# Dynamical Behaviors of Stochastic Delayed One-Predator and Two-Competing-Prey Systems with Holling Type IV and Crowley-Martin Type Functional Responses

Authors: Changjian Wang; Zuoliang Xiong; Rensheng He; Hongwei Yin
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

This paper is devoted to stochastic delayed one-predator and two-competing-prey systems with two kinds of different functional responses. By establishing appropriate Lyapunov functions, the globally positive solution and stochastic boundedness are investigated. In some case, the stochastic permanence and extinction are also obtained. Moreover, sufficient conditions of the global asymptotic stability of the system are established. Finally, some numerical examples are provided to explain our conclusions.


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## Body

## 1. Introduction

It is well known that predator-prey system, cooperative system, and competitive system are three kinds of important ecological systems. The dynamic relationship among species is a significant theme whether in ecology or in mathematical ecology because of its importance and universal existence with many concerned biological systems (see [1]). A lot of systems about predator-prey behaviors have been proposed (see [2–4]).A main objective for ecologists is to find the relationships among species. And the consumption rate of each predator on prey is an important component of the relationships between predator and prey, that is, predator’s functional response. In order to describe different situations when predators search or compete for food, many significant functional responses have been proposed, such as L-V and Holling II–IV types (see [5–7]). A suitable functional response is not only related to the density of prey, but also to the predator. A statistics from 19 predator-prey systems indicates that Crowley-Martin type, Beddington-DeAngelis type, and Hassell-Varley type predator-dependent functions can provide a better description in some case. In [8], the following predator-prey system bas been studied:(1)dxdt=xA-Bx-CyA1+B1x+C1y+B1C1xy,dydt=y-D-Ey+FxA1+B1x+C1y+B1C1xy,where x is the density of prey and y is the density of predator at time t.However, interaction of multiple species often occurs in nature and their relationships are much more complex than that of the two species (see [9, 10]). Therefore, it is more realistic to study the multiple species predator-prey systems. Motivated by above, we consider the following systems: (2)dy1=y1a1-b1y1-c2y21+y2-c3z1+my1+nz+ky1zdt,dy2=y2a2-b2y2-c1y11+y1-c4z1+m1y22dt,dz=ya3-b3z+d1y11+my1+nz+ky1z+d2y21+m1y22dt,where y1,y2 and z denote two competed prey and predator densities, respectively. And the parameters a1, a2, and a3 are the intrinsic growth of three species; b1, b2, and b3 are the intraspecific competition rate of three species, respectively. c1 and c2 are the interspecific competition rates of two competed species, c3 and c4 are the predators’ capturing rates, and d1 and d2 are the rates of conversion of nutrients into the production of predator. And all the parameters in system (2) are constants. It is very necessary to point out that c3z/1+my1+nz+ky1z is a special functional response; when m=n=k=0 it becomes a linear mass-action function response (or Holling type I functional response), when n=k=0 it becomes a Holling type II functional response, when k=0 it becomes a modified Holling type II functional response, and when mn=k it becomes a Crowley-Martin functional response.In the real world, population dynamics are often affected by white noise from the environment, which relate to climate, geographical distribution, geological features, human disaster, human intervention, and other environmental factors. Therefore, the flow of biological energy is a process of fluctuation. The oscillation of population biomass is directly related to the birth and death rate of random perturbation. Up to now, there have been many works considering the effect of random perturbation (see [11–13]). In this paper, we assume that white noise affects the intrinsic birth rate, capture rate of predator, and conversion rate of the predator population. On the other hand, the development trend of the real biological system is not only related to the status of the system, but also depends on the history of the system more or less, which is called time delays. Moreover, time delay widely exists in biological systems; for example, in the predator-prey system, the process for the conversion of prey to predators is not immediately translated into the predator population but after a certain period of time to digest the transformation. So a more realistic predator-prey model should consider the effects of time delays (see [14, 15]). As a matter of fact, delay differential systems have much more complicated dynamical behaviors than the differential equations without delays. Therefore, the following stochastic delay systems are considered: (3)dy1=y1a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1zdt+σ1y1dB1t+σ2y1z1+my1+nz+ky1zdB2t,dy2=y2a2-b2y2-c1y1t-τ11+y1t-τ1-c4z1+m1y22dt+σ3y2dB3t,dz=za3-b3z+d1y1t-τ31+my1t-τ3+nzt-τ3+ky1t-τ3zt-τ3+d2y2t-τ31+m1y22t-τ3dt+σ4zdB4t+σ5y1z1+my1+nz+ky1zdB5t.σi2 are the intensities of the noises, i=1,2,3,4,5. Bi(t) are standard Brownian motions which are defined on a complete probability space (Ω,F,P),i=1,2,3,4,5. Let τ=max{τ1,τ2,τ3} and C=C([-τ,0],R+3) be the set of continuous functions from [-τ,0] to R+3 with initial condition γ∈C[-τ,0],R+3 and the norm γ=sup-τ≤θ≤0ξ(θ)<+∞.Any biological system, whether it is population, biological communities, ecosystems, or the biosphere, its dynamic behavior is one of the main objects of the study, such as the resistance of ecosystem, persistence, recoverability, variability, and consistency. Therefore, we use mathematical theory and methods to study the dynamics of biological populations, which can not only protect the ecological balance but also can improve the ecological environment for human survival. According to what we know, few current literatures are found to discuss stochastic delayed predator-prey systems with Holling type IV and Crowley-Martin type functional responses at the same time. In this paper, it is the first time to obtain the condition of global asymptotic stability of system (3).This paper is carried out as follows. In Section2 and in Section 3, global positive solution and stochastically ultimate boundedness of system (3) are investigated. In Sections 4 and 5, we study the stochastic permanence and extinction, respectively. In Section 6, we obtain the fact that system (3) is globally asymptotically stable. In the end, in Section 7, some numerical examples are provided to explain our findings.


