# The Principle of Competitive Exclusion about a Stochastic Lotka-Volterra Model with Two Predators Competing for One Prey

Authors: Zhongwei Cao; Qiumei Zhang; Yanan Zhao
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

This paper concerns a stochastic Lotka-Volterra model with two predators competing for one prey. The sufficient conditions which guarantee the principle of competitive exclusion for this perturbed model are given by using Lyapunov analysis methods. Numerical simulations for a set of parameter values are presented to illustrate the analytical findings.


---

## Body

## 1. Introduction

Volterra has argued that the coexistence of two or more predators competing for fewer prey resources is impossible, which was later known as the principle of competitive exclusion. The principle of competitive exclusion was reexamined by Koch [1] in 1974 who found via numerical simulation that the coexistence of two predators competing exploitatively for a single prey species in a constant and uniform environment was in fact possible when the predator functional response to the prey density was assumed according to nonlinear function, and such coexistence occurred along what appeared to be a periodic orbit in the positive octant of R3 rather than an equilibrium. The similar themes were discussed in [2–8]. The authors in [6] studied the global dynamics of 3-dimensional Lotka-Volterra models with two predators competing for a single prey species in a constant and uniform environment. They obtained sufficient and necessary conditions for the principle of competitive exclusion to hold and gave the global dynamical behavior of the three species. They assumed that the two predator species compete purely exploitatively with no interference between rivals, the growth rate of the prey species is logistic or linear in the absence of predation respectively, and the predator’s functional response is linear. Based on the above assumptions the model can be written as follows:(1)dStdt=Stb1-a11St-a12Xt-a13Yt,dXtdt=Xt-b2+b21St,dYtdt=Yt-b3+b31St,where S(t),X(t),Y(t) are the densities of the prey and the ith predator (i=1,2) population, respectively. b1 is the intrinsic rate of growth of the prey, and 1/a11 is the carrying capacity of the prey, which describes the richness of resources for prey. a12,a13 are the effects of the ith predation on the prey, b2,b3 are the natural death rates of the ith predator in the absence of prey, and b21,b31 are the efficiency and propagation rates of the ith predator in the presence of prey.The above discussion rests on the assumption that the environmental parameters involved with the model system are all constants irrespective to time and environmental fluctuations. We consider the effect of environmental fluctuation on the model system. There are two ways to develop the stochastic model corresponding to an existing deterministic one. Firstly, one can replace the environmental parameters involved with the deterministic model system by some random parameters; see [9, 10]. Secondly, one can add the randomly fluctuating driving force directly to the deterministic growth equations of prey and predator populations without altering any particular parameter; see [11–14]. In the present study we follow the second method. To incorporate the effect of randomly fluctuating environment, we introduce stochastic perturbation terms in the growth equations of both prey and predator population:(2)dSt=Stb1-a11St-a12Xt-a13Ytdt+σ1StdB1t,dXt=Xt-b2+b21Stdt+σ2XtdB2t,dYt=Yt-b3+b31Stdt+σ3YtdB3t,where B1(t),B2(t),B3(t) are independent Brownian motions defined on a complete probability space (Ω,F,Ftt≥0,P) with a filtration Ftt≥0 satisfying the usual conditions, and σ12,σ22,σ32 are the intensities of environmental white noise.The aim is to study the dynamics of 3-dimensional Lotka-Volterra models with two predators competing for a single prey species by stochastic perturbation. Zhang and Jiang [15] give the sufficient conditions which guarantee that the principle of coexistence holds for this perturbed model via Markov semigroup theory. Furthermore, they prove that the densities of the solution can converge in L1 to an invariant density under appropriate conditions. In this paper, we study the principle of competitive exclusion associated with system (2). The paper is organized as follows. In Section 2, we give the sufficient conditions to guarantee that the principle of competitive exclusion holds for system (2). We make simulations to illustrate our analytical results in Section 3.Remark 1.
In [15] (Theorem 2.1), Zhang and Jiang show that, for any initial value S0,X0,Y0∈R+3, system (2) has a unique positive solution (S(t),X(t),Y(t))  a.s.


