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

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  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 [R.sup.3] rather than an equilibrium. The similar themes were discussed in [2-8]. The authors in  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:

dS(t)/dt = S(t)([b.sub.1] - [a.sub.11] S(t) - [a.sub.12] X(t) - [a.sub.13] Y(t)), dX(t)/dt = X(t)(-[b.sub.2] + [b.sub.21] S(t)), dY(t)/dt = Y(t)(-[b.sub.3] + [b.sub.31] S(t)), (1)

where S(t), X(t), Y(t) are the densities of the prey and the ith predator (i = 1,2) population, respectively. [b.sub.1] is the intrinsic rate of growth of the prey, and 1/[a.sub.11] is the carrying capacity of the prey, which describes the richness of resources for prey. [a.sub.12], [a.sub.13] are the effects of the ith predation on the prey, [b.sub.2], [b.sub.3] are the natural death rates of the ith predator in the absence of prey, and [b.sub.21], [b.sub.31] 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:

dS(t) = S(t)([b.sub.1] - [a.sub.11] S(t) - [a.sub.12] X(t) - [a.sub.13] Y(t)) dt + [[sigma].sub.1] S(t) d[B.sub.1](t), dX(t) = X(t)(-[b.sub.2] + [b.sub.21] S(t)) dt + [[sigma].sub.2] X(t) d[B.sub.2](t), dY(t) = Y(t)(-[b.sub.3] + [b.sub.31] S(t)) dt + [[sigma].sub.3] Y(t) d[B.sub.3](t), (2)

where [B.sub.1](t), [B.sub.2](t), [B.sub.3](t) are independent Brownian motions defined on a complete probability space ([OMEGA], F, [{[F.sub.t]}.sub.t[greater than or equal to]0], P) with a filtration [{[F.sub.t]}.sub.t[greater than or equal to]0] satisfying the usual conditions, and [[sigma].sup.2.sub.1], [[sigma].sup.2.sub.2], [[sigma].sup.2.sub.3] 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  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 [L.sup.1] 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  (Theorem 2.1), Zhang and Jiang show that, for any initial value (S(0), X(0), Y(0)) [member of] [R.sup.3.sub.+], 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

<x(t)> = [1/t] [[integral].sup.t.sub.0] x(r) dr. (3)

Lemma 2. Assume ([b.sub.2] [conjunction] [b.sub.3]) > (1/2)([[sigma].sup.2.sub.2] [disjunction] [[sigma].sup.2.sub.3]). Then for any initial value (S(0), X(0), Y(0)) [member of] [R.sup.3.sub.+], the solution of system (2) has the property

[mathematical expression not reproducible]. (4)

where [theta] is a positive constant that satisfies 2 < [theta] < 1 + 2(([b.sub.2] [conjunction] [b.sub.3])/([[sigma].sup.2.sub.2] [disjunction] [[sigma].sup.2.sub.3])).

Proof. Let

U(t) = S(t) + [[a.sub.12]/[b.sub.21]] X(t) + [[a.sub.13]/[b.sub.31]]Y(t) := S(t) + [X.sub.1](t) + [Y.sub.1](t). (5)

By Ito's formula and (2), we have

dU(t) = [S(t) ([b.sub.1] - [a.sub.11] S(t)) - [b.sub.2][X.sub.1](t) - [b.sub.3][Y.sub.1](t)] dt + [[sigma].sub.1] S(t) d[B.sub.1](t) + [[sigma].sub.2][X.sub.1](t) d[B.sub.2](t) + [[sigma].sub.3] [Y.sub.1](t) d[B.sub.3](t). (6)

Define the function W by W(U) = [(1 + U).sup.[theta]]. Then

dW(U) = LW(U) dt + [theta] [(1 + U(t)).sup.[theta]-1] [[[sigma].sub.1] S(t) d[B.sub.1](t) + [[sigma].sub.2] [X.sub.1](t) d[B.sub.2](t) + [[sigma].sub.3] [Y.sub.1](t) d[B.sub.3](t)], (7)

where

[mathematical expression not reproducible] (8)

here Z = [X.sub.1] + [Y.sub.1].

