# Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control

Authors: Jinlei Liu; Wencai Zhao
Journal: Complexity
 (2019)

---

## Abstract

In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.


---

## Body

## 1. Introduction

In nature, time delays exist in many ecosystems [1–5]. For example, maturity stage is a common phenomenon in biological population, and many diseases have a long incubation period. The mathematical model describing this phenomenon with time delay is called the delay differential equation. In 1999, Saito et al. [6] studied a Lotka–Volterra predator-prey model with discrete delays, which can be defined as follows:(1)dxtdt=xtr1+axt+αxt−τ1−βyt−τ2,dytdt=ytr2+ayt+βxt−τ1+αyt−τ2,where xt and yt stand for the population density of prey and predator at time t, respectively. rii=1,2 represent the intrinsic growth rate of corresponding population. τ1 and τ2 are discrete time delays. aa<0, α and β are constants.Due to the environmental changes and increased human activities, many rare species are at risk of extinction. How to protect endangered species of floras and faunas and maintain the diversity of ecosystems is an important issue that needs to be solved urgently. In the process of marine fishery production, overfishing often results in the exhaustion of fishery resources. It is rewarding for humans to develop and utilize the ecological system of the population rationally, which also contributes to the sustainability of the system [7–14]. In 2003, Gopalsamy and Weng [15] studied the following population competition model with feedback control:(2)dx1tdt=x1tb1−a11x1t−a12x2t−α1u1t−τ,dx2tdt=x2tb2−a21x1t−a22x2t−α2u2t−τ,du1tdt=−η1u1t+a1x1t−τ,du2tdt=−η2u2t+a2x2t−τ,where u1t and u2t are the feedback control variables, bi>0, aij>0, αi>0, ηi>0, and ai>0i,j=1,2. They also discussed the existence of positive equilibrium point and global attraction of the model. In 2013, Li et al. [16] introduced feedback control variables into the two-species competition system and discussed the extinction and global attraction of equilibrium points. They found that if the two-species competition model is globally stable, the system retains the stable property after adding feedback controls and the position of equilibrium point is changed. If the two-species competition model is extinct, by choosing the suitable values of feedback control variables, they can make extinct species become globally stable, or still keep the property of extinction. In 2017, Shi et al. [17] discussed a Lotka–Volterra predator-prey model with discrete delays and feedback control as follows:(3)dx1tdt=x1tr1−a11x1t−a12x2t−τ1,dx2tdt=x2tr2+a21x1t−τ2−a22x2t−cut,dutdt=−eut+fx2t,where ut is the feedback control variable, e and f denote the feedback control coefficients, aiii=1,2 denote the intraspecific competition rates, aiji≠j,i,j=1,2 stand for the capturing rates of the prey and predator populations, τ1 is the time of catching prey, and τ2 is maturation delay of predator. Shi et al. [17] show that(i)
The solutionx1t,x2t,ut of system (3) is ultimately bounded(ii)
When the conditionsr1/r2>a12/a22+cf/e,a11/a21>a12/a22 are established, system (3) has a unique globally asymptotically stable positive equilibrium point x1∗,x2∗,u∗, where x1∗=er1a22−r2a12+r1cf/ea11a22+a12a21+cfa11, x2∗=er2a11+r1a21/ea11a22+a12a21+cfa11, and u∗=f/ex2∗In fact, in nature, ecosystems are inevitably affected by various environmental noises [18–28]. Mathematical models with environmental disturbances can usually be described by stochastic differential equations. Stochastic noise can generally be divided into two categories: one type is a small number of strong interference, usually called colored noise or electrical noise, which can be described by the Markov chain [29–31]; the other type is the sum of many small, independent random interference, called white noise, which is usually represented by Brownian motion [32–35]. Assume that the population’s intrinsic growth rate ri is disturbed by white noise:(4)ri⟶ri+σiB˙it,i=1,2.Then, model (3) is transformed into(5)dx1t=x1tr1−a11x1t−a12x2t−τ1dt+σ1x1tdB1t,dx2t=x2tr2+a21x1t−τ2−a22x2t−cutdt+σ2x2tdB2t,dut=−eut+fx2tdt,and satisfies the initial conditions(6)xiθ=ϕiθ,uθ=ψθ,θ∈−τ,0,i=1,2,where Biti=1,2 denote the independent standard Brownian motion, σi2 denote the intensity of white noise, τ=maxτ1,τ2, ϕi0>0,ψ0>0, and ϕiθ and ψθ are both nonnegative continuous functions on −τ,0.Due to the interference of stochastic noise, system (5) does not possess an equilibrium point. An interesting question is: Does model (5) still have stability? What is the influence of white noise on system (5)? This paper mainly studies the dynamical properties of stochastic systems (5) also satisfying initial conditions (6). The second part proves the suitability of the system. The third part discusses the oscillation of the stochastic model near the positive equilibrium point x1∗,x2∗,u∗ of the corresponding deterministic model. The fourth and fifth parts, respectively, obtain the conditions for the persistence and extinction of the stochastic system. Finally, the correctness of the theoretical derivation is verified by numerical simulation.