## 2. The Principle of Competitive Exclusion for System (2)

In this section, we show that system (2) allows the competitive exclusion of two competing predators for some values of parameters, which implies that the competitive exclusion of two predators competing for a single prey species is possible when the predator functional response to the prey density is linear.For simplicity, define(3)xt=1t∫0txrdr.Lemma 2.
Assume(b2∧b3)>1/2(σ22∨σ32). Then for any initial value (S(0),X(0),Y(0))∈R+3, the solution of system (2) has the property (4)limsupt→∞⁡log⁡St+a12/b21Xt+a13/b31Ytlog⁡t≤1θa.s.where θ is a positive constant that satisfies 2<θ<1+2b2∧b3/σ22∨σ32.Proof.
Let(5)Ut=St+a12b21Xt+a13b31Yt≔St+X1t+Y1t. ByItô’s formula and (2), we have(6)dUt=Stb1-a11St-b2X1t-b3Y1tdt+σ1StdB1t+σ2X1tdB2t+σ3Y1tdB3t. Define the function W by W(U)=(1+U)θ. Then(7)dWU=LWUdt+θ1+Utθ-1σ1StdB1t+σ2X1tdB2t+σ3Y1tdB3t,where (8)LWU=θ1+Uθ-1Sb1-a11S-b2X1-b3Y1+θθ-121+Uθ-2σ12S2+σ22X12+σ32Y12=θ1+Uθ-21+USb1-a11S-b2X1-b3Y1+θ-12σ12S2+σ22X12+σ32Y12≤θ1+Uθ-21+S+Zb1S-a11S2-b2∧b3Z+θ-12σ12S2+σ22∨σ32Z2)≤θ1+Uθ-2-a11S3+b1+θ-12σ12S2+b1SZ+b1S-b2∧b3-θ-12σ22∨σ32Z2; here Z=X1+Y1.
Now, choose a constantλ>0 sufficiently small such that it satisfies (9)b2∧b3-θ-12σ22∨σ32-3λθ≫0. ApplyingItô’s formula, we have (10)EeλtWUt=WU0+E∫0tLeλsWUsds, where (11)LeλtWU=λeλtWUt+eλtLWU≤θeλt1+Uθ-2-a11S3+b1+θ-12σ12S2+b1SZ+b1S-b2∧b3-θ-12σ22∨σ32Z2+λθ1+S+Z2≤θeλt1+Uθ-2-a11S3+b1+θ-12σ12+3λθS2+b1SZ+b1S+3λθ-b2∧b3-θ-12σ22∨σ32-3λθZ2≤θeλt1+Uθ-2-a11S3+cS2+b1S+3λθ-12b2∧b3-θ-12σ22∨σ32-3λθZ2≤θeλtH;here c≔b1+θ-1/2σ12+3λ/θ+1/2b12b2∧b3-θ-1/2σ22∨σ32-3λ/θ-1 and H≔sup(S,Z)∈R+21+S+Zθ-2{-a11S3+cS2+b1S+3λ/θ-1/2[(b2∧b3)-θ-1/2(σ22∨σ32)-3λ/θ]Z2}. This implies (12)Eeλt1+Uθ≤1+U0θ+θeλtHλ. Then (13)limsupt→∞⁡E1+Uθ≤θHλ. There exists a positive constant M such that (14)E1+Uθ≤M,t≥0. For sufficiently small δ>0, k=1,2,…, (7) implies that (15)Esupk-1δ≤t≤kδ⁡1+Utθ≤E1+Uk-1δθ+I1+I2≤M+I1+I2, where (16)I1=Esupk-1δ≤t≤kδ⁡∫k-1δtθ1+Usθ-2-a11S3s+b1+θ-12σ12S2s+b1SsZs+b1Ss-b2∧b3-θ-12σ22∨σ32Z2sds≤Esupk-1δ≤t≤kδ⁡∫k-1δtθ1+Usθ-2-a11S3s+b1+θ-12σ12+12b12b2∧b3-θ-12σ22∨σ32-1S2s+b1Ss-12b2∧b3-θ-12σ22∨σ32Z2sds≤θH0Esupk-1δ≤t≤kδ∫k-1δt1+Usθds≤θH0E∫k-1δkδ1+Usθds≤θH0δEsupk-1δ≤t≤kδ⁡1+Utθ; here H0≔sup(S,Z)∈R+2{-a11S3(s)  +  (b1+θ-1/2σ12  +  1/2b12b2∧b3-θ-1/2σ22∨σ32-1)S2(s)  +  b1S(s)  −  1/2[(b2∧b3)  −  θ-1/2(σ22∨σ32)]Z2(s)}, and(17)I2=Esupk-1δ≤t≤kδ⁡∫k-1δtθ1+Utθ-1σ1SsdB1s+σ2X1sdB2s+σ3Y1sdB3s≤32E∫k-1δkδθ21+Us2θ-1σ12S2s+σ22X12s+σ32Y12sds1/2≤32θσ12∨σ22∨σ321/2δ1/2Esupk-1δ≤t≤kδ⁡1+Ut2θ1/2≤32θσ12∨σ22∨σ321/2δ1/2Esupk-1δ≤t≤kδ⁡1+Utθ, according to the Burkholder-Davis-Gundy inequality. Therefore, (18)Esupk-1δ≤t≤kδ⁡1+Utθ≤E1+Uk-1δθ+θH0δ+32θσ12∨σ22∨σ321/2δ1/2Esupk-1δ≤t≤kδ⁡1+Utθ. In particular, choose δ>0 such that θH0δ+32θσ12∨σ22∨σ321/2δ1/2≤1/2, and so (19)Esupk-1δ≤t≤kδ⁡1+Utθ≤2E1+Uk-1δθ≤2M. Let ε>0 be arbitrary. Then, by the Chebyshev’s inequality, we have (20)Pω:supk-1δ≤t≤kδ⁡1+Utθ>kδ1+ε≤2Mkδ1+ε,k=1,2,… In view of the Borel-Cantelli lemma, we see that, for almost all ω∈Ω,(21)supk-1δ≤t≤kδ⁡1+Utθ≤kδ1+εholds for all but finitely many k. Hence there exists an integer k0(ω)>1/δ+2, for almost all ω∈Ω, for which (21) holds whenever k≥k0. Consequently, for almost all ω∈Ω, if k≥k0 and (k-1)δ≤t≤kδ, (22)log⁡1+Utθlog⁡t≤1+εlog⁡kδlog⁡k-1δ. Therefore, (23)limsupt→∞⁡log⁡1+Utθlog⁡t≤1+εa.s. Letting ε→0, we obtain the desired assertion (24)limsupt→∞⁡log⁡1+Utθlog⁡t≤1a.s. and (25)limsupt→∞⁡log⁡Utlog⁡t≤limsupt→∞⁡log⁡1+Utlog⁡t≤1θa.s.
Hence the proof of this lemma is complete.Theorem 3.
Let(S(t),X(t),Y(t)) be the solution of system (2) with initial value (S(0),X(0),Y(0))∈R+3. Assume (b2∧b3)>1/2(σ22∨σ32). The principle of competitive exclusion holds for system (2) if one of the following conditions holds.
(i) Ifb1>1/2σ12, b3/b31≥b2/b21, b3+1/2σ32/b31>b2/b21+Kσ/a111/2 and b1-1/2σ12/a11>b2+1/2σ22/b21, then (26)liminft→∞⁡Xt≥1a12b1-12σ12-a11b2+1/2σ22b21>0a.s.limsupt→∞⁡log⁡Ytt≤-b3-12σ32+b31b2b21+b31Kσa111/2<0a.s. where K(σ)=S¯σ12/2+a12X¯σ22/2b21 and (S¯,X¯,0) is the boundary equilibrium of system (1).
(ii) Ifb1>1/2σ12, b2/b21≥b3/b31, b2+1/2σ22/b21>b3/b31+K0σ/a111/2 and b1-1/2σ12/a11>b3+1/2σ32/b31, then (27)limsupt→∞⁡log⁡Xtt≤-b2-12σ22+b21b3b31+b21K0σa111/2<0a.s.liminft→∞⁡Yt≥1a13b1-12σ12-a11b3+1/2σ32b31>0a.s. where K0(σ)=S~σ12/2+a13Y~σ32/2b31 and (S~,0,Y~) is the boundary equilibrium of system (1).Proof.