Now, choose a constant [lambda] > 0 sufficiently small such that it satisfies

([b.sub.2] [conjunction] [b.sub.3]) - [theta] - 1/2([[sigma].sup.2.sub.2] [disjunction] [[sigma].sup.2.sub.3]) - 3[lambda]/[theta] [much greater than] 0. (9)

Applying Ito's formula, we have

E[[e.sup.[lambda]t]W(U(t))] = W(U(0)) + E [[integral].sup.t.sub.0] L[[e.sup.[lambda]s]W(U(s))] ds, (10)

where

[mathematical expression not reproducible]; (11)

here c := [b.sub.1] + (([theta] - 1)/2)[[sigma].sup.2.sub.1] + 3[lambda]/[theta] + (1/2)[[b.sup.2.sub.1] [([b.sub.2] [conjunction] [b.sub.3]) - (([theta] - 1)/2)([[sigma].sup.2.sub.2] [disjunction] [[sigma].sup.2.sub.3]) - 3[lambda]/[theta]].sup.-1] and [mathematical expression not reproducible]. This implies

E[[e.sup.[lambda]t] [(1 + U).sup.[theta]]] [less than or equal to] [(1 + U(0)).sup.[theta]] + [theta][e.sup.[lambda]t]H/[lambda]. (12)

Then

[mathematical expression not reproducible]. (13)

There exists a positive constantMsuch that

E [[(1 + U).sup.[theta]]] [less than or equal to] M, t [greater than or equal to] 0. (14)

For sufficiently small [delta] > 0, k = 1, 2,..., (7) implies that

[mathematical expression not reproducible], (15)

where

[mathematical expression not reproducible]; (16)

here [mathematical expression not reproducible], and

[mathematical expression not reproducible], (17)

according to the Burkholder-Davis-Gundy inequality. Therefore,

[mathematical expression not reproducible]. (18)

In particular, choose [delta] > 0 such that [theta][H.sub.0][delta] + [square root of (32)][theta][([[sigma].sup.2.sub.1] [disjunction] [[sigma].sup.2.sub.2] [disjunction] [[sigma].sup.2.sub.3]).sup.1/2][[delta].sup.1/2] [less than or equal to] 1/2, and so

[mathematical expression not reproducible]. (19)

Let [epsilon] > 0 be arbitrary. Then, by the Chebyshev's inequality, we have

[mathematical expression not reproducible] (20)

In view of the Borel-Cantelli lemma, we see that, for almost all [omega] [member of] [OMEGA],

[mathematical expression not reproducible] (21)

holds for all but finitelymany k. Hence there exists an integer [k.sub.0]([omega]) > 1/[delta] + 2, for almost all [omega] [member of] [OMEGA], for which (21) holds whenever k [greater than or equal to] [k.sub.0]. Consequently, for almost all [omega] [member of] [OMEGA], if k [greater than or equal to] [k.sub.0] and (k - 1)[delta] [less than or equal to] t [less than or equal to] k[delta],

log [(1 + U(t)).sup.[theta]]/log t [less than or equal to] (1 + [epsilon]) log (k[delta])/log ((k - 1) [delta]). (22)

Therefore,

[mathematical expression not reproducible]. (23)

Letting [epsilon] [right arrow] 0, we obtain the desired assertion

[mathematical expression not reproducible]. (24)

and

[mathematical expression not reproducible]. (25)

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)) [member of] [R.sup.3.sub.+]. Assume ([b.sub.2] [conjunction] [b.sub.3]) > (1/2)([[sigma].sup.2.sub.2] [disjunction] [[sigma].sup.2.sub.3]). The principle of competitive exclusion holds for system (2) if one of the following conditions holds.

(i) If [b.sub.1] > (1/2)[[sigma].sup.2.sub.1], [b.sub.3]/[b.sub.31] [greater than or equal to] [b.sub.2]/[b.sub.21], ([b.sub.3] + (1/2)[[sigma].sup.2.sub.3])/[b.sub.31] > [b.sub.2]/[b.sub.21] + [(K([sigma])/[a.sub.11]).sup.1/2] and ([b.sub.1] - (1/2)[[sigma].sup.2.sub.1])/[a.sub.11] > ([b.sub.2] + (1/2)[[sigma].sup.2.sub.2])/[b.sub.21], then

[mathematical expression not reproducible]. (26)

where K([sigma]) = [bar.S][[sigma].sup.2.sub.1]/2 + [a.sub.12] [bar.X][[sigma].sup.2.sub.2]/2 [b.sub.21] and ([bar.S], [bar.X], 0) is the boundary equilibrium of system (1).