## 2. Existence and Uniqueness of Global Positive Solutions

The stochastic differential equation is expressed as(7)dxt=fxt,tdt+gxt,tdBt,x∈Rn.If the Lyapunov functionVx,t∈C2,1Rn×R+,R, the stochastic differential equation of Vx,t along system (7) is defined as [36](8)dVxt,t=Vtxt,t+Vxxt,tfxt,t+12trgTxt,tVxxxt,tgxt,tdt+Vxxt,tgxt,tdBta.s,where LV=Vtx,t+Vxx,tfx,t+1/2trgTx,tVxxx,tgx,t represent diffusion operator.Theorem 1.
For any given initial condition (6), model (5) has a unique global positive solution x1t,x2t,ut, and the solution will remain in ℝ+3 with probability one.Proof.
Since the coefficients of system (5) satisfy the locally Lipschitz condition, for any given initial condition (6), model (5) has a unique local positive solution x1t,x2t,ut in interval t∈0,τe, where τe is the explosion time.
To prove that this solution is global, we only need to proveτe=∞ a.s. Let k0>0 be a sufficiently large constant for any initial value x10,x20, and u0 lying within the internal 1/k0,k0. For each integer k≥k0, define the stopping time(9)τk=inft∈0,τe:minx1t,x2t,ut≤1kormaxx1t,x2t,ut≥k.
Obviously,τk is increasing as k⟶∞. Let τ∞=limk⟶∞τk; therefore, τ∞≤τe a.s. Now, we need to verify τ∞=∞ a.s. Otherwise, there are two constants T>0 and ϵ∈0,1 such that Pτ∞≤T>ϵ. So, there is a positive integer k1≥k0, such that(10)Pτk≤T>ϵ,k≥k1.
Define aC2−functionV: ℝ+3⟶ℝ+ by(11)Vx1,x2,u=x1−1−lnx1+a11a22a212x2−1−lnx2+12a11∫t−τ2tx12sds+a12∫t−τ1tx2sds+a11a22c2fa212u2+u−1−lnu=V1x1,x2+V2u,where(12)V1x1,x2=x1−1−lnx1+a11a22a212x2−1−lnx2+12a11∫t−τ2tx12sds+a12∫t−τ1tx2sds,V2u=a11a22c2fa212u2+u−1−lnu.
The nonnegativity of this function can be obtained from(13)x−1−lnx≥0,x>0.
Applying Itô’s formula yields(14)dVx1,x2,u=LVx1,x2,udt+σ1x1−1dB1t+a11a22a212σ2x2−1dB2t,where(15)LV=LV1+LV2,LV1=x1t−1r1−a11x1t−a12x2t−τ1+12σ12+a11a22a212x2t−1r2+a21x1t−τ2−a22x2t−cut+a11a222a212σ22+12a11x12t−12a11x12t−τ2+a12x2t−a12x2t−τ1=r1+a11x1t−12a11x12t−a12x1tx2t−τ1−r1+12σ12+a11a22a212r2+a22x2t−a22x22t−cx2tut+cut+a11a22a21x2t−1x1t−τ2+a11a22a212−r2+12σ22−12a11x12t−τ2+a12x2t≤r1+a11x1t−12a11x12t+a11a22a212r2x2t−12a22x22t−cx2tut+cut+a12x2t−r1+12σ12+a11a22a212−r2+12σ22+12a22,LV2=a11a22cfa212ut−eut+fx2t+1−1ut−eut+fx2t≤−a11a22cefa212u2t+a11a22ca212utx2t−eut+fx2t+e.
Therefore,(16)LV=LV1+LV2≤r1+a11x1t−12a11x12t+a11a22a212r2x2t−12a22x22t−cx2tut+cut+a12x2t−r1+12σ12+a11a22a212−r2+12σ22+12a22−a11a22cefa212u2t+a11a22ca212utx2t−eut+fx2t+e≤max−12a11x12t+r1+a11x1t+a11a22a212max−12a22x22t+r2+a212f+a12a11a22x2t+a11a22a212max−cefu2t+c−a212ea11a22ut+a11a22a21212a22+12σ22−r2+12σ12+e−r1≤K,where K is a positive constant. So, we get(17)dV≤Kdt+σ1x1−1dB1t+a11a22a212σ2x2−1dB2t.
Integrating (17) from 0 to τk∧T and taking expectation on both sides, we have(18)EVx1τk∧T,x2τk∧T,uτk∧T≤EVx10,x20,u0+KT.
SetΩk=τk≤T, and from inequality (10), we have PΩk≥ϵ. Note that, for every ω∈Ωk, there is at least one of x1τk,ω,x2τk,ω, or uτk,ω equaling either k or 1/k, and then, we have(19)Vx1τk∧T,x2τk∧T,uτk∧T≥k−1−lnk∧1k−1−ln1k.
It can be obtained by (18)(20)EVx10,x20,u0+KT≥E1ΩkωVx1τk,ω,x2τk,ω,uτk,ω≥εk−1−lnk∧1k−1+lnk,where 1Ωk is the indicator function of Ωk, and letting k⟶∞ yields(21)∞>EVx10,x20,u0+KT=∞.
This is a contradiction; we must haveτ∞=∞, and we have completed the proof.


