# Dynamics Analysis of a Stochastic Leslie–Gower Predator-Prey Model with Feedback Controls

Authors: Huailan Ren; Wencai Zhao
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This study focuses on the investigation of a stochastic Leslie–Gower predator-prey model with feedback controls and Holling type II functional responses. First, the existence and uniqueness of a global positive solution to the system under white noise interference are proved. Second, the conditions for the existence of the system’s positive recurrence are established by constructing suitable Lyapunov functions. Additionally, the persistence and extinction of prey and predator in the system are discussed, and the impacts of noise interference and feedback controls on the system are revealed. Finally, we validate the theoretical results by numerical simulations.


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## Body

## 1. Introduction

The relationship between biological populations is usually expressed and analyzed by differential equations [1–5]. Among numerous studies of biological populations, Leslie [1] presented a classical predator-prey model as follows:(1)dN1t=r1N1t1−N1tK−pN1tN2tdt,dN2t=N2tr2−hN2tN1tdt,with N10>0 and N20>0, where N1t and N2t are the density of the prey and the predator at time t, respectively. K denotes the carrying capacity of prey, and r1 and r2 are the intrinsic growth rates of the prey and the predator. pN1t is the functional response of the predator to the prey. h is the measure of food quantity that the prey provides for conversion into one predator birth. The term r2N1t/h is the predator’s environmental capacity, and hN2t/N1t is called the Leslie–Gower term. Leslie accentuated that the decrease in the number of predators was related to the per capita availability of the prey to be hunted, and the environmental carrying capacity of the predator was proportional to the number of preys.On the basis of model (1), Aziz-Alaoui and Daher Okiye [6] proposed a modified Leslie–Gower predator-prey model with Holling type II functional responses as follows:(2)dN1t=N1tr11−N1tK−aN2tn+N1tdt,dN2t=N2tr2−bN2tm+N1tdt,where a is the maximum of the average predation rate and b is the maximum digestibility rate of predators. n and m are the environment protecting effects of the prey and the predator, respectively. Combined with the Lyapunov function methods, the global stability of the coexisting interior equilibrium point of model (2) was investigated.As is known to all, ecosystems are inevitably disturbed by the environmental noise [7–15]. Strong environmental disturbances can cause population fluctuations and even extinction. Based on model (2), Ji et al. [16] presented the following Leslie–Gower predator-prey model with stochastic perturbation:(3)dN1t=N1tr11−N1tK−aN2tm+N1tdt+σ1N1tdB1t,dN2t=N2tr2−bN2tm+N1tdt+σ2N2tdB2t.To simplify the problem, they chosen=m (assuming the predator and prey receive the same protection from the environment) in model (3). Here, σi2i=1,2 denotes the intensity of white noise, and Biti=1,2 are the standard Brownian motions independent of each other in the complete probability space Ω,ℱ,ℱt≥0,ℙ. Also, the conditions for the persistence in mean and extinction of the system were provided. In recent years, many research results have been achieved in the study of stochastic ecosystems [17–25]. For instance, Miao et al. [26] verified the stationary distribution of an n-species stochastic differential equation model. Lahrouz et al. [27] studied a stochastic Leslie–Gower predator-prey model with regime switching where the effects of white noise and colored noise on the model were considered comprehensively, and the long-term dynamic behavior of the population and sufficient conditions for the existence of stationary distribution were investigated. In [28], a predator-prey model of stochastic Holling type II schemes with Markov switching and prey harvesting was proposed, and they obtained the conditions for the existence of the stationary distribution and got the optimal harvesting strategy.How to regulate the ecosystem reasonably to ensure its balance and sustainable development is of importance for issue facing mankind [29–33]. Maltby et al. [34] considered ecosystem management to be a control of physical, chemical, and biological processes, which linked the regulation of organisms and their nonliving environment and human activities to create an ideal ecosystem state. In recent years, widely research has been conducted on the biological mathematical models with feedback controls [35–40]. By introducing control variables, Gopalsamy and Weng [41] presented a competition model with time delay and feedback controls following by the discussion on the existence and global attractiveness of the model’s positive equilibrium. In [42], a Lotka–Volterra two-species competition model was studied, and the influence of feedback controls on the global stability of the system was explored.Synthesizing the above literatures, we construct the following stochastic Leslie–Gower predator-prey model with feedback controls:(4)dN1t=N1tr11−N1tK−aN2tm+N1t−c1u1tdt+σ1N1tdB1t,dN2t=N2tr2−bN2tm+N1t−c2u2tdt+σ2N2tdB2t,du1t=−f1u1t+g1N1tdt,du2t=−f2u2t+g2N2tdt,where u1t and u2t are the feedback control variables and c1,c2,f1,f2,g1,andg2 are all positive constants from the realistic biological significance.The rest of this paper is structured as follows: in Section2, we prove the existence and uniqueness of global positive solutions to system (4). In Section 3, we obtain the conditions for the existence of positive recurrence of the solutions to system (4). In Section 4, we analyze the survival conditions of the system and obtain sufficient conditions for persistence in mean and extinction. In Section 5, we use MATLAB to perform some numerical simulations to supplement and illustrate the result.