## 2. Existence of Global Positive Solution

Theorem 1.
For arbitrary initial dataγ∈C[-τ,0],R+3, system (3) has a unique positive solution on τ≥0 which will remain in R+3 with probability 1.Proof.
Define aC3-function V1→R+ by (4)V1y1,y2,z=y1-1-ln⁡y1+y2-1-ln⁡y2+z-1-ln⁡z.Clearly, V1(y1,y2,z) is nonnegative when y1>0,y2>0,z>0. Now we continue to define (5)V2y1,y2,z=V1y1,y2,z+c1∫t-τ1ty1sds+c2∫t-τ2ty2sds,where c1,c2 are the same in system (3). By the same method in [13], we can complete the proof and omit it here.


## 3. Stochastically Ultimate Boundedness

Stochastically ultimate boundedness of system (3) is studied in this part. Firstly, we present a useful lemma in the following.Lemma 2.
For any initial dataγ∈C[-τ,0],R+3, y1t,y2t,zt is a positive solution of system (3); there exist three positive constants K1p, K2p, and K3p, p>1, which satisfy (6)limt→∞Ey1tp=K1p,limt→∞Ey2tp=K2p,limt→∞Eztp=K3p.Proof.
We defineV1y1=y1p,V2y2=y2p, and V3z=zp. By Itô’s formula, we have (7)dV1=py1p-1dy1+12pp-1y1p-2dy12=py1pdy1a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1z+12p-1σ12+σ22z221+my1+nz+ky1z2dt+σ1py1pdB1t+σ2py1pz1+my1+nz+ky1zdB2t⩽py1pa1-b1y1+12p-1+p-1σ222n2+σ1py1pdB1t+σ2py1pz1+my1+nz+ky1zdB2t.Also, we can obtain that (8)dV2=py2p-1dy2+12pp-1y2p-2dy22=py2pa2-b2y2-c1y1t-τ11+y1t-τ1-c4z1+m1y22+σ3py2pdB3t⩽py2pa2-b2y2+12p-1σ32dt+py2pσ3dB3t,(9)dV3=pzp-1dz+12pp-1zp-2dy2=pzpa3-b3z+d1y1t-τ31+my1t-τ3+nzt-τ3+ky1t-τ3zt-τ3+d2y2t-τ31+m1y22t-τ3+12p-1σ42+σ52y1221+my1+nz+ky1z2dt+σ4pypdB4t+σ5x1yp1+mx1+nz+kx1zdB5t⩽pzpa3+d1m+d222+12m1-b3z+12p-1σ42+σ52m2dt+σ4pzpdB4t+σ5y1zp1+my1+nz+ky1zdB5t.By taking expectation in both sides of inequality (9), we have that (10)dEy1ptdt⩽pa1+p-1σ122+p-1σ222n2Ey1pt-b1pEy1p+1t⩽pa1+p-1σ122+p-1σ222n2Ey1pt-b1pEy1ptp+1/p⩽pEy1pta1+p-1σ122+p-1σ222n2-b1Ey1pt1/p.By the same way, we can obtain that (11)dEy2ptdt⩽pEy2pta2+p-1σ322-b2Ey2pt1/p,(12)dEzptdt⩽pEzpta3+d1m+d222+12m1+12p-1σ42+σ52m2-b3Ezpt1/p.Therefore, we have (13)limn→∞Ey1tp=K1p,limn→∞Ey2tp=K2p,limn→∞Eztp=K3p,where (14)K1p=a1+p-1σ12/2+p-1σ22/2n2b1p,K2p=a2+p-1σ32/2b2p,K3p=a3+d1/m+d22/2+1/2m1+1/2p-1σ42+σ52/m2b3p.This completes the proof.Theorem 3.
For arbitrary initial dataγ∈C[-τ,0],R+3, by the definition of stochastic boundedness, one has the fact that system (3) is stochastically ultimately bounded.Proof.
From Lemma2, we can see that E|y1(t),y2(t),z(t)|p⩽K(p). For any ϵ∈(0,1), let G=[K(p)/ϵ]1/p,p>1; then using Chebyshev inequality, we can obtain that (15)Py1t,y2t,zt>G<Ey1t,y2t,ztpGp⩽ϵ,where K(p)=3p/2(K1(p)+K2(p)+K3(p)).
This completes the proof of Theorem3.