(i) Ifb1-1/2σ12/a11>b2+1/2σ22/b21, system (1) has the boundary equilibrium (S¯,X¯,0). Then (28)S¯=b2b21,X¯=b1b21-a11b2a12b21.Define (29)VS,X,Y=S-S¯-S¯log⁡SS¯+a12b21X-X¯-X¯log⁡XX¯+a13b31Y. ByItô’s formula and (2), we have (30)dV=-a11S-S¯2-a12S-S¯X-X¯-a13S-S¯Y+S¯σ122dt+σ1S-S¯dB1t+a12b21b21S-S¯X-X¯+X¯σ222dt+σ2X-X¯dB2t+a13b31-b3+b31S¯Y+b31S-S¯Ydt+σ3YdB3t≔LVdt+σ1S-S¯dB1t+a12b21σ2X-X¯dB2t+a13b31σ3YdB3t, where (31)LV=-a11S-S¯2-a12S-S¯X-X¯-a13S-S¯Y+S¯σ122+a12b21b21S-S¯X-X¯+X¯σ222+a13b31-b3+b31S¯Y+b31S-S¯Y=-a11S-S¯2+S¯σ122+a12X¯σ222b21+a13b31-b3+b31S¯Y≤-a11S-S¯2+Kσ, according to b3/b31≥b2/b21. Integrating both sides of it from 0 to t yields(32)Vt-V0≤-a11∫0tSs-S¯2ds+Kσt+∫0tσ1Ss-S¯dB1s+a12b21σ2Xs-X¯dB2s+a13b31σ3YsdB3s.
Let(33)Mt=∫0tσ1Ss-S¯dB1s+a12b21σ2Xs-X¯dB2s+a13b31σ3YsdB3s, which is a real-valued continuous local martingale, and M(0)=0.
By Lemma2, for arbitrary small 0<ε<1/2-1/θ, there exists a constant T=T(ω) and a set Ωε such that P(Ωε)≥1-ε and, for t≥T, ω∈Ωε, (34)Ut≤t1/θ+ε. Then, (35)∫0∞dM,Mt1+t2≤∫0∞σ12S-S¯2+a122/b212σ22X-X¯2+a132/b312σ32Y21+t2dt≤2∫0∞U21+t2dt+2∫0∞σ12S¯2+a122/b212σ22X¯21+t2dt<∞. Let ε→0; by the strong law of large numbers [16], (36)limt→∞⁡Mtt=0a.s.
It then follows from (32) that(37)limsupt→∞⁡1t∫0tSs-S¯2ds≤Kσa11a.s.
ByItô’s formula, we have (38)dlog⁡Y=-b3+b31S-12σ32dt+σ3dB3t=-b3-12σ32+b31S¯+b31S-S¯dt+σ3dB3t, and then (39)log⁡Ytt=log⁡Y0t-b3-12σ32+b31S¯+b31St-S¯+σ3B3tt≤log⁡Y0t-b3-12σ32+b31S¯+b311t∫0tSs-S¯2ds1/2+σ3B3tt, which together with limt→∞B3(t)/t=0 and (37) implies (40)limsupt→∞⁡log⁡Ytt≤-b3-12σ32+b31b2b21+b31Kσa111/2a.s. From the condition b3+1/2σ32/b31>b2/b21+Kσ/a111/2, it is easy to see that (41)limsupt→∞⁡log⁡Ytt≤-b3-12σ32+b31b2b21+b31Kσa111/2<0a.s.; namely, Y(t) tends to zero exponentially almost surely.
We derive from (2) that(42)log⁡Stt=log⁡S0t+b1-12σ12-a11St-a12Xt-a13Yt+σ1B1ttand(43)log⁡Xtt=log⁡X0t-b2-12σ22+b21St+σ2B2tt.
From Lemma2, we know that (44)limsupt→∞⁡log⁡Utt≤0a.s. which implies(45)limsupt→∞⁡log⁡Xtt≤limsupt→∞⁡log⁡Utt≤0a.s.(46)limsupt→∞⁡log⁡Stt≤limsupt→∞⁡log⁡Utt≤0a.s.
Substituting (45) into (43) yields that(47)limsupt→∞⁡St≤b2+1/2σ22b21a.s.It then follows from (42), (46), and (47), using limt→∞Yt=0 and the condition b1-1/2σ12/a11>b2+1/2σ22/b21 as well, that (48)liminft→∞⁡Xt≥1a12b1-12σ12-a11b2+1/2σ22b21>0a.s.
(ii) The proof is similar to the former part of the proof.Theorem 4.