## 3. Asymptotic Property

Due to the interference of white noise, the solution of system (5) will have stochastic oscillation. Next, we discuss the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model. In order to study the problem conveniently, the hypothesis is(22)A1:r1r2>a12a22+cf/e,a11a21>a12a22.Theorem 2.
For any given initial condition (6), if hypothesis A1 is established, the solution x1t,x2t,ut of system (5) has the property that(23)limsupt⟶+∞1tE∫0tx1θ−x1∗2+x2θ−x2∗2+uθ−u∗2dθ≤nm,where(24)m=mina11fa11a22−a12a21a11a22+a12a21,a12fa11a22−a12a212a11a21,a12cea21,n=fx1∗σ122+a12fx2∗σ222a21,where x1∗,x2∗,u∗ is the positive equilibrium point of the corresponding deterministic model (3).Proof.
Define the function(25)Vx1,x2,u=fx1−x1∗−x1∗lnx1x1∗+a12fa21x2−x2∗−x2∗lnx2x2∗+a12c2a21u−u∗2+a12f2ω2∫t−τ2tx1s−x1∗2ds+a12fω12∫t−τ1tx2s−x2∗2ds,where ω1 and ω2 are positive constants. Define(26)V1=x1−x1∗−x1∗lnx1x1∗,V2=x2−x2∗−x2∗lnx2x2∗,V3=u−u∗2,V4=a12f2ω2∫t−τ2tx1s−x1∗2ds+a12fω12∫t−τ1tx2s−x2∗2ds.
By Itô’s formula, we obtain(27)dV1=LV1dt+x1−x1∗σ1dB1t,where(28)LV1=x1t−x1∗r1−a11x1t−a12x2t−τ1+12x1∗σ12=x1t−x1∗−a11x1t−x1∗−a12x2t−τ1−x2∗+12x1∗σ12≤−a11x1t−x1∗2+a122ω1x1t−x1∗2+a12ω12x2t−τ1−x2∗2+12x1∗σ12.
Similarly,(29)dV2=LV2dt+x2−x2∗σ2dB2t,where(30)LV2=x2t−x2∗r2+a21x1t−τ2−a22x2t−cut+12x2∗σ22=x2t−x2∗a21x1t−τ2−x1∗−a22x2t−x2∗−cut−u∗+12x2∗σ22≤a212ω2x1t−τ2−x1∗2+a21ω22x2t−x2∗2−a22x2t−x2∗2−cx2t−x2∗ut−u∗+12x2∗σ22.
In the same way,(31)dV3=2ut−u∗−eut+fx2tdt=−2eut−u∗2+2fut−u∗x2t−x2∗dt,dV4=a12f2ω2x1t−x1∗2−a12f2ω2x1t−τ2−x1∗2+a12fω12x2t−x2∗2−a12fω12x2t−τ1−x2∗2dt.
Therefore, we have(32)LV=fLV1+a12fa21LV2+a12c2a21dV3dt+dV4dt≤−a11fx1t−x1∗2+a12f2ω1x1t−x1∗2+a12fω12x2t−τ1−x2∗2+12fx1∗σ12+a12f2ω2x1t−τ2−x1∗2+a12fω22x2t−x2∗2−a12a22fa21x2t−x2∗2−a12cfa21x2t−x2∗ut−u∗+a12fx2∗2a21σ22−a12cea21ut−u∗2+a12cfa21ut−u∗x2t−x2∗+a12f2ω2x1t−x1∗2−a12f2ω2x1t−τ2−x1∗2+a12fω12x2t−x2∗2−a12fω12x2t−τ1−x2∗2=−fa11−a122ω1−a122ω2x1t−x1∗2−fa12a22a21−a12ω22−a12ω12x2t−x2∗2−a12cea21ut−u∗2+fx1∗2σ12+a12fx2∗2a21σ22.
Letω1=ω2=a11a22+a12a21/2a11a21, we obtain(33)LV≤−a11fa11a22−a12a21a11a22+a12a21x1t−x1∗2−a12fa11a22−a12a212a11a21x2t−x2∗2−a12cea21ut−u∗2+fx1∗2σ12+a12fx2∗2a21σ22.
Therefore,(34)dVx1,x2,u=LVdt+fx1−x1∗σ1dB1t+a12fa21x2−x2∗σ2dB2t.
Integrate both sides of (18) from 0 to t and take the expectation, and then we get(35)EVt−EV0≤−E∫0ta11fa11a22−a12a21a11a22+a12a21x1θ−x1∗2dθ−E∫0ta12fa11a22−a12a212a11a21x2θ−x2∗2dθ−E∫0ta12cea21uθ−u∗2dθ+fx1∗σ12t2+a12fx2∗σ22t2a21.
Divide both sides byt and take the limit superior, and then we have(36)lim supt⟶+∞1tE∫0ta11fa11a22−a12a21a11a22+a12a21x1θ−x1∗2dθ+lim supt⟶+∞1tE∫0ta12fa11a22−a12a212a11a21x2θ−x2∗2dθ+lim supt⟶+∞1tE∫0ta12cea21uθ−u∗2dθ≤fx1∗σ122+a12fx2∗σ222a21.
Obviously,(37)lim supt⟶+∞1tE∫0tx1θ−x1∗2+x2θ−x2∗2+uθ−u∗2dθ≤nm,where(38)m=mina11fa11a22−a12a21a11a22+a12a21,a12fa11a22−a12a212a11a21,a12cea21,n=fx1∗σ122+a12fx2∗σ222a21.
We have completed the proof.
Theorem 2 shows that if the conditionA1 holds, the solution oscillates around the equilibrium point x∗, and the amplitude of oscillation is positively correlated with the intensity σ12andσ22 of environmental noise. In particular, if σ12=σ22=0, the influence of environmental noise is not taken into account:(39)x1t⟶x1∗,x2t⟶x2∗,ut⟶u∗fort⟶+∞.
The equilibrium pointx∗ is globally asymptotically stable. This is the conclusion of reference [17].