## 4. Stochastic Permanence

Theorem 4.
If(16)12maxσ12+σ22+σ52n2,σ32,σ42+σ22+σ52m2<mina1-c2,a2-c1,a3-c3mholds, by the definition of stochastic permanence, we say that system (3) is stochastically permanent.Proof.
DefineV(y1,y2,z)=y1+y2+z for (y1(t),y2(t),z(t))∈R+3; then (17)dVy1,y2,z=y1a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1zdt+y2a2-b2y2-c1y1t-τ11+y1t-τ1-c4z1+m1y22dt+za3-b3z+d1y1t-τ31+my1t-τ3+nzt-τ3+ky1t-τ3zt-τ3+d2y2t-τ31+m1y22t-τ3dt+σ1y1dB1t+σ3y2dB3t+σ4zdB4t+y1z1+my1+nz+ky1zσ2dB2t+σ5dB5t.Let W(y1,y2,z)=1/V(y1,y2,z); using Itô’s formula, we can obtain(18)dW=-1V2dV+1V3dV=-W2y1a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1z+y2a2-b2y2-c1y1t-τ11+y1t-τ1-c4z1+m1y22+za3-b3z+d1y1t-τ31+my1t-τ3+nzt-τ3+kx1t-τ3zt-τ3+d2y2t-τ31+m1y22t-τ3dt+W3σ12y12+σ32y22+σ42z2+σ22y12z21+my1+nz+ky1z2+σ52y12z21+my1+nz+ky1z2dt-W2σ1y1dB1t+σ3y2dB3t+σ4zdB4t+y1z1+my1+nz+ky1zσ2dB2t+σ5dB5t=LWdt-W2σ1y1dB1t+σ3y2dB3t+σ4zdB4t+y1z1+my1+nz+ky1zσ2dB2t+σ5dB5t,where (19)LW=-W2y1a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1z+y2a2-b2y2-c1y1t-τ11+y1t-τ1-c4z1+m1y22+za3-b3z+d1y1t-τ31+my1t-τ3+nzt-τ3+kx1t-τ3zt-τ3+d2y2t-τ31+m1y22t-τ3+W3σ12y12+σ32y22+σ42z2+σ22+σ52y12z21+my1+nz+ky1z2.If (16) holds, we can find a positive constant μ which satisfies the following condition: (20)1+μ2maxσ12+σ22+σ52n2,σ32,σ42+σ22+σ52m2<mina1-c2,a2-c1,a3-c3m.Using Itô’s formula, we can obtain(21)d1+Wμ=μ1+Wμ-1LW+μμ-12W41+Wμ-2σ12y12+σ32y22+σ42z2+σ22+σ52y12z21+my1+nz+ky1z2dt+μW21+Wμ-1σ1y1dB1t+σ3y2dB3t+σ4zdB4t+y1z1+my1+nz+ky1zσ2dB2t+σ5dB5t.Hence, (22)L1+Wμ=μ1+Wμ-1LW+μμ-12W41+Wμ-2σ12y12+σ32y22+σ42z2+σ22+σ52y12z21+mx1+nz+kx1z2.Under the condition of (20), we can choose another positive constant k making it satisfy the following condition: (23)μ1+μ2maxσ12+σ22+σ52n2,σ32,σ42+σ22+σ52m2<mina1-c2,a2-c1,a3-c3m-k.Using Itô’s formula, we can obtain (24)dekt1+Wμ=ektd1+Wμ+kekt1+Wμdt.Hence, (25)Lekt1+Wμ=ektLW+kekt1+Wμ+μμ-12W41+Wμ-2σ12y12+σ32y22+σ42z2+σ22+σ52y12z21+my1+nz+ky1z2⩽kekt1+Wμ-2k+μmax⁡b1,b2+c42m1,c42+b3+2k-μmin⁡a1-c2,a2-c1,a3-c3m+μmax⁡σ12+σ22+σ52n2,σ32,σ42+σ22+σ52m2+μmax⁡b1,b2+c42m1,c42+b3W+k-μmin⁡a1-c2,a2-c1,a3-c3m+μ1+μ2maxσ12+σ22+σ52n2,σ32,σ42+σ22+σ52m2W2.Obviously, we can find a positive constant K which satisfies (26)Lekt1+Wμ⩽Kekt.Then (27)dekt1+Wμ⩽Kekt-ektμW21+Wμ-1σ1y1dB1t+σ3y2dB3t+σ4zdB4t+y1z1+my1+nz+ky1zσ2dB2t+σ5dB5t.We can integrate above inequality and then take expectation (28)Eekt1+Wμ⩽1+W0μ+Kkekt=1+W0μ+M1ekt,where M1=K/k, so (29)limt→+∞sup⁡EWtμ⩽limt→+∞sup⁡E1+Wtμ⩽M1.Since (y1(t)+y2(t)+z(t))μ⩽3μ(y1(t)+y2(t)+z(t))μ/2=3μ|y1(t),y2(t),z(t)|, where (y1(t),y2(t),z(t))∈R+3, therefore (30)limt→+∞sup⁡E1y1t,y2t,ztμ⩽3μlimt→+∞sup⁡EWtμ⩽3μM1.From Theorem 3, we have (31)limt→+∞sup⁡Ey1t,y2t,ztp⩽Kp,for any ε>0; let χ=(K(p)/ε)1/p, using Chebyshev’s inequality, we can obtain the conclusion.


