COMPLEXITY Complexity 1099-0526 1076-2787 Hindawi 10.1155/2019/4873290 4873290 Research Article Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control https://orcid.org/0000-0002-9811-2179 Liu Jinlei 1 https://orcid.org/0000-0003-4689-695X Zhao Wencai 1 2 Jalili Mahdi 1 College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 2 State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 2019 16112019 2019 02 06 2019 26 09 2019 24 10 2019 16112019 2019 Copyright © 2019 Jinlei Liu and Wencai Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

National Natural Science Foundation of China 11371230 Shandong University of Science and Technology 2014TDJH102
1. Introduction

In nature, time delays exist in many ecosystems . 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.  studied a Lotka–Volterra predator-prey model with discrete delays, which can be defined as follows:(1)dxtdt=xtr1+axt+αxtτ1βytτ2,dytdt=ytr2+ayt+βxtτ1+αytτ2,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 . In 2003, Gopalsamy and Weng  studied the following population competition model with feedback control:(2)dx1tdt=x1tb1a11x1ta12x2tα1u1tτ,dx2tdt=x2tb2a21x1ta22x2tα2u2tτ,du1tdt=η1u1t+a1x1tτ,du2tdt=η2u2t+a2x2tτ,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.  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.  discussed a Lotka–Volterra predator-prey model with discrete delays and feedback control as follows:(3)dx1tdt=x1tr1a11x1ta12x2tτ1,dx2tdt=x2tr2+a21x1tτ2a22x2tcut,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, aijij,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.  show that

The solution x1t,x2t,ut of system (3) is ultimately bounded

When the conditions r1/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=er1a22r2a12+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 . 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 ; the other type is the sum of many small, independent random interference, called white noise, which is usually represented by Brownian motion . Assume that the population’s intrinsic growth rate ri is disturbed by white noise:(4)riri+σiB˙it,i=1,2.

Then, model (3) is transformed into(5)dx1t=x1tr1a11x1ta12x2tτ1dt+σ1x1tdB1t,dx2t=x2tr2+a21x1tτ2a22x2tcutdt+σ2x2tdB2t,dut=eut+fx2tdt,and satisfies the initial conditions(6)xiθ=ϕiθ,uθ=ψθ,θτ,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,xRn.

If the Lyapunov function Vx,tC2,1Rn×R+,R, the stochastic differential equation of Vx,t along system (7) is defined as (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 +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 t0,τe, where τe is the explosion time.

To prove that this solution is global, we only need to prove τe= 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 kk0, define the stopping time(9)τk=inft0,τe:minx1t,x2t,ut1k  or  maxx1t,x2t,utk.

Obviously, τk is increasing as k. Let τ=limkτk; therefore, ττe a.s. Now, we need to verify τ= a.s. Otherwise, there are two constants T>0 and ϵ0,1 such that PτT>ϵ. So, there is a positive integer k1k0, such that(10)PτkT>ϵ,kk1.

Define a C2functionV: +3+ by(11)Vx1,x2,u=x11lnx1+a11a22a212x21lnx2+12a11tτ2tx12sds+a12tτ1tx2sds+a11a22c2fa212u2+u1lnu=V1x1,x2+V2u,where(12)V1x1,x2=x11lnx1+a11a22a212x21lnx2+12a11tτ2tx12sds+a12tτ1tx2sds,V2u=a11a22c2fa212u2+u1lnu.

The nonnegativity of this function can be obtained from(13)x1lnx0,x>0.