## 2. Existence and Uniqueness of Global Positive Solutions

In order to study the dynamic behaviors of system (4), it is necessary to first determine whether there is a global positive solution for any given initial conditions.Theorem 1.
For any given initial valueN10,N20,u10,u20∈ℝ+4, the model (4) has a unique positive solution N1t,N2t,u1t,u2t on t≥0 and the solution will remain in ℝ+4 with probability one.Proof.
Since the coefficients of the model (4) satisfy the local Lipschitz condition, for any initial value N10,N20,u10,u20∈ℝ+4, there exists a unique local solution N1t,N2t,u1t,u2t∈ℝ+4 on t∈0,τe, where τe is the explosion time. We want to prove this solution is global, i.e., to show the τe=∞ a.s. Define a number h0≥1 which is large enough to make Ni0,ui0∈1/h0,h0i=1,2. For each integer h≥h0, we define the stopping time τh as follows:(5)τh=inft∈0,τe:minN1t,N2t,u1t,u2t≤1hormaxN1t,N2t,u1t,u2t≥h,where the sequence τh is increasing with respect to h. Let limh⟶∞τh=τ∞, by the definition of stopping time, τ∞≤τe a.s. Next, we need to prove τ∞=∞ so that τe=∞. If this statement is false, then there exist constants T>0 and ε∈0,1 such that(6)ℙτ∞≤T>ε.
Thus, there is an integerh1≥h0 such that(7)ℙτh≤T≥ε,for  allh≥h1.
For the solutionN1t,N2t,u1t,u2t of model (4), we define a C2 function V: ℝ+4⟶ℝ+ by(8)VN1,N2,u1,u2=g1N1−1−lnN1+g2N2−1−lnN2+c1u1−122+c2u2−122=V1+V2,where(9)V1=g1N1−1−lnN1+g2N2−1−lnN2,V2=c1u1−122+c2u2−122.
By using the generalized Itô formula, we get(10)dV=LVdt+g1N1−1σ1dB1t+g2N2−1σ2dB2t,where(11)LV=LV1+LV2,LV1=g1N1−1r11−N1K−aN2m+N1−c1u1+g1σ122+g2N2−1r2−bN2m+N1−c2u2+g2σ222=g1r1N1−g1r1KN12−g1aN1N2m+N1−g1c1u1N1−g1r1+g1r1KN1+g1aN2m+N1+g1c1u1+g1σ122+g2r2N2−g2bN22m+N1−g2c2u2N2−g2r2+g2bN2m+N1+g2c2u2+g2σ222≤−g1r1KN12+g1r1+g1r1KN1+g1c1u1+g1a+g2bm+g2r2N2−g1c1u1N1−g2c2u2N2+g2c2u2+g1σ122+g2σ222,LV2=c1u1−1−f1u1+g1N1+c2u2−1−f2u2+g2N2=−f1c1u12+f1c1u1+c1g1u1N1−c1g1N1−c2f2u22+f2c2u2+c2g2u2N2−c2g2N2≤−f1c1u12+f1c1u1+c1g1u1N1−c2f2u22+f2c2u2+c2g2u2N2,LV≤max−g1r1KN12+g1r1+g1r1KN1+max−f1c1u12+g1c1+c1f1u1+max−c2f2u22+g2c2+f2c2u2+g1a+g2bm+g2r2N2+g1σ122+g2σ222=M+g1a+g2bm+g2r2N2+g1σ122+g2σ222,where(12)M=max−g1r1KN12+g1r1+g1r1KN1+max−f1c1u12+g1c1+c1f1u1+max−c2f2u22+g2c2+f2c2u2.
Applying the Itô formula to (4), one can get(13)N2t=N20er2t−∫0tbN22sm+N1ser2t−sds−∫0tc2u2sN2ser2t−sds+σ2∫0ter2t−sN2sdB2s≤N20er2t+σ2∫0ter2t−sN2sdB2s.
Therefore,(14)dV≤M+g1a+g2bm+g2r2N20er2t+σ2∫0ter2t−sN2sdB2s+g1σ122+g2σ222dt+g1N1−1σ1dB1t+g2N2−1σ2dB2t.
Integrating both sides of (14) from 0 to τh∧T and taking the expectation, we obtain(15)EVN1τh∧T,N2τh∧T,u1τh∧T,u2τh∧T≤VN10,N20,u10,u20+M+g1σ122+g2σ222Eτh∧T+g1a+g2bm+g2r2N20∫0τh∧TEer2tdt≤VN10,N20,u10,u20+M+g1σ12+g2σ222T+N20r2g1a+g2bm+g2r2er2T.
For allh≥h1, let Ωh=τh≤T, we have ℙΩh≥ε. Note that, for any ω∈Ωh, at least one of N1τh,ω,N2τh,ω,u1τh,ω, or u2τh,ω equals 1/h or h, and then(16)VN1τh,ω,N2τh,ω,u1τh,ω,u2τh,ω≥gh−1−lnh∧1h−1+lnh,where(17)g=ming1,g2.
Combining inequality (15), which implies that(18)VN10,N20,u10,u20+M+g1σ122+g2σ222T+1r2N20g1a+g2bm+g2r2er2T≥E1ΩhωVN1τh,ω,N2τh,ω,u1τh,ω,u2τh,ω≥gh−1−lnh∧1h−1+lnhℙΩh≥εgh−1−lnh∧1h−1+lnh,where 1Ωh is the indicator function of Ωh. As h⟶∞, there is(19)∞>VN10,N20,u10,u20+M+g1σ122+g2σ222T+1r2N20g1a+g2bm+g2r2er2T=∞.
Obviously, this is contradictory. The existence and uniqueness of global positive solutions are proved.