## 5. Extinction

The extinction of system (3) will be investigated in this part.Definition 5.
For arbitrary initial dataγ∈C[-τ,0],R+3, if (32)limt→+∞supln⁡y1tt<0,limt→+∞supln⁡y2tt<0,limt→+∞supln⁡ztt<0hold, then system (3) is extinct.Theorem 6.
For arbitrary initial dataγ∈C[-τ,0],R+3, assume that (33)a1-σ122<0,a2-σ322<0,a3+d1m+d222+12m1-σ422<0holds; by the definition of extinction, system (3) is said to be extinct.Proof.
We establish three Lyapunov functions as follows:(34)V1y1t=ln⁡y1t,V2y2t=ln⁡y2t,V3zt=ln⁡zt.Making use of the generalized Itô’s formula results in (35)dV1y1=dy1y1-dy122y12=a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1z-12σ12+σ22z21+my1+nz+ky1z2dt+σ1dB1t+σ2z1+my1+nz+ky1zdB2t,dV2y2=dy2y2-dy222y22=a2-b2y2-c1y1t-τ11+y1t-τ1-c4z1+m1y22-σ322dt+σ3dB3t,dV3z=dzz-dz22z2=a3-b3z+d2y2t-τ31+m1y22t-τ3+d1y1t-τ31+my1t-τ3+nzt-τ3+ky1t-τ3zt-τ3-12σ42+σ52y121+my1+nz+ky1z2dt+σ4dB4t+σ5y11+my1+nz+ky1zdB5t.Hence, we can integrate them from 0 to t: (36)ln⁡y1t=ln⁡y10+∫0ta1-b1y1s-c2y2s-τ21+y2s-τ2-c3zs1+my1s+nzs+ky1szs-12σ12+σ22z2s1+my1s+nzs+ky1szs2ds+∫0tσ1dB1s+M1t,ln⁡y2t=ln⁡y20+∫0ta2-b2y2s-c1y1s-τ11+y1s-τ1-c4zs1+m1y22s-σ322ds+∫0tσ3dB3s,ln⁡zt=ln⁡z0+∫0ta3-b3zs+d2y2s-τ31+m1y22s-τ3+d1y1s-τ31+my1s-τ3+nzs-τ3+ky1s-τ3zs-τ3-12σ42+σ52y12s1+my1s+nzs+ky1szs2ds+∫0tσ4dB4s+M2t,where M1t=∫0t‍σ2zs/1+my1s+nzs+ky1szsdB2s and   M2t=∫0t‍σ5y1s/1+my1s+nzs+ky1sztdB5s are martingales. And (37)M1t,M1t=∫0tσ2ys1+my1s+nys+ky1szs2ds,M2t,M2t=∫0tσ5y1s1+my1s+nzs+ky1szs2dsare the quadratic variations, respectively. Using the exponential martingale inequality, for arbitrary positive constants N,η, and υ, we can obtain (38)Psup0⩽t⩽NMt-12Mt,Mt>β=e-ηυ.Choosing N=n,η=1, and υ=2ln⁡n, we have that (39)Psup0⩽t⩽nM1t-12M1t,M1t>2ln⁡n⩽1n2,Psup0⩽t⩽nM2t-12M2t,M2t>2ln⁡n⩽1n2.Applying the Borel-Cantelli lemma, there exists a random integer n0=n0(ω) for almost all ω∈Ω such that (40)sup0⩽t⩽nM1t-12M1t,M1t⩽2ln⁡n,sup0⩽t⩽nM2t-12M2t,M2t⩽2ln⁡n,for n⩾n0.
Thus(41)M1t⩽2ln⁡n+12∫0tσ2zs1+my1s+nzs+ky1szs2ds,M2t⩽2ln⁡n+12∫0tσ5y1s1+my1s+nzs+ky1szs2ds,for all 0⩽t⩽n,n⩾n0 a.s. We substitute it into (36) (42)ln⁡y1t-ln⁡y10⩽a1-12σ12t+∫0tσ1dB1s+2ln⁡n,ln⁡y2t-ln⁡y20⩽a2-12σ32t+∫0tσ3dB3s,ln⁡zt-ln⁡z0⩽a3+d1m+d222+12m1-σ422t+∫0tσ4dB4s+2ln⁡n.We divide t on both sides of (42) and let t→+∞: (43)limt→+∞supln⁡y1tt⩽a1-σ122<0,limt→+∞supln⁡y2tt⩽a2-σ322<0,limt→+∞supln⁡ztt⩽a3+d1m+d222+12m1-σ422<0.The desired assertion is derived.