Applying Itô’s formula yields(14)dVx1,x2,u=LVx1,x2,udt+σ1x11dB1t+a11a22a212σ2x21dB2t,where(15)LV=LV1+LV2,LV1=x1t1r1a11x1ta12x2tτ1+12σ12+a11a22a212x2t1r2+a21x1tτ2a22x2tcut+a11a222a212σ22+12a11x12t12a11x12tτ2+a12x2ta12x2tτ1=r1+a11x1t12a11x12ta12x1tx2tτ1r1+12σ12+a11a22a212r2+a22x2ta22x22tcx2tut+cut+a11a22a21x2t1x1tτ2+a11a22a212r2+12σ2212a11x12tτ2+a12x2tr1+a11x1t12a11x12t+a11a22a212r2x2t12a22x22tcx2tut+cut+a12x2tr1+12σ12+a11a22a212r2+12σ22+12a22,LV2=a11a22cfa212uteut+fx2t+11uteut+fx2ta11a22cefa212u2t+a11a22ca212utx2teut+fx2t+e.

Therefore, (16)LV=LV1+LV2r1+a11x1t12a11x12t+a11a22a212r2x2t12a22x22tcx2tut+cut+a12x2tr1+12σ12+a11a22a212r2+12σ22+12a22a11a22cefa212u2t+a11a22ca212utx2teut+fx2t+emax12a11x12t+r1+a11x1t+a11a22a212max12a22x22t+r2+a212f+a12a11a22x2t+a11a22a212maxcefu2t+ca212ea11a22ut+a11a22a21212a22+12σ22r2+12σ12+er1K,where K is a positive constant. So, we get(17)dVKdt+σ1x11dB1t+a11a22a212σ2x21dB2t.

Integrating (17) from 0 to τkT and taking expectation on both sides, we have(18)EVx1τkT,x2τkT,uτkTEVx10,x20,u0+KT.

Set Ωk=τkT, and from inequality (10), we have PΩkϵ. Note that, for every ωΩk, there is at least one of x1τk,ω,x2τk,ω, or uτk,ω equaling either k or 1/k, and then, we have(19)Vx1τkT,x2τkT,uτkTk1lnk1k1ln1k.

It can be obtained by (18)(20)EVx10,x20,u0+KTE1ΩkωVx1τk,ω,x2τk,ω,uτk,ωεk1ln  k1k1+lnk,where 1Ωk is the indicator function of Ωk, and letting k yields(21)>EVx10,x20,u0+KT=.

This is a contradiction; we must have τ=, 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+1tE0tx1θx12+x2θx22+uθu2dθnm,where(24)m=mina11fa11a22a12a21a11a22+a12a21,a12fa11a22a12a212a11a21,a12cea21,n=fx1σ122+a12fx2σ222a21,where x1,x2,u is the positive equilibrium point of the corresponding deterministic model (3).

Proof.

Define the function(25)Vx1,x2,u=fx1x1x1ln x1x1+a12fa21x2x2x2ln x2x2+a12c2a21uu2+a12f2ω2tτ2tx1sx12ds+a12fω12tτ1tx2sx22ds,where ω1 and ω2 are positive constants. Define(26)V1=x1x1x1lnx1x1,V2=x2x2x2lnx2x2,V3=uu2,V4=a12f2ω2tτ2tx1sx12ds+a12fω12tτ1tx2sx22ds.

By Itô’s formula, we obtain(27)dV1=LV1dt+x1x1σ1dB1t,where(28)LV1=x1tx1r1a11x1ta12x2tτ1+12x1σ12=x1tx1a11x1tx1a12x2tτ1x2+12x1σ12a11x1tx12+a122ω1x1tx12+a12ω12x2tτ1x22+12x1σ12.

Similarly,(29)dV2=LV2dt+x2x2σ2dB2t,where(30)LV2=x2tx2r2+a21x1tτ2a22x2tcut+12x2σ22=x2tx2a21x1tτ2x1a22x2tx2cutu+12x2σ22a212ω2x1tτ2x12+a21ω22x2tx22a22x2tx22cx2tx2utu+12x2σ22.

In the same way,(31)dV3=2utueut+fx2tdt=2eutu2+2futux2tx2dt,dV4=a12f2ω2x1tx12a12f2ω2x1tτ2x12+a12fω12x2tx22a12fω12x2tτ1x22dt.