## 4. Persistence

In nature, whether ecosystems can survive or not is our main concern. Before discussing the persistence of stochastic system, we give the following assumption:(40)A2:μ=maxσ1,σ2<minx1∗mn0,x2∗mn0,u∗mn0,n0=fx1∗2+a12fx2∗2a21.Theorem 3.
For any given initial condition (6), if assumptions A1andA2 hold at the same time, the solution x1t,x2t,ut of system (5) is persistent that(41)lim inft⟶∞1tE∫0tx1θdθ>0,lim inft⟶∞1tE∫0tx2θdθ>0,lim inft⟶∞1tE∫0tuθdθ>0.Proof.
According to (37), we have(42)lim supt⟶+∞1tE∫0tx1θ−x1∗2dθ≤nm,lim supt⟶+∞1tE∫0tx2θ−x2∗2dθ≤nm,lim supt⟶+∞1tE∫0tuθ−u∗2dθ≤nm.
As we know,x1t≥0 and x1∗>0, from 2x1tx1∗≥x1∗2−x1t−x1∗2, one can get(43)x1t≥x1∗2−x1t−x1∗22x1∗.
By the conditionμ<x1∗m/n0, we have(44)lim inft⟶∞1tE∫0tx1θdθ≥x1∗2−lim supt⟶∞1tE∫0tx1θ−x1∗22x1∗dθ≥x1∗2−n2mx1∗≥x1∗2−μ2n02mx1∗>0.
Similarly, whenμ<minx2∗m/n0,u∗m/n0,(45)lim inft⟶∞1tE∫0tx2θdθ>0lim inft⟶∞1tE∫0tuθdθ>0.