## 6. Global Asymptotic Stability

Definition 7.
Let(y1(t),y2(t),z(t)) and (y1∗(t),y2∗(t),z∗(t)) be two arbitrary solutions of system (3) with initial data γ,γ∗∈([-τ,0],R+3), respectively. If (44)Plimt→+∞Ey1t,y2t,zt-y1∗t,y2∗t,z∗t=0=1,we say that system (3) is globally asymptotically stable in expectation.Lemma 8.
For any initial dataγ∈C[-τ,0],R+3, (y1(t),y2(t),z(t)) is a solution of system (3); then almost every sample track of (y1(t),y2(t),z(t)) is uniformly continuous for t⩾-τ.Proof.
Considering system (3), we have that (45)y1t=y10+∫0tf1sds+∫0tg1sdB1s+∫0tg2sdB2s, where (46)f1s=y1sa1-b1y1s-c2y2s-τ21+y2s-τ2-c3zs1+my1s+nzs+ky1szs,g1s=σ1y1s,g2s=σ2y1szs1+my1s+nzs+ky1szs. Letting ϑ>2, we obtain that(47)Ef1tϑ=Ey1a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1zϑ⩽12Ey12ϑ+1242ϑ-1a12ϑ+b12ϑEy12ϑ+c22ϑ∣Ey2t-τ22ϑ+c32ϑEz2ϑ⩽12K12ϑ+1242ϑ-1a12ϑ+b12ϑK12ϑ+c22ϑK22ϑ+c32ϑK32ϑ≜L1ϑ,Eg1sϑ=Eσ1ϑy1ϑ=σ1ϑEy1ϑ⩽σ1pK1ϑ≜L2ϑ,Eg2sϑ=Eσ2y1z1+my1+nz+ky1zϑ⩽Eσ2y1nϑ⩽σ2nϑK1ϑ≜L3ϑ. For another, using the moment inequality of stochastic integral (see [16]), for 0⩽t1<t2<+∞ and ϑ>2, we have that (48)E∫t1t2g1sdB1sϑ⩽ϑϑ-12ϑ/2t2-t1ϑ-2/2∫t1t2Eg1sϑds⩽ϑϑ-12ϑ/2t2-t1ϑ/2L2ϑ,(49)E∫t1t2g2sdB2sϑ⩽ϑϑ-12ϑ/2t2-t1ϑ-2/2∫t1t2Eg2sϑds⩽ϑϑ-12ϑ/2t2-t1ϑ/2L3ϑ. Thus for t2-t1<1 and 1/ϑ+1/q=1, we can obtain (50)Ey1t2-y1t1ϑ=E∫t1t2f1sds+∫t1t2g1sdB1s+∫t1t2g2sdB2sϑ⩽3ϑ-1E∫t1t2f1sdsϑ+E∫t1t2g1sdB1sϑ+E∫t1t2g1sdB1sϑ⩽3ϑ-11+ϑϑ-12ϑ/2L4ϑt2-t1ϑ/2, where L4(ϑ)=max{L1(ϑ),L2(ϑ)+L3(ϑ)}. From Lemma 2.3 of [13], we have that almost every sample path of y1(t) is uniformly continuous. In the same method, the uniform continuity of y2(t) and z(t) can be obtained.Lemma 9 (see [17]).
g ( t ) ⩾ 0 is a uniformly continuous and integrable function defined on [0,+∞); then limt→∞g(t)=0.Theorem 10.
If(51)A≔-b1+c1+2d1+2σ52m+mn+kc3n2+mn+kσ22n3<0,B≔-b2+c2+3d22+2c4m1a2+σ32a3+d1/m+d22/2+1/2m1+σ42+σ52/m2b2b3<0,C≔-b3+2c3+2c4+2σ22n+mn+kd1m2+mn+kσ52m3<0,then system (3) is globally asymptotically stable.Proof.
( y 1 ( t ) , y 2 ( t ) , z ( t ) ) and (y1∗(t),y2∗(t),z∗(t)) are two arbitrary solutions of system (3) with initial data γ,γ∗∈([-τ,0],R+3), respectively. Define (52)Vt=ln⁡y1-ln⁡y1∗+ln⁡y2-ln⁡y2∗+ln⁡z-ln⁡z∗+c1∫t-τ1ty1s-y1∗sds+c2∫t-τ2ty2s-y2∗sds+2d1∫t-τ3ty1s-y1∗sds+3d22∫t-τ3ty2s-y2∗sds+mn+kd1m2∫t-τ3tzs-z∗sds.Calculating the right differential