## 3. Existence of Positive Recurrence

For the stochastic model with feedback controls (4), we are primarily concerned with the long-term dynamic behavior of the population. For example, under what conditions can the path from one point return to a given region for a limited time? This requires us to discuss the positive recurrence of the solution N1t,N2t,u1t,u2t of model (4).LetXt be a regular time-homogeneous Markov process in El (the l-dimensional Euclidean space) of the stochastic equation:(20)dXt=bXdt+∑r=1lgrXdBrt.The diffusion matrix is defined as follows:(21)Ax=aijx,aijx=∑r=1lgrixgrjx.Definition 1.
(see [43]). The Markov process Xt is said to be positive recurrent with a bounded domain U⊂El, if Eτx<∞ for any x∉U, here τx is the time when the trajectory from x arrives at U, namely, τx=inft:Xt∈U.Lemma 1.
(see [43]). Suppose that the process Xt almost surely exits from each bounded domain in a finite time. Then, a sufficient condition for positive recurrence is that there exists a nonnegative function Vt,x in the domain t>0×Uc such that(22)VR=inft>0,x≥RVt,x⟶∞,asR⟶∞,LV≤0.First, we consider the equilibrium point of the corresponding deterministic system. From model (4), we set(23)r11−N1K−aN2m+N1−c1u1=0,r2−bN2m+N1−c2u2=0,−f1u1+g1N1=0,−f2u2+g2N2=0.By calculating (23), the following equation is obtained:(24)u1=g1N1f1,u2=g2N2f2,N2=r2f2m+N1f2b+c2g2m+N1,c2g2c1g1K+r1f1N12+f2b+c2g2mc1g1K+r1f1−f1r1c2g2KN1+ar2f1f2K−mf1r1c2g2K−f1r1f2bK=0.Let(25)ξ=f2b+c2g2mc1g1K+r1f1−r1f1c2g2K.Ifξ satisfies(26)Δ=ξ2−4c2g2c1g1K+r1f1ar2f1f2K−mr1f1c2g2K−r1f1f2bK≥0,and(27)ar2f2<r1bf2+mc2g2,then there exists a unique positive solution N1∗,N2∗,u1∗,u2∗ to equations (23). That is, the deterministic system corresponding to the stochastic model (4) has a unique positive equilibrium point.Theorem 2.
Let conditions (26) and (27) hold. If the hypothesis(28)H1gi=cii=1,2H2r1mK>a2b,a=bN2∗m+N1∗H3δ<minr1mK−a2bN1∗+N1∗σ12+N2∗σ224r1m/K−a2/b2,bN2∗2,mf1u1∗2,mf2u2∗2δ=N1∗σ12+N2∗σ22216r1m/K−a2/b+12N1∗σ12+N2∗σ22N1∗+m,holds, then the solution to model (4) is positive recurrence with respect to the elliptic domain U.Proof.
LetN1t,N2t,u1t,u2t be a solution to (4) for any given initial value.