## 5. Extinction

Define(46)Δ=a21r1−12σ12+a11r2−12σ22,Δ1=ea11a22+a12a21−a11cf,Δ2=a12Δea11a22+a12a21−a11cf−r1−12σ12ea11a22+a12a212.For the extinction of system (5), we have the following conclusions.Theorem 4.
For any given initial condition (6), the solution x1t,x2t,ut of system (5) has the following properties:(i)
whenr1−1/2σ12<0andr2−1/2σ22<0, the population is extinct(ii)
whenr1−1/2σ12>0, and Δ<0, population x1 is persistent and x2 and u are extinct(iii)
whenr1−1/2σ12>0,Δ>0, Δ1>0, and Δ2>0, population x1 is extinct and x2 and u are persistentBefore proving Theorem 4, consider the following auxiliary system [37]:(47)dz1t=z1tr1−a11z1tdt+σ1z1tdB1t,dz2t=z2tr2+a21z1t−τ2−a22z2tdt+σ2z2tdB2t,and it satisfies the initial condition: ϕθ=ϕ1θ,ϕ2θ∈C−τ,0,R+2.Lemma 1.
Ifr1−1/2σ12>0, the solution zt of system (47) has the following properties:(i)
limt⟶+∞t−1∫0tz1sds=r1−1/2σ12/a11(ii)
ifΔ<0,limt⟶+∞z2t=0(iii)
ifΔ>0,limt⟶+∞t−1∫0tz2sds=Δ/a11a22,a.sProof.
By Itô’s formula, we obtain(48)lnz1t−lnz10=r1−12σ12t−a11∫0tz1sds+σ1B1t,(49)lnz2t−lnz20=r2−12σ22t+a21∫0tz1s−τ2ds−a22∫0tz2sds+σ2B2t=r2−12σ22t+a21∫0tz1sds−a21∫t−τ2tz1sds−∫−τ20z1sds−a22∫0tz2sds+σ2B2t.
Dividing both sides of (48) and (49) by t, we have(50)t−1lnz1tz10=r1−12σ12−a11t−1∫0tz1sds+t−1σ1B1t,(51)t−1lnz2tz20+t−1a21∫t−τ2tz1sds−∫−τ20z1sds=r2−12σ22+a21t−1∫0tz1sds−a22t−1∫0tz2sds+t−1σ2B2t.
By using Lemma 2 of [37], from equation (48), it follows that(52)lim supt⟶+∞t−1∫0tz1sds≤r1−1/2σ12a11,lim inft⟶+∞t−1∫0tz1sds≥r1−1/2σ12a11,and therefore,(53)limt⟶+∞t−1∫0tz1sds=r1−1/2σ12a11.
Substitute (53) into (50), and from limt⟶+∞t−1Bit=0i=1,2, we get(54)limt⟶+∞t−1lnz1t=0.
On the other hand, computing (50) × a21 + (51) × a11, we have(55)a21t−1lnz1tz10+a11t−1lnz2tz20+t−1a11a21∫t−τ2tz1sds−∫−τ20z1sds=a21r1−12σ12+a11r2−12σ22−a11a22t−1∫0tz2sds+t−1a21σ1B1t+a11σ2B2t.
By Lemma 2 in literature [37], when Δ<0,(56)limt⟶+∞z2t=0.
IfΔ>0,(57)limt⟶+∞t−1∫0tz2sds=Δa11a22a.s.
Proof of lemma is completed.This is where we prove Theorem 4.Proof.
By using Itô’s formula for system (5), we have(58)lnx1t−lnx10=r1−12σ12t−a11∫0tx1sds−a12∫0tx2s−τ1ds+σ1B1t=r1−12σ12t−a11∫0tx1sds−a12∫0tx2sds+a12∫t−τ1tx2sds−∫−τ10x2sds+σ1B1t,(59)lnx2t−lnx20=r2−12σ22t+a21∫0tx1s−τ2ds−a22∫0tx2sds−c∫0tusds+σ2B2t=r2−12σ22t+a21∫0tx1sds−a21∫t−τ2tx1sds−∫−τ20x1sds−a22∫0tx2sds−c∫0tusds+σ2B2t.