d+V(t), by Itô’s formula(53)d+V=sgny1-y1∗a1-b1y1-c2y2t-τ21+y2t-τ2-c3z1+my1+nz+ky1z-σ22z221+my1+nz+ky1z2dt+σ1dB1t+σ2z1+my1+nz+ky1zdB2t-a1-b1y1∗-c2y2∗t-τ21+y2∗t-τ2-c3z∗1+my1∗+nz∗+ky1∗z∗-σ22z∗221+my1∗+nz∗+ky1∗z∗2dt+σ1dB1t+σ2z∗1+my1∗+nz∗+ky1∗z∗dB2t+sgny2-y2∗a2-b2y2-c1y1t-τ11+y1t-τ1-c4y1+m1y22dt+σ3dB3t-a2-b2y2∗-c1y1∗t-τ11+y1∗t-τ1-c4z∗1+m1y2∗2dt+σ3dB3t+sgnz-z∗a3-b3z+d2y2t-τ31+m1y22t-τ3+d1y1t-τ31+my1t-τ3+nzt-τ3+ky1t-τ3zt-τ3-σ52y1221+my1+nz+ky1z2dt+σ4dB4t+σ5y11+my1+nz+ky1zdB5t-a3-b3z∗+d2y2∗t-τ31+m1y2∗2t-τ3+d1y1∗t-τ31+my1∗t-τ3+nz∗t-τ3+ky1∗t-τ3z∗t-τ3-σ52y1∗221+my1∗+nz∗+ky1∗z∗2dt+σ4dB4t+σ5y1∗1+my1∗+nz∗+ky1∗z∗dB5t+c1y1-y1∗-y1t-τ1-y1∗t-τ1+c2y2-y2∗-y2t-τ2-y2∗t-τ2+2d1y1-y1∗-y1t-τ3-y1∗t-τ3+3d22y2-y2∗-y2t-τ3-y2∗t-τ3+mn+kd1m2z-z∗-zt-τ3-z∗t-τ3.We can take integration from 0 to t and then take expectation(54)EVt-EV0=E∫0tsgny1s-y1∗s-b1y1s-y1∗sy1∗s21+my1∗s+nz∗s+ky1∗sz∗s2-c2y2s-τ21+y2s-τ2-y2∗s-τ21+y2∗s-τ2-c3zs1+my1s+nzs+ky1szs-z∗s1+my1∗s+nz∗s+ky1∗sz∗s-σ222z2s1+my1s+nzs+ky1szs2-z∗s21+my1∗s+nz∗s+ky1∗sz∗s2+sgny2s-y2∗szs1+m1y22s-z∗s1+m1y2∗s2-b2y2s-y2∗s-c1y1s-τ11+y1s-τ1-y1∗s-τ11+y1∗s-τ1-c4zs1+m1y22s-z∗s1+m1y2∗s2+sgnzs-z∗s-y1∗s21+my1∗s+nz∗s+ky1∗sz∗s2-b3zs-z∗s+d1y1s-τ31+my1s-τ3+nzs-τ3+ky1s-τ3zs-τ3-y1∗s-τ31+my1∗s-τ3+nz∗s-τ3+ky1∗s-τ3z∗s-τ3+d2y2s-τ31+m1y22s-τ3-y2∗s-τ31+m1y2∗2s-τ3-σ522y12s1+my1s+nzs+ky1szs2-y1∗s21+my1∗s+nz∗s+ky1∗sz∗s2+c1y1s-y1∗s-y1s-τ1-y1∗s-τ1+c2y2s-y2∗s-y2s-τ2-y2∗s-τ2+2d1y1s-y1∗s-y1s-τ3-y1∗s-τ3+3d22y2s-y2∗s-y2s-τ3-y2∗s-τ3+mn+kd1m2·zs-z∗s-zs-τ3-z∗s-τ3-y1∗s21+my1∗s+nz∗s+ky1∗sz∗s2ds.Hence, (55)dEVtdt⩽-b1+c1+2d1+2σ52m+mn+kc3n2+mn+kσ22n3Ey1-y1∗+-b2+c2+3d22+c4m1Ez31/3Ey231/3+Ez31/3Ey2∗31/3Ey2-y2∗+-b3+2c3+2c4+2σ22n+mn+kd1m2+mn+kσ52m3Ez-z∗.By Lemma 2, we have (56)Ey1t31/3⩽a1+σ12+σ22/n2b1,Ey2t31/3⩽a2+σ32b2,Ezt31/3⩽a3+d1/m+d22/2+1/2m1+σ42+σ52/m2b3.Thus (57)dEVtdt⩽-b1+c1+2d1+2σ52m+mn+kc3n2+mn+kσ22n3Ey1t-y1∗t+-b2+c2+3d22+2c4m1a2+σ32a3+σ42+d1/m+d22/2+1/2m1+σ52/m2b2b3Ey2t-y2∗t+-b3+2c3+2c4+2σ22n+mn+kdm2+mn+kσ52m3Ezt-z∗t=AEy1t-y1∗t+BEy2t-y2∗t+CEzt-z∗t.Integrating both sides of (57) yields that(58)Vt-V0⩽A∫0tEy1s-y1∗sds+B∫0tEy2s-y2∗sds+C∫0tEzs-z∗sds.Therefore (59)Vt-A∫0tEy1s-y1∗sds+B∫0tEy2s-y2∗sds+C∫0tEzs-z∗sds⩽V0<+∞.Then we have (60)Ey1t,y2t,zt-y1∗t,y2∗t,z∗t⩽Ey1t-y1∗t+y2t-y2∗t+zt-z∗t∈L10,+∞.Thus it follows from Lemma 9 that(61)limt→∞y1t-y1∗t=0,limt→∞y2t-y2∗t=0,limt→∞zt-z∗t=0.