N
1
0
,
N
2
0
,
u
1
0
,
u
2
0
∈
R
+
4. Define a C2 function(29)V=N1−N1∗−N1∗lnN1N1∗+N2−N2∗−N2∗lnN2N2∗+u1−u1∗22+u2−u2∗22=V1+V2+V3,where(30)V1=N1−N1∗−N1∗lnN1N1∗,V2=N2−N2∗−N2∗lnN2N2∗,V3=u1−u1∗22+u2−u2∗22.
Obviously,VR=inft>0,x≥RVt,x⟶∞ as R⟶∞, where x=N1,N2,u1,u2T. By using the Itô formula, we get(31)LV1=N1−N1∗r1−r1N1K−aN2m+N1−c1u1+N1∗σ122=N1−N1∗r1KN1∗+aN2∗m+N1∗+c1u1∗−r1N1K−aN2m+N1−c1u1+N1∗σ122=−r1KN1−N1∗2+aN2∗N1−N1∗m+N1∗m+N1−aN2−N2∗m+N1N1−N1∗−c1u1−u1∗N1−N1∗+N1∗σ122=−r1KN1−N1∗2+aN2∗N1−N1∗2m+N1∗m+N1−aN2−N2∗N1−N1∗m+N1−c1u1−u1∗N1−N1∗+N1∗σ122,LV2=N2−N2∗bN2∗m+N1∗+c2u2∗−bN2m+N1−c2u2+N2∗σ222=N2−N2∗bN2∗N1−N1∗m+N1∗m+N1−bN2−N2∗m+N1−c2u2−u2∗+N2∗σ222=bN2∗N1−N1∗N2−N2∗m+N1∗m+N1−bN2−N2∗2m+N1−c2u2−u2∗N2−N2∗+N2∗σ222,LV3=u1−u1∗−f1u1+g1N1+u2−u2∗−f2u2+g2N2=u1−u1∗−f1u1+g1N1+f1u1∗−g1N1∗+u2−u2∗−f2u2+g2N2+f2u2∗−g2N2∗=u1−u1∗−f1u1−u1∗+g1N1−N1∗+u2−u2∗−f2u2−u2∗+g2N2−N2∗=−f1u1−u1∗2+g1N1−N1∗u1−u1∗−f2u2−u2∗2+g2u2−u2∗N2−N2∗.
Therefore,(32)LV=−r1KN1−N1∗2+aN2∗N1−N1∗2m+N1∗m+N1−aN2−N2∗N1−N1∗m+N1−c1u1−u1∗N1−N1∗+N1∗σ122+bN2∗N1−N1∗N2−N2∗m+N1∗m+N1−bN2−N2∗2m+N1−c2u2−u2∗N2−N2∗+N2∗σ222−f1u1−u1∗2+g1N1−N1∗u1−u1∗−f2u2−u2∗2+g2u2−u2∗N2−N2∗=−r1/Km+N1−aN2∗/m+N1∗m+N1N1−N1∗2+bN2∗m+N1∗−aN2−N2∗N1−N1∗m+N1+g1−c1u1−u1∗N1−N1∗+g2−c2u2−u2∗N2−N2∗−bN2−N2∗2m+N1−f1u1−u1∗2−f2u2−u2∗2+N1∗σ122+N2∗σ222.
From the hypotheses (H1) and (H2), we get(33)LV≤−r1/Km−a2/bm+N1N1−N1∗2−bN2−N2∗2m+N1−f1u1−u1∗2−f2u2−u2∗2+N1∗σ122+N2∗σ222.
It can be written as(34)m+N1LV≤−r1mK−a2bN1−N1∗2−bN2−N2∗2−m+N1f1u1−u1∗2−m+N1f2u2−u2∗2+N1∗σ12+N2∗σ222m+N1≤−r1mK−a2bN1−N1∗2−bN2−N2∗2−mf1u1−u1∗2−mf2u2−u2∗2+N1∗σ12+N2∗σ222m+N1=−r1mK−a2bN1−N1∗−N1∗σ12+N2∗σ224r1m/K−a2/b2−bN2−N2∗2−mf1u1−u1∗2−mf2u2−u2∗2+N1∗σ12+N2∗σ22216r1m/K−a2/b+12N1∗σ12+N2∗σ22N1∗+m.
Obviously, the ellipsoid(35)−r1mK−a2bN1−N1∗−N1∗σ12+N2∗σ224r1m/K−a2/b2−bN2−N2∗2−mf1u1−u1∗2−mf2u2−u2∗2+δ≥0,lies entirely in R+4. For any x∈El/U, where U is a neighborhood of the ellipsoid, we have LV≤−M (M is a positive constant).
From Lemma 1, we obtain the existence of positive recurrence to model (4). Because the diffusion matrix Ax of system (4) does not satisfy the uniformly elliptical condition in U, we only prove that the system is positive recurrent but does not obtain the ergodic property.