(ii) If [b.sub.1] > (1/2)[[sigma].sup.2.sub.1], [b.sub.2]/[b.sub.21] [greater than or equal to] [b.sub.3]/[b.sub.31], ([b.sub.2] + (1/2)[[sigma].sup.2.sub.2])/[b.sub.21] > [b.sub.3]/[b.sub.31] + [([K.sub.0]([sigma])/[a.sub.11]).sup.1/2] and ([b.sub.1] - (1/2)[[sigma].sup.2.sub.1])/[a.sub.11] > [b.sub.3] + (1/2)[[sigma].sup.2.sub.3])/[b.sub.31], then

[mathematical expression not reproducible]. (27)

where [K.sub.0]([sigma]) = [??][[sigma].sup.2.sub.1]/2 + [a.sub.13][??][[sigma].sup.2.sub.3]/2[b.sub.31] and ([??], 0, [??]) is the boundary equilibrium of system (1).

Proof. (i) If ([b.sub.1] -(1/2)[[sigma].sup.2.sub.1])/[a.sub.11] > ([b.sub.2] + (1/2)[[sigma].sup.2.sub.2])/[b.sub.21], system (1) has the boundary equilibrium ([bar.S], [bar.X], 0). Then

[bar.S] = [b.sub.2]/[b.sub.21], [bar.X] = [b.sub.1][b.sub.21] - [a.sub.11][b.sub.2]/[a.sub.12][b.sub.21]. (28)

Define

[mathematical expression not reproducible]. (29)

By Ito's formula and (2), we have

[mathematical expression not reproducible], (30)

where

[mathematical expression not reproducible], (31)

according to [b.sub.3]/[b.sub.31] [greater than or equal to] [b.sub.2]/[b.sub.21]. Integrating both sides of it from 0 to t yields

[mathematical expression not reproducible]. (32)

Let

[mathematical expression not reproducible], (33)

which is a real-valued continuous local martingale, and M(0) = 0.

By Lemma 2, for arbitrary small 0 < [epsilon] < 1/2 - 1/[theta], there exists a constant T = T([omega]) and a set [[OMEGA].sub.[epsilon]] such that P([[OMEGA].sub.[epsilon]]) [greater than or equal to] 1-[epsilon] and, for t [greater than or equal to] T, [omega] [member of] [[OMEGA].sub.[epsilon]],

U(t) [less than or equal to] [t.sup.1/[theta]+[epsilon]]. (34)

Then,

[mathematical expression not reproducible]. (35)

Let [epsilon] [right arrow] 0; by the strong law of large numbers ,

[mathematical expression not reproducible]. (36)

It then follows from (32) that

[mathematical expression not reproducible]. (37)

By Ito's formula, we have

d log Y = (-[b.sub.3] + [b.sub.31] S - 1/2 [[sigma].sup.2.sub.3])dt + [[sigma].sub.3]d[B.sub.3](t) = [-[b.sub.3] - 1/2 [[sigma].sup.2.sub.3] + [b.sub.31]S + [b.sub.31](S - [bar.S])] dt + [[sigma].sub.3] d[B.sub.3](t), (38)

and then

[mathematical expression not reproducible], (39)

which together with [lim.sub.t[right arrow][infinity]][B.sub.3](t)/t = 0 and (37) implies

[mathematical expression not reproducible]. (40)

From the condition ([b.sub.3] + (1/2)[[sigma].sup.2.sub.3])/[b.sub.31] > [b.sub.2]/[b.sub.21] + [(K([sigma])/[a.sub.11]).sup.1/2], it is easy to see that

[mathematical expression not reproducible]; (41)

namely, Y(t) tends to zero exponentially almost surely.