## 7. Numerical Simulation

We give some illustrative examples to evidence our results in this part. Using the Milstein method mentioned in ([18]), we get the discretization equations of system (3):(62)y1,i+1=y1,i+y1,ia1-b1y1,i-c2y2,i-s21+y2,i-s2-c3zi1+my1,i+nzi+ky1,iziΔt+σ1y1,iΔtη1,i+σ2y1,izi1+my1,i+nzi+ky1,iziΔtη2,i+σ122y1,iη1,i2-1Δt+σ22y1,izi21+nzi21+my1,i+nzi+ky1,izi3η2,i2-1Δt,y2,i+1=y2,i+y2,ia2-b2y2,i-c1y1,i-s11+y1,i-s1-c4zi1+m1y2,i2Δt+σ3y2,iΔtη3,i+σ322y2,iη3,i2-1Δt,zi+1=zi+zia3-b3zi+d1y1,i-s31+my1,i-s3+nzi-s3+ky1,i-s3zi-s3+d2y2,i-s31+m1y2,i-s32Δt+σ4ziΔtη4,i+σ5y1,izi1+my1,i+nzi+ky1,iziΔtη5,i+σ422ziη4,i2-1Δt+σ52y1,i2zi1+m1y1,i21+my1,i+nzi+ky1,izi3η5,i2-1Δt,where η1,i,η2,i,η3,i,η4,i, and η5,i are independent Gaussian random variables which follow N(0,1). Let step size Δt=0.01 and steps = 20000. We choose (63)a1=0.4;b1=0.5;c2=0.2;c3=0.2;m=1;n=0.8;k=0.1;a2=0.5;b2=0.55;c1=0.1;c4=0.1;m1=1;a3=0.6;b3=0.7;d1=0.02;d2=0.01;τ1=0.1;τ2=0.2;τ3=0.3.We assume that the parameters are the same above in the following discussion. Let the initial data be (64)γ1θ=1.5,1.3,1,-τ⩽θ⩽0,where τ=max{τ1,τ2,τ3}. We choose σ1=σ2=σ3=σ4=σ5=0.1; from Theorem 3 we know that system (62) is stochastically ultimately bounded (see Figures 1–3: blue curves). To understand the influence of white noise, we choose σ1=σ2=σ3=σ4=σ5=0; we obtain deterministic system of system (62) and it is also ultimately bounded (see Figures 1–3: red curves).Figure 1
Solution of model (62) for γ1(θ)=(1.5,1.3,1). Blue: σi=0.1(i=1,2,3,4,5). Red: σi=0(i=1,2,3,4,5).Figure 2
Solution of model (62) for γ1(θ)=(1.5,1.3,1). Blue: σi=0.1(i=1,2,3,4,5). Red: σi=0(i=1,2,3,4,5).Figure 3
Solution of model (62) for γ1(θ)=(1.5,1.3,1). Blue: σi=0.1(i=1,2,3,4,5). Red: σi=0(i=1,2,3,4,5).For stochastic permanence, we also chooseσ1=σ2=σ3=σ4=σ5=0.1; by calculating we obtain that (65)min⁡a1-c2,a2-c1,a3-c3m-12max⁡σ12+σ22+σ52n2,σ32,σ42+σ22+σ52m2=2871600,which satisfies the condition (16) (see Figure 4).Figure 4
Solution of model (62) for γ1(θ)=(1.5,1.3,1) and σi=0.1(i=1,2,3,4,5).In the following we choose larger noisesσ1=0.95,σ2=0.1,σ3=1.05,σ4=1.5, and σ5=0.1, which satisfy the condition of Theorem 6; then system (62) will go to extinction (see Figure 5).Figure 5
Solution of model (62) for γ1(θ)=(1.5,1.3,1) and σ1=0.95,σ2=0.1,σ3=1.05,σ4=1.5, and σ5=0.1.In the end, we discuss the global asymptotic stability. Let(66)γ1∗=0.7,0.8,0.6,-τ⩽θ⩽0;we also choose σ1=σ2=σ3=σ4=σ5=0.1; by calculating we have that (67)A≔-b1+c1+2d1+2σ52m+mn+kc3n2+mn+kσ22n3=-52712800,B≔-b2+c2+3d22+2c4m1a2+σ32a3+d1/m+d22/2+1/2m1+σ42+σ52/m2b2b3=-1745279,C≔-b3+2c3+2c4+2σ22n+mn+kd1m2+mn+kσ52m3=-6125,which satisfy the condition of Theorem 10; then system (62) is globally asymptotically stable. By the same way, we can obtain the fact that the deterministic system of (3) is also globally stable (see Figures 6–8).Figure 6