## 4. Persistence and Extinction

In the above section, we obtained the existence conditions for the positive recurrence of system (4). In this section, we discuss the persistence and extinction of species of the stochastic model (4) with feedback controls.For convenience, let us define some notations as follows:(36)Δ1=r1f1−c1g1Kr1−σ12/2f1r1,Δ2=bf2−c2g2mr2−σ22/2bf2.Theorem 3.
For any given initial conditionN10,N20,u10,u20∈R+4, the solution N1t,N2t,u1t,u2t to model (4) have the following properties:(i)
Whenr1−σ12/2<0 and r2−σ22/2<0, N1 and N2 are extinct, and u1 and u2 are also extinct(ii)
Whenr1−σ12/2>0, r2−σ22/2<0, and Δ1>0, N2 is extinct, and N1 and u1 are persistent in mean(iii)
Whenr1−σ12/2<0, r2−σ22/2>0, and Δ2>0, N1 is extinct, and N2 and u2 are persistent in mean.Proof.
Applying the Itô formula to system (4), we obtain(37)lnN1t−lnN10=r1−σ122t−r1K∫0tN1sds−a∫0tN2sm+N1s−c1∫0tu1sds+∫0tσ1dB1s,(38)lnN2t−lnN20=r2−σ222t−b∫0tN2sm+N1sds−c2∫0tu2sds+∫0tσ2dB2s.
Dividing both sides of (37) and (38) by t, we get(39)t−1lnN1t=t−1lnN10+r1−σ122−t−1r1K∫0tN1sds−t−1a∫0tN2sm+N1sds−t−1c1∫0tu1sds+t−1M1t,(40)t−1lnN2t=t−1lnN20+r2−σ222−t−1b∫0tN2sm+N1sds−t−1c2∫0tu2sds+t−1M2t,where(41)M1t=∫0tσ1dB1s,M2t=∫0tσ2dB2s.
Based on the strong law of large number for martingales, we obtain(42)limt⟶∞M1tt=0,limt⟶∞M2tt=0a.s.Case 1.
According to formula (39), we gain(43)t−1lnN1t≤t−1lnN10+r1−σ122−t−1r1K∫0tN1sds+t−1M1t.
By Lemma 2.3 in [44], when r1−σ12/2<0,(44)limt⟶∞N1t=0a.s.
According to the third equation of system (4), we can acquire(45)limt⟶∞u1t=0a.s.
Similarly, by equation (40), when r2−σ22/2<0, there is(46)limt⟶∞N2t=0a.s.
Furthermore, from the fourth equation of model (4), we can obtain(47)limt⟶∞u2t=0a.s.Case 2.
We suppose thatZ1t is a solution to the following equation:(48)dZ1t=Z1tr1−r1Z1tK−aN2tm+Z1tdt+σ1Z1tdB1t,with Z10=N10>0. According to the comparison theorem for the stochastic equation, for t≥0, we get(49)Z1t≥N1t,a.s.