We derive from (2) that

log S(t)/t = log S(0)/t + [b.sub.1] - 1/2 [[sigma].sup.2.sub.1] - [a.sub.11]<S(t)> - [a.sub.12]<X(t)> - [a.sub.13] <Y(t)> + [[sigma].sub.1][B.sub.1](t)/t (42)

and

log X(t)/t = log X(0)/t - [b.sub.2] - 1/2 [[sigma].sup.2.sub.2] + [b.sub.21]<S (t)> + [[sigma].sub.2][B.sub.2](t)/t. (43)

From Lemma 2, we know that

[mathematical expression not reproducible]. (44)

which implies

[mathematical expression not reproducible]. (45)

[mathematical expression not reproducible]. (46)

Substituting (45) into (43) yields that

[mathematical expression not reproducible]. (47)

It then follows from (42), (46), and (47), using [lim.sub.t[right arrow][infinity]]<Y(t)> = 0 and the condition ([b.sub.1] - (1/2)[[sigma].sup.2.sub.1])/[a.sub.11] > ([b.sub.2] + (1/2)[[sigma].sup.2.sub.2])/[b.sub.21] as well, that

[mathematical expression not reproducible]. (48)

(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)) [member of] [R.sup.3.sub.+]. The principle of competitive exclusion does not hold for system (2) if [b.sub.1] > (1/2)[[sigma].sup.2.sub.1], ([b.sub.1] - (1/2)[[sigma].sup.2.sub.1])/[a.sub.11] < ([b.sub.2] + (1/2)[[sigma].sup.2.sub.2])/[b.sub.21], and ([b.sub.1] - (1/2)[[sigma].sup.2.sub.1])/[a.sub.11] < ([b.sub.3] + (1/2)[[sigma].sup.2.sub.3])/[b.sub.31]. In this case

[mathematical expression not reproducible]. (49)

[mathematical expression not reproducible]. (50)

if [b.sub.1] < (1/2)[[sigma].sup.2.sub.1].

Proof. Since the solution of system (2) is positive, by a classical comparison theorem of stochastic differential equations, it is clear that S(t) [less than or equal to] [phi](t) a.s., where [phi](t) is the solution of the stochastic logistic equation:

d[phi](t) = [phi](t)([b.sub.1] - [a.sub.11] [phi](t)) dt + [[sigma].sub.1] [phi](t) d[B.sub.1](t). (51)

From the result in , we know

[mathematical expression not reproducible]. (52)

provided [b.sub.1] > [[sigma].sup.2.sub.1]/2.

By Ito's formula, we have

d log X = (-[b.sub.2] + [b.sub.21] S - [[sigma].sup.2.sub.2]/2)dt + [[sigma].sub.2]d[B.sub.2](t) [less than or equal to] (-[b.sub.2] + [b.sub.21][phi] - [[sigma].sup.2.sub.2]/2)dt + [[sigma].sub.2]d[B.sub.2](t), (53)

and then

log X(t)/t [less than or equal to] log X(0)/t - [b.sub.2] - [[sigma].sup.2.sub.2]/2 + [b.sub.21] <[phi](t)> + [[sigma].sub.2][[B.sub.2](t)/t]. (54)

From (52) and the condition ([b.sub.1] - (1/2)[[sigma].sup.2.sub.1])/[a.sub.11] < ([b.sub.2] + (1/2)[[sigma].sup.2.sub.2])/[b.sub.21], it is easy to see that

[mathematical expression not reproducible]; (55)

namely, X(t) tends to zero exponentially almost surely.

By Ito's formula, we have

d log Y = (-[b.sub.3] + [b.sub.31] S - [[sigma].sup.2.sub.3]/2)dt + [[sigma].sub.3]d[B.sub.3] (t) [less than or equal to] (-[b.sub.3] + [b.sub.31][phi] - [[sigma].sup.2.sub.3]/2)dt + [[sigma].sub.3]d[B.sub.3](t), (56)

so

log Y(t)/t [less than or equal to] log Y(0)/t - [b.sub.3] - [[sigma].sup.2.sub.3]/2 + [b.sub.31] <[phi](t)> + [[sigma].sub.3][B.sub.3](t)/t. (57)

Together with (52) and the condition ([b.sub.1] - (1/2)[[sigma].sup.2.sub.1])/[a.sub.11] < ([b.sub.3] + (1/2)[[sigma].sup.2.sub.3])/[b.sub.31], we have

[mathematical expression not reproducible]; (58)

namely, Y(t) tends to zero exponentially almost surely.