We first prove (i): from equation (58), we can get(60)lnx1t−lnx10≤r1−12σ12t−a11∫0tx1sds+σ1B1t.
By the conditionr1−1/2σ12<0 and Lemma 2 in literature [37],(61)limt⟶+∞x1t=0a.s.
From (57), (61), and the condition r2−1/2σ22<0, we have(62)limt⟶+∞x2t=0a.s.
Further, from the third equation of model (5), we can get(63)limt⟶+∞ut=0a.s.
Secondly, it proves (ii): comparing model (5) with auxiliary system (37), one can get x1t≤z1tandx2t≤z2t. By Lemma 1, when r1−1/2σ12>0andΔ<0, limt⟶+∞z2t=0a.s.; therefore, limt⟶+∞x2t=0a.s. So, we have limt⟶+∞ut=0. Then, the limit system of model (5) is as follows:(64)dx1t=x1tr1−a11x1tdt+σ1x1tdB1t.
From Lemma 1, we can conclude that(65)limt⟶+∞t−1∫0tx1sds=r1−1/2σ12a11a.s.
Therefore, populationx1 is persistent and x2 and u are extinct.
Finally, we prove (iii): by Lemma 1, whenr1−1/2σ12>0andΔ>0, limt⟶+∞t−1∫0tz2sds=Δ/a11a22,a.s. And, by (54),(66)limsupt⟶+∞t−1lnx1t≤limt⟶+∞t−1lnz1t=0.
Computing (58) × a21 + (59) × a11,(67)a21t−1lnx1tx10+a11t−1lnx2tx20+t−1a11a21∫t−τ2tx1sds−∫−τ20x1sds=t−1a12a21∫t−τ1tx2sds−∫−τ10x2sds+a21r1−12σ12+a11r2−12σ22−a11ct−1∫0tusds−a11a22+a12a21t−1∫0tx2sds+t−1a21σ1B1t+a11σ2B2t.
Therefore,(68)a21t−1lnx1tx10+a11t−1lnx2tx20+t−1a11a21∫t−τ2tx1sds−∫−τ20x1sds−t−1a12a21∫t−τ1tx2sds−∫−τ10x2sds≤a21r1−12σ12+a11r2−12σ22−a11a22+a12a21t−1∫0tx2sds+t−1a21σ1B1t+a11σ2B2t.
Lett⟶+∞ and the condition Δ=a21r1−1/2σ12+a11r2−1/2σ22>0 is satisfied, so(69)lim supt⟶+∞t−1∫0tx2sds≤Δa11a22+a12a21a.s.
Considering the third equation of model (5),(70)ut−u0t=−et−1∫0tusds+ft−1∫0tx2sds;hence,(71)lim supt⟶+∞t−1∫0tusds≤fΔea11a22+a12a21a.s.
Substitute (71) into equation (67), and by the condition Δ1=ea11a22+a12a21−a11cf>0, we have(72)lim inft⟶+∞t−1∫0tx2sds≥Δ−a11cfΔ/ea11a22+a12a21a11a22+a12a21=ea11a22+a12a21−a11cfΔea11a22+a12a212>0a.s.
It is concluded that predatorx2t is persistent.
Whent⟶+∞ and substituting (72) into equation (70), one can see that(73)lim inft⟶+∞t−1∫0tusds≥fea11a22+a12a21−a11cfΔe2a11a22+a12a212>0a.s.
So,ut is persistent.
The conditionΔ2>0 means that(74)r1−12σ12<a12ea11a22+a12a21−a11cfΔea11a22+a12a212.
Substituting (72) into equation (58) and by the condition Δ2>0, we have(75)limt⟶+∞x1t=0a.s.
So,x1t is extinction. The proof of Theorem 4 is complete.