The path curve ofy1(t) and y1∗(t). Red: solution of model (62) for γ1(θ)=(1.5,1.3,1). Green: solution of model (62) for γ1∗=(0.7,0.8,0.6).Figure 7
The path curve ofy2(t) and y2∗(t). Red: solution of model (62) for γ1(θ)=(1.5,1.3,1). Green: solution of model (62) for γ1∗=(0.7,0.8,0.6).Figure 8
The path curve ofz(t) and z∗(t). Red: solution of model (62) for γ1(θ)=(1.5,1.3,1). Green: solution of model (62) for γ1∗=(0.7,0.8,0.6).


## 8. Conclusions and Discussion

In this work, stochastic delayed one-predator and two-competing-prey systems with two different kinds of functional responses have been studied. Globally positive solution, stochastically ultimate boundedness, and the stochastic permanence and extinction for system (3) are investigated. Moreover, sufficient criteria for the global asymptotic stability of the system are established. In the end, some numerical examples are provided to explain our results. Through the study of the dynamic behavior of system (3), by comparing Theorems 4 and 6, we can obtain that if the environment noise is small, the stochastic system can maintain permanent while the system can be extinct under sufficiently large environmental noise (see Figures 4 and 5). Therefore, from Theorem 10, we can choose the appropriate parameters in a suitable environment noise intensity to make system (3) asymptotically stable.Recently, predator-prey models have been investigated extensively for their theoretical and practical significance. In the current literatures, most of this work is restricted to two-dimensional predator-prey system; few has been done on predator-prey system with interspecific competition in preys. In system (3), we study a three-dimensional hybrid system where the predator can capture two kinds of preys with different functional responses. In fact, the population models are often subjected to the influence of environmental noises inevitably. In most of predator-prey models, there is only a white noise which affects intrinsic rate of increase of predator or prey. In system (3), we consider the white noise not only has effect on the intrinsic rate of population growth but also on capture rate of predator and conversion rate of the predator population. Three time delays are introduced to make the model closer to the reality. It is interesting to point out that system (3) contains two kinds of different mathematical models; if we remove the predator, system (3) is reduced to a competitive model in [13]; if we remove any one of the two preys system (3) is reduced to a simple predator-prey system. Some meaningful questions deserve further investigation. One may investigate the stationary distribution of system (3). Moreover, it is worth considering the corresponding nonautonomous system of system (3). One may discuss dynamics behaviors contained predators with a mutual cooperation in other ecosystems.



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