From the conditionr2−σ22/2<0, we can get limt⟶∞N2t=0. For arbitrary small ε>0, we have N2t<ε for t>T(T is a constant) and(50)Z1tr1−r1Z1tK−aεmdt+σ1Z1tdB1t≤dZ1t≤Z1tr1−r1Z1tKdt+σ1Z1tdB1t.
Byr1−σ12/2>0, we have(51)liminft⟶∞t−1∫∫0tZ1sds≥r1−σ12/2−a/mεKr1,limsupt⟶∞t−1∫0tZ1sds≤r1−σ12/2Kr1a.s.
For the arbitrariness ofε, we get(52)limsupt⟶∞t−1∫0tN1sds≤limt⟶∞t−1∫0tZ1sds=r1−σ12/2Kr1a.s.
According to the third equation of system (4), we can derive(53)limsupt⟶∞t−1∫0tu1sds≤g1Kr1−σ12/2f1r1a.s.
Substituting (53) into equation (39) leads to(54)t−1lnN1t≥t−1lnN10+r1−σ122−t−1r1K∫0tN1sds−c1g1Kr1−σ12/2f1r1−amε+t−1M1t,=t−1lnN10+r1f1−c1g1Kr1−σ12/2f1r1−t−1r1K∫0tN1sds−amε+t−1M1t.
When(55)Δ1=r1f1−c1g1Kr1−σ12/2f1r1>0,we acquire(56)liminft⟶∞t−1∫0tN1sds≥Δ1Kr1a.s.
Considering the third equation of (4) again, we can see(57)liminft⟶∞t−1∫0tu1sds≥Δ1Kg1r1f1>0a.s.
That is,N1 and u1 are persistent in mean.Case 3.
LetZ2t be a solution of the following equation:(58)dZ2t=Z2tr2−bZ2tm+N1tdt+σ2Z2tdB2t=Z2tr2−bZ2tm+N1tmm+N1t+bN1tZ2tmm+N1tdt+σ2Z2tB2t=Z2tr2−bmZ2t+bN1tZ2tmm+N1tdt+σ2Z2tdB2t,with Z20=N20>0. Using the comparison theorem for the stochastic equation, we get that, for t≥0,(59)Z2t≥N2ta.s.
By the sure reason as the proof of Case (i), we know thatlimt⟶+∞N1t=0 holds when r1−σ12/2<0. Hence for arbitrary small ε>0, there is N1t/m+N1t≤ε for t≥t0. Thus, we can obtain(60)Z2tr2−bZ2tmdt+σ2Z2tdB2t≤dZ2t≤Z2tr2−b1−εZ2tmdt+σ2Z2tdB2t.
Sincer2−σ22/2>0, then we have(61)liminft⟶∞t−1∫0tZ2sds≥r2−σ22/2b/m,limsupt⟶∞t−1∫0tZ2sds≤r2−σ22/2b1−ε/ma.s.
From the arbitrariness ofε, one can get(62)limsupt⟶∞t−1∫0tN2sds≤limt⟶∞t−1∫0tZ2sds=mr2−σ22/2ba.s.
Then, we have from the fourth equation in system (4)(63)limsupt⟶∞t−1∫0tu2sds≤g2mr2−σ22/2bf2a.s.
Combining (40) with (63) leads to(64)t−1lnN2t≥t−1lnN20+r2−σ222−t−1bm∫0tN2sds−c2g2mr2−σ22/2bf2+t−1M2t.
ConsideringΔ2=bf2−c2g2mr2−σ22/2/bf2>0, we can conclude(65)liminft⟶∞t−1∫0tN2sds≥Δ2mb>0a.s.
Combined with the fourth equation of system (4), we can obtain(66)liminft−1∫0tu2sds≥Δ2g2mbf2>0a.s.
This completes the proof.□