Let(S(t),X(t),Y(t)) be the solution of system (2) with initial value (S(0),X(0),Y(0))∈R+3. The principle of competitive exclusion does not hold for system (2) if b1>1/2σ12, b1-1/2σ12/a11<b2+1/2σ22/b21, and b1-1/2σ12/a11<b3+1/2σ32/b31. In this case (49)limt→∞⁡St=b1-1/2σ12a11a.s.limsupt→∞⁡log⁡Xtt≤-b2-12σ22+b21b1-1/2σ12a11<0a.s.limsupt→∞⁡log⁡Ytt≤-b3-12σ32+b31b1-1/2σ12a11<0a.s. In addition, (50)limt→∞⁡St=0,limt→∞⁡Xt=0andlimt→∞⁡Yt=0a.s. if b1<1/2σ12.Proof.
Since the solution of system (2) is positive, by a classical comparison theorem of stochastic differential equations, it is clear that S(t)≤ϕ(t)a.s., where ϕ(t) is the solution of the stochastic logistic equation: (51)dϕt=ϕtb1-a11ϕtdt+σ1ϕtdB1t. From the result in [17], we know(52)limt→∞ϕt=b1-σ12/2a11,limt→∞log⁡ϕtt=0a.s.provided b1>σ12/2.
ByItô’s formula, we have (53)dlog⁡X=-b2+b21S-σ222dt+σ2dB2t≤-b2+b21ϕ-σ222dt+σ2dB2t, and then (54)log⁡Xtt≤log⁡X0t-b2-σ222+b21ϕt+σ2B2tt.
From (52) and the condition b1-1/2σ12/a11<b2+1/2σ22/b21, it is easy to see that(55)limsupt→∞⁡log⁡Xtt≤-b2-σ222+b21b1-1/2σ12a11<0a.s.;namely, X(t) tends to zero exponentially almost surely.
ByItô’s formula, we have (56)dlog⁡Y=-b3+b31S-σ322dt+σ3dB3t≤-b3+b31ϕ-σ322dt+σ3dB3t, so (57)log⁡Ytt≤log⁡Y0t-b3-σ322+b31ϕt+σ3B3tt. Together with (52) and the condition b1-1/2σ12/a11<b3+1/2σ32/b31, we have(58)limsupt→∞⁡log⁡Ytt≤-b3-σ322+b31b1-1/2σ12a11<0a.s.;namely, Y(t) tends to zero exponentially almost surely.
We derive from (42) that (59)a11St=b1-σ122-log⁡St-log⁡S0t-a12Xt-a13Yt+σ1B1tt≥b1-σ122-log⁡ϕt-log⁡S0t-a12Xt-a13Yt+σ1B1tt. This together with (52), (55), and (58) implies that (60)liminft→∞⁡St≥b1-σ12/2a11a.s. From (61)b1-σ12/2a11≤liminft→∞⁡St≤limsupt→∞⁡St≤limsupt→∞⁡ϕt=b1-σ12/2a11a.s., we obtain (62)limt→∞⁡St=b1-σ12/2a11a.s.
Ifb1<12σ12, it is easy to see that (63)limt→∞St=0,limt→∞Xt=0andlimt→∞Yt=0a.s.from (52), (55), and (58).Remark 5.
By Lemma 2.5 in [12], they study the persistence and extinction of each species, to reveal the effects of stochastic noises on the persistence and extinction of each species. However, we give the sufficient conditions to guarantee that the principle of competitive exclusion holds for the model by constructing suitable stochastic Lyapunov functions, which is the biggest difference between this paper and [12].


