


## 6. Conclusions and Discussion

In this paper, we study a stochastic predator-prey system with regime switching and Holling-type II functional responses. Theorems1 and 2 give the sufficient conditions of stationary distribution and the stochastic permanence of (2). Finally, some examples are given to verify the main results. Our numerical examples reveal that switching and random factors bring much influence to the dynamics of this system.By comparison analysis, we give Remarks1 and 2 to show that our main results improve or generalize the corresponding results in [3]. The main method applied in this paper is by constructing some suitable functionals instead of stochastic analysis techniques to study the stationary distribution, which is less applied for switching system. In the process of our analysis, Holling-type II functional response brings some difficulties and we apply inequality techniques to overcome them.As Arditi and Ginzburg [23] pointed out that the predator-dependent functional response can provide better description in some cases, then how to deal with predator-dependent functional response such as Beddington–DeAngelis type? Furthermore, studies have shown that the mental state of the adolescent prey can be mediated by fear induced from predators and the alternation causes deleterious outcomes on their adult’s survival [24] and then how fear will impact our system? All these are interesting for us to study in the future.



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