Therefore, we have(32)LV=fLV1+a12fa21LV2+a12c2a21dV3dt+dV4dta11fx1tx12+a12f2ω1x1tx12+a12fω12x2tτ1x22+12fx1σ12+a12f2ω2x1tτ2x12+a12fω22x2tx22a12a22fa21x2tx22a12cfa21x2tx2utu+a12fx22a21σ22a12cea21utu2+a12cfa21utux2tx2+a12f2ω2x1tx12a12f2ω2x1tτ2x12+a12fω12x2tx22a12fω12x2tτ1x22=fa11a122ω1a122ω2x1tx12fa12a22a21a12ω22a12ω12x2tx22a12cea21utu2+fx12σ12+a12fx22a21σ22.

Let ω1=ω2=a11a22+a12a21/2a11a21, we obtain(33)LVa11fa11a22a12a21a11a22+a12a21x1tx12a12fa11a22a12a212a11a21x2tx22a12cea21utu2+fx12σ12+a12fx22a21σ22.

Therefore,(34)dVx1,x2,u=LVdt+fx1x1σ1dB1t+a12fa21x2x2σ2dB2t.

Integrate both sides of (18) from 0 to t and take the expectation, and then we get(35)EVtEV0E0ta11fa11a22a12a21a11a22+a12a21x1θx12dθE0ta12fa11a22a12a212a11a21x2θx22dθE0ta12cea21uθu2dθ+fx1σ12t2+a12fx2σ22t2a21.

Divide both sides by t and take the limit superior, and then we have(36)lim supt+1tE0ta11fa11a22a12a21a11a22+a12a21x1θx12dθ+lim supt+1tE0ta12fa11a22a12a212a11a21x2θx22dθ+lim supt+1tE0ta12cea21uθu2dθfx1σ122+a12fx2σ222a21.

Obviously,(37)lim supt+1tE0tx1θx12+x2θx22+uθu2dθnm,where(38)m=mina11fa11a22a12a21a11a22+a12a21,a12fa11a22a12a212a11a21,a12cea21,n=fx1σ122+a12fx2σ222a21.

We have completed the proof.

Theorem 2 shows that if the condition A1 holds, the solution oscillates around the equilibrium point x, and the amplitude of oscillation is positively correlated with the intensity σ12 and  σ22 of environmental noise. In particular, if σ12=σ22=0, the influence of environmental noise is not taken into account:(39)x1tx1,x2tx2,utufor t+.

The equilibrium point x is globally asymptotically stable. This is the conclusion of reference .

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<minx1mn0,x2mn0,umn0,n0=fx12+a12fx22a21.

Theorem 3.

For any given initial condition (6), if assumptions A1 and A2 hold at the same time, the solution x1t,x2t,ut of system (5) is persistent that(41)lim inft1tE0tx1θdθ>0,lim inft1tE0tx2θdθ>0,lim inft1tE0tuθdθ>0.

Proof.

According to (37), we have(42)lim supt+1tE0tx1θx12dθnm,lim supt+1tE0tx2θx22dθnm,lim supt+1tE0tuθu2dθnm.

As we know, x1t0 and x1>0, from 2x1tx1x12x1tx12, one can get(43)x1tx12x1tx122x1.

By the condition μ<x1m/n0, we have(44)lim inft1tE0tx1θdθx12lim supt1tE0tx1θx122x1dθx12n2mx1x12μ2n02mx1>0.

Similarly, when μ<minx2m/n0,um/n0,(45)lim inft1tE0tx2θdθ>0lim inft1tE0tuθdθ>0.