## 3. Numerical Simulations

We present some examples to confirm and visualize the observed results by using Milstein’s Higher Order Method [18].Example 1.
Choose parametersb1=1,b2=1.2,b3=1.5,a11=0.3,a12=0.2,a13=0.3,b21=0.4,b31=0.3. These values imply that b2/b21<b1/a11<b3/b31. For deterministic system (1), the predator species X(t) survives, and the predator species Y(t) goes to extinction.
Next, we observe the role of competition in the stochastic system (2) under environmental disturbance.
(i) We takeσ1=0.1,σ2=0.1,σ3=0.1. In this case, (64)1=b1>12σ12=0.005,5=b3b31≥b2b21=3,5.017=b3+1/2σ32b31>b2b21+Kσa111/2=3.23 and (65)3.317=b1-1/2σ12a11>b2+1/2σ22b21=3.0125.We conclude from Theorem 3, for the initial value S(0)=1,X(0)=1,Y(0)=1, that the solution of system (2) obeys (66)liminft→∞⁡Xt≥1a12b1-12σ12-a11b2+1/2σ22b21=0.45625>0a.s.limsupt→∞⁡log⁡Ytt≤-b3-12σ32+b31b2b21+b31Kσa111/2=-0.536<0a.s. The numerical simulations in Figure 1 support these results clearly, illustrating survival of the predator species X(t) and extinction of the predator species Y(t).
Furthermore, we choose the same parameters as in Figure1 but change the intensities of the white noise (σ12=σ22=σ32=0.12 and σ12=σ22=σ32=0.062), which also satisfy the conditions in Theorem 3. As expected, Figures 2 and 3 show the solution (the red lines) is fluctuating around a small zone. By comparing Figures 2 and 3, we can see with a decrease of the white noise, the zone which the solution is fluctuating in is getting small. From their density functions (on the right of this figure), we consider that (S(t),X(t)) has a stationary distribution.
(ii) We chooseσ1=0.1,σ2=0.6,σ3=0.3. The conditions in Theorem 4(67)1=b1>12σ12=0.005,3.317=b1-1/2σ12a11<b2+1/2σ22b21=3.45 and (68)3.317=b1-1/2σ12a11<b3+1/2σ32b31=5.15 are satisfied. In this case, (69)limt→∞St=b1-1/2σ12a11=3.317a.s.limsupt→∞⁡log⁡Xtt≤-b2-12σ22+b21b1-1/2σ12a11=-0.0533<0a.s.limsupt→∞⁡log⁡Ytt≤-b3-12σ32+b31b1-1/2σ12a11=-0.5499<0a.s. The numerical simulations, in Figure 4, support these results clearly, illustrating extinction of the competing predator species X(t) and Y(t).
(iii) We chooseσ1=1.5,σ2=0.8,σ3=1.0. In this case, 1=b1<1/2σ12=1.125. Then (70)limt→∞⁡St=0,limt→∞⁡Xt=0andlimt→∞⁡Yt=0. The numerical simulations in Figure 5 support these results clearly, illustrating extinction of the prey species S(t) and the competing predator species X(t),Y(t).Figure 1
Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with σ1=0.1,σ2=0.1,σ3=0.1, with initial value S(0)=1,X(0)=1,Y(0)=1.Figure 2
( S ( t ) , X ( t ) ) has a stationary distribution. In the left, the red lines represent the solution of system (2), and the blue lines represent the solution of the corresponding undisturbed system. The pictures on the right are the density functions of system (2).Figure 3
There also exists a stationary distribution of(S(t),X(t)), and the fluctuation is reduced with the decreasing of the white noise. The lines have the same meaning as Figure 2.Figure 4
Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with σ1=0.1,σ2=0.6,σ3=0.3, with initial value S(0)=1,X(0)=1,Y(0)=1.Figure 5
Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with σ1=1.5,σ2=0.8,σ3=1.0, with initial value S(0)=1,X(0)=1,Y(0)=1.Example 2.
Choose parametersb1=1.5,b2=0.9,b3=1.2,a11=0.3,a12=0.2,a13=0.3,b21=0.3,b31=0.4. These values imply that b2/b21=b3/b31<b1/a11. For deterministic system (1), two competing predators coexist at a positive equilibrium.
However, we chooseσ1=0.1,σ2=0.1,σ3=0.6. The conditions in Theorem 3(71)1.5=b1>12σ12=0.005,3=b3b31≥b2b21=3,3.45=b3+1/2σ32b31>max⁡b2b21+Kσa111/2=3.28 and (72)4.983=b1-1/2σ12a11>b2+1/2σ22b21=3.017are satisfied. For stochastic system (2), the numerical simulations in Figure 6, support these results clearly, illustrating survival of the predator species X(t) and extinction of the predator species Y(t).Figure 6
Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with σ1=0.1,σ2=0.1,σ3=0.6, with initial value S(0)=1,X(0)=1,Y(0)=1.


## 4. Conclusion

In this paper, we have proposed and analyzed the principle of competitive exclusion about a Lotka-Volterra model with two predators competing for one prey by stochastic perturbation. Based on this model, we mainly have investigated that system (2) allowed the competitive exclusion of two competing predators for some values of parameters when the predator functional response to the prey density is linear. Theorem 3 shows that when σ12<2b1, the principle of competitive exclusion holds for system (2) under certain conditions. In this case, the predator species Y(t) (or X(t)) goes to extinction, and the predator species X(t) (or Y(t)) survives. Theorem 4 shows that if σ12<2b1, the principle of competitive exclusion does not hold for system (2) under certain conditions. In this case, both competing predators go to extinction. If σ12>2b1, it is easy to see that limt→∞S(t)=0,limt→∞X(t)=0andlimt→∞Y(t)=0. Furthermore, from the numerical simulation, if conditions about the principle of competitive exclusion hold, we can see that there exists a stationary distribution of (S(t),X(t)) for this system when the white noise is small.In this paper, we only considered the white noise. In fact, there are some random perturbations, for example, the telephone noise. Recently, stochastic models with the telephone noise have been studied by many authors; see [19–22]. In the future study, we will consider a stochastic Lotka-Volterra model with two predators competing for one prey perturbed by the telephone noise.



---

*Source: 7312581-2018-07-17.xml*