## 5. Numerical Simulations

In this section, we use MATLAB to perform some numerical simulations to illustrate our main theoretical results by choosing the fixed parameters as follows:a=0.7, b=0.7, m=4, K=0.5, r1=0.4, and r2=0.2.In Figure1, we choose f1=2, f2=2, g1=3, g2=3, c1=0.3, c2=0.3, σ1=1.1, and σ2=0.75. And we calculate that r1−σ12/2=−0.205<0andr2−σ22/2=−0.081<0; in view of (i) of Theorem 3, the populations N1 and N2 are extinct, and the feedback control variables u1 and u2 are also extinct.Figure 1
For populationsN1andN2 and feedback control variables u1andu2, we choose the parameters as f1=2, f2=2, g1=3, g2=3, c1=0.3, c2=0.3, σ1=1.1, and σ2=0.75 and the initial value N10,N20,u10,u20=0.4,0.4,0.2,0.2.In Figure2, we choose f1=3, f2=2, g1=2, g2=3, c1=0.2, c2=0.3, σ1=0.1, and σ2=0.75, where r1−σ12/2=0.395>0andr2−σ22/2=−0.081<0 and Δ1=0.329>0. By (ii) of Theorem 3, the population N2 is extinct, and the population N1 and the feedback control variable u1 are persistent.Figure 2
For populationsN1andN2 and feedback control variables u1andu2, we choose the parameters as f1=3, f2=2, g1=2, g2=3, c1=0.2, c2=0.3, σ1=0.1, and σ2=0.75 and the initial value N10,N20,u10,u20=0.4,0.4,0.2,0.2.In Figure3, we choose f1=2, f2=3, g1=3, g2=2, c1=0.3, c2=0.2, σ1=1.1, and σ2=0.1, where we have r2−σ22/2=0.195>0,r1−σ12/2=−0.205<0, and Δ2=0.121>0. According to (iii) in Theorem 3, the population N1 is extinct, and the population N2 and the feedback control variable u2 are persistent.Figure 3
For populationsN1andN2 and feedback control variables u1andu2, we choose the parameters as f1=2, f2=3, g1=3, g2=2, c1=0.3, c2=0.2, σ1=1.1, and σ2=0.1 and the initial value N10,N20,u10,u20=0.4,0.4,0.2,0.2.In Figure4, we choose f1=2, f2=2, g1=3, g2=3, c1=0.3, c2=0.3, σ1=0.1, and σ2=0.1. Calculated by the MATLAB program, which satisfy the conditions Δ=ξ2−4c2g2c1g1K+r1f1ar2f1f2K−mr1f1c2g2K−r1f1f2bK≥0, ar2f2<r1bf2+mc2g2, and H1–H3 of Theorem 2, we can get that the solution of system (4) is positive recurrence. And from Figure 4, we can find that N1 and N2 are persistent in mean.Figure 4
For populationsN1andN2 and feedback control variables u1andu2, we choose the parameters as f1=2, f2=2, g1=3, g2=3, c1=0.3, c2=0.3, σ1=0.1, and σ2=0.1 and the initial value N10,N20,u10,u20=0.4,0.4,0.2,0.2.
(a)
(b)
(c)
(d)


## 6. Conclusions

This paper presents a stochastic Leslie–Gower predator-prey model with feedback controls and Holling type II functional response. Firstly, we prove the existence and uniqueness of global positive solution. Then, we discuss the existence of positive recurrence of system (4) by constructing appropriate Lyapunov functions. Moreover, we obtain the sufficient conditions for the persistence and extinction of populations in Theorem 3. The results imply that the white noise intensity has effects on dynamic behaviors of the system, and the larger white noise intensity will lead to the extinction of the biological population. In addition, the persistence and extinction of the populations will also be affected by the value of the feedback control variable coefficients.



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