5. Extinction

Define(46)Δ=a21r112σ12+a11r212σ22,Δ1=ea11a22+a12a21a11cf,Δ2=a12Δea11a22+a12a21a11cfr112σ12ea11a22+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:

when r11/2σ12<0 and r21/2σ22<0, the population is extinct

when r11/2σ12>0, and Δ<0, population x1 is persistent and x2 and u are extinct

when r11/2σ12>0,Δ>0, Δ1>0, and Δ2>0, population x1 is extinct and x2 and u are persistent

Before proving Theorem 4, consider the following auxiliary system :(47)dz1t=z1tr1a11z1tdt+σ1z1tdB1t,dz2t=z2tr2+a21z1tτ2a22z2tdt+σ2z2tdB2t,and it satisfies the initial condition: ϕθ=ϕ1θ,ϕ2θCτ,0,R+2.

Lemma 1.

If r11/2σ12>0, the solution zt of system (47) has the following properties:

limt+t10tz1sds=r11/2σ12/a11

if Δ<0,limt+z2t=0

if Δ>0,limt+t10tz2sds=Δ/a11a22,  a.s

Proof.

By Itô’s formula, we obtain(48)ln z1tln z10=r112σ12ta110tz1sds+σ1B1t,(49)ln z2tln z20=r212σ22t+a210tz1sτ2dsa220tz2sds+σ2B2t=r212σ22t+a210tz1sdsa21tτ2tz1sdsτ20z1sdsa220tz2sds+σ2B2t.

Dividing both sides of (48) and (49) by t, we have(50)t1lnz1tz10=r112σ12a11t10tz1sds+t1σ1B1t,(51)t1lnz2tz20+t1a21tτ2tz1sdsτ20z1sds=r212σ22+a21t10tz1sdsa22t10tz2sds+t1σ2B2t.

By using Lemma 2 of , from equation (48), it follows that(52)lim supt+t10tz1sdsr11/2σ12a11,lim inft+t10tz1sdsr11/2σ12a11,and therefore,(53)limt+t10tz1sds=r11/2σ12a11.

Substitute (53) into (50), and from limt+t1Bit=0i=1,2, we get(54)limt+t1ln z1t=0.

On the other hand, computing (50) × a21 + (51) × a11, we have(55)a21t1ln z1tz10+a11t1ln z2tz20+t1a11a21tτ2tz1sdsτ20z1sds=a21r112σ12+a11r212σ22a11a22t10tz2sds+t1a21σ1B1t+a11σ2B2t.

By Lemma 2 in literature , when Δ<0,(56)limt+z2t=0.

If Δ>0,(57)limt+t10tz2sds=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)lnx1tlnx10=r112σ12ta110tx1sdsa120tx2sτ1ds+σ1B1t=r112σ12ta110tx1sdsa120tx2sds+a12tτ1tx2sdsτ10x2sds+σ1B1t,(59)lnx2tlnx20=r212σ22t+a210tx1sτ2dsa220tx2sdsc0tusds+σ2B2t=r212σ22t+a210tx1sdsa21tτ2tx1sdsτ20x1sdsa220tx2sdsc0tusds+σ2B2t.

We first prove (i): from equation (58), we can get(60)lnx1tlnx10r112σ12ta110tx1sds+σ1B1t.

By the condition r11/2σ12<0 and Lemma 2 in literature ,(61)limt+x1t=0 a.s.

From (57), (61), and the condition r21/2σ22<0, we have(62)limt+x2t=0 a.s.

Further, from the third equation of model (5), we can get(63)limt+ut=0 a.s.

Secondly, it proves (ii): comparing model (5) with auxiliary system (37), one can get x1tz1t and x2tz2t. By Lemma 1, when r11/2σ12>0 and Δ<0, limt+z2t=0 a.s.; therefore, limt+x2t=0  a.s. So, we have limt+ut=0. Then, the limit system of model (5) is as follows:(64)dx1t=x1tr1a11x1tdt+σ1x1tdB1t.

From Lemma 1, we can conclude that(65)limt+t10tx1sds=r11/2σ12a11 a.s.

Therefore, population x1 is persistent and x2 and u are extinct.