## 6. Conclusions and Numerical Simulations

This paper proposes a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control. We firstly study the existence and uniqueness of global positive solution. By constructing appropriate Lyapunov functions and applying Itô’s formula, we discuss the asymptotic behavior of stochastic system at the positive equilibrium point of the corresponding deterministic model. Finally, this paper gives the conditions for the persistence and extinction of stochastic system. Theorem 3 shows that the system is persistent if the intensityσii=1,2 of random disturbance and the coefficient c of feedback control variable satisfy the condition A2. Theorem 4 indicates that(1)
If the coefficientc of the feedback control cutx2t remains unchanged and the intensity σii=1,2 of random disturbance increases, the population cannot resist the disturbance of the external environment and extinct(2)
If the intensity of random disturbanceσii=1,2 remains unchanged and the coefficient c of feedback control cutx2t is small, Δ1>0 and Δ2>0 are satisfied, which will cause the continuous increase of predator number for a period of time, thus leading to the extinction of the prey population.Therefore, utilizing the feedback control measures to limit the predator quantity within a certain range is beneficial for the sustained existence of the population.In order to verify the correctness of the theoretical analysis, we carry out the following numerical simulations. Choose the parameters in system (5) as follows:(76)r1=1.5,r2=1.2,e=0.5,f=0.2,a11=0.2,a21=0.2,a22=0.4,τ1=1,τ2=1.LetΔt=0.01 and the initial value x10=7,x20=6,andu0=5.(i)
Setc=0.4,a12=0.25,σ1=0.05,andσ2=0.05, and the positive equilibrium point of the corresponding deterministic model is x1∗,x2∗,u∗=10/3,10/3,4/3. It is proved that conditions A1andA2 are satisfied and Theorem 2 and Theorem 3 are valid. System (5) oscillates slightly near the point 10/3,10/3,4/3 and persistently (see Figure 1).(ii)
Setc=0.1,a12=0.25,σ1=1.8,andσ2=1.6, one can get r1−1/2σ12=−0.12<0andr2−1/2σ22=−0.08<0. According to Theorem 4 (i), system (5) is extinct (see Figure 2).(iii)
Setc=0.1,a12=0.25,σ1=0.05,andσ2=2.5, we have r1−1/2σ12=1.4988>0 and Δ=a21r1−1/2σ12+a11r2−1/2σ22=−0.0852<0. According to Theorem 4 (ii), population x1 is persistent and x2 and u are extinct (see Figure 3).(iv)
Setc=0.1,a12=0.55,σ1=0.05andσ2=0.05, we obtain r1−1/2σ12=1.4988>0, Δ=a21r1−1/2σ12+a11r2−1/2σ22=0.5395>0, Δ1=ea11a22+a12a21−a11cf=0.091>0, and Δ2=a12Δea11a22+a12a21−a11cf−r1−1/2σ12ea11a22+a12a212=0.027>0. According to Theorem 4 (iii), population x1 is extinct and x2 and u are persistent (see Figure 4).Figure 1
(a) Time series diagram ofx1t,x2t,ut and (b) its phase diagram.
(a)
(b)Figure 2
Time series diagram ofx1t,x2t,ut.Figure 3
Time series diagram ofx1t,x2t,ut.Figure 4
(a) Time series diagram ofx1t,x2t,ut. (b) Phase diagram of x2t,ut.
(a)
(b)



---

*Source: 4873290-2019-11-16.xml*