Finally, we prove (iii): by Lemma 1, when r11/2σ12>0 and Δ>0, limt+t10tz2sds=Δ/a11a22,  a.s. And, by (54),(66)limsupt+t1ln  x1tlimt+t1ln  z1t=0.

Computing (58) × a21 + (59) × a11,(67)a21t1lnx1tx10+a11t1lnx2tx20+t1a11a21tτ2tx1sdsτ20x1sds=t1a12a21tτ1tx2sdsτ10x2sds+a21r112σ12+a11r212σ22a11ct10tusdsa11a22+a12a21t10tx2sds+t1a21σ1B1t+a11σ2B2t.

Therefore,(68)a21t1lnx1tx10+a11t1lnx2tx20+t1a11a21tτ2tx1sdsτ20x1sdst1a12a21tτ1tx2sdsτ10x2sdsa21r112σ12+a11r212σ22a11a22+a12a21t10tx2sds+t1a21σ1B1t+a11σ2B2t.

Let t+ and the condition Δ=a21r11/2σ12+a11r21/2σ22>0 is satisfied, so(69)lim supt+t10tx2sdsΔa11a22+a12a21 a.s.

Considering the third equation of model (5),(70)utu0t=et10tusds+ft10tx2sds;hence,(71)lim supt+t10tusdsfΔea11a22+a12a21 a.s.

Substitute (71) into equation (67), and by the condition Δ1=ea11a22+a12a21a11cf>0, we have(72)lim inft+t10tx2sdsΔa11cfΔ/ea11a22+a12a21a11a22+a12a21=ea11a22+a12a21a11cfΔea11a22+a12a212>0  a.s.

It is concluded that predator x2t is persistent.

When t+ and substituting (72) into equation (70), one can see that(73)lim inft+t10tusdsfea11a22+a12a21a11cfΔe2a11a22+a12a212>0 a.s.

So, ut is persistent.

The condition Δ2>0 means that(74)r112σ12<a12ea11a22+a12a21a11cfΔea11a22+a12a212.

Substituting (72) into equation (58) and by the condition Δ2>0, we have(75)limt+x1t=0  a.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 intensity σii=1,2 of random disturbance and the coefficient c of feedback control variable satisfy the condition A2. Theorem 4 indicates that

If the coefficient c 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

If the intensity of random disturbance σii=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.

Let Δt=0.01 and the initial value x10=7,x20=6, and u0=5.

Set c=0.4,a12=0.25,σ1=0.05, and σ2=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 A1 and A2 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).

Set c=0.1,a12=0.25,σ1=1.8, and σ2=1.6, one can get r11/2σ12=0.12<0 and r21/2σ22=0.08<0. According to Theorem 4 (i), system (5) is extinct (see Figure 2).

Set c=0.1,a12=0.25,σ1=0.05, and σ2=2.5, we have r11/2σ12=1.4988>0 and Δ=a21r11/2σ12+a11r21/2σ22=0.0852<0. According to Theorem 4 (ii), population x1 is persistent and x2 and u are extinct (see Figure 3).

Set c=0.1,a12=0.55,σ1=0.05 and σ2=0.05, we obtain r11/2σ12=1.4988>0, Δ=a21r11/2σ12+a11r21/2σ22=0.5395>0, Δ1=ea11a22+a12a21a11cf=0.091>0, and Δ2=a12Δea11a22+a12a21a11cfr11/2σ12ea11a22+a12a212=0.027>0. According to Theorem 4 (iii), population x1 is extinct and x2 and u are persistent (see Figure 4).

(a) Time series diagram of x1t,x2t,ut and (b) its phase diagram.

Time series diagram of x1t,x2t,ut.

Time series diagram of x1t,x2t,ut.

(a) Time series diagram of x1t,x2t,ut. (b) Phase diagram of x2t,ut.

Data Availability

All data sets used in this study are hypothetical.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11371230) and SDUST Research Fund (2014TDJH102).