We derive from (42) that

[mathematical expression not reproducible]. (59)

This together with (52), (55), and (58) implies that

[mathematical expression not reproducible]. (60)

From

[mathematical expression not reproducible], (61)

we obtain

[mathematical expression not reproducible]. (62)

If [b.sub.1] < 1/2 [[sigma].sup.2.sub.1], it is easy to see that

[mathematical expression not reproducible]. (63)

from (52), (55), and (58).

Remark 5. By Lemma 2.5 in , 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 .

3. Numerical Simulations

We present some examples to confirm and visualize the observed results by using Milstein's Higher Order Method .

Example 1. Choose parameters [b.sub.1] = 1, [b.sub.2] = 1.2, [b.sub.3] = 1.5, [a.sub.11] = 0.3, [a.sub.12] = 0.2, [a.sub.13] = 0.3, [b.sub.21] = 0.4, [b.sub.31] = 0.3. These values imply that [b.sub.2]/[b.sub.21] < [b.sub.1]/[a.sub.11] < [b.sub.3]/[b.sub.31]. 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 take [[sigma].sub.1] = 0.1, [[sigma].sub.2] = 0.1, [[sigma].sub.3] = 0.1. In this case,

1 = [b.sub.1] > 1/2 [[sigma].sup.2.sub.1] = 0.005, 5 = [b.sub.3]/[b.sub.31] [greater than or equal to] [b.sub.2]/[b.sub.21] = 3, 5.017 = [b.sub.3] + (1/2) [[sigma].sup.2.sub.3]/[b.sub.31] > [b.sub.2]/[b.sub.21] + [(K([sigma])/[a.sub.11]).sup.1/2] = 3.23 (64)

and

3.317 = [b.sub.1] - (1/2) [[sigma].sup.2.sub.1]/[a.sub.11] > [b.sub.2] + (1/2) [[sigma].sup.2.sub.2]/[b.sub.21] = 3.0125. (65)

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

[mathematical expression not reproducible]. (66)

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 Figure 1 but change the intensities of the white noise ([[sigma].sup.2.sub.1] = [[sigma].sup.2.sub.2] = [[sigma].sup.2.sub.3] = 0.12 and [[sigma].sup.2.sub.1] = [[sigma].sup.2.sub.2] = [[sigma].sup.2.sub.3] = 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 choose [[sigma].sub.1] = 0.1, [[sigma].sub.2] = 0.6, [[sigma].sub.3] = 0.3. The conditions in Theorem 4

1 = [b.sub.1] > 1/2 [[sigma].sup.2.sub.1] = 0.005, 3.317 = [b.sub.1] - (1/2) [[sigma].sup.2.sub.1]/[a.sub.11] < [b.sub.2] + (1/2) [[sigma].sup.2.sub.2]/[b.sub.21] = 3.45 (67)

and

3.317 = [b.sub.1] - (1/2) [[sigma].sup.2.sub.1]/[a.sub.11] < [b.sub.3] + (1/2) [[sigma].sup.2.sub.3]/[b.sub.31] = 5.15 (68)

are satisfied. In this case,

[mathematical expression not reproducible]. (69)

The numerical simulations, in Figure 4, support these results clearly, illustrating extinction of the competing predator species X(t) and Y(t).

(iii) We choose [[sigma].sub.1] = 1.5, [[sigma].sub.2] = 0.8, [[sigma].sub.3] = 1.0. In this case, 1 = [b.sub.1] < (1/2)[[sigma].sup.2.sub.1] = 1.125. Then

[mathematical expression not reproducible]. (70)

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).

Example 2. Choose parameters [b.sub.1] = 1.5, [b.sub.2] = 0.9, [b.sub.3] = 1.2, [a.sub.11] = 0.3, [a.sub.12] = 0.2, [a.sub.13] = 0.3, [b.sub.21] = 0.3, [b.sub.31] = 0.4. These values imply that [b.sub.2]/[b.sub.21] = [b.sub.3]/[b.sub.31] < [b.sub.1]/[a.sub.11]. For deterministic system (1), two competing predators coexist at a positive equilibrium.

However, we choose [[sigma].sub.1] = 0.1, [[sigma].sub.2] = 0.1, [[sigma].sub.3] = 0.6. The conditions in Theorem 3

1.5 = [b.sub.1] > 1/2 [[sigma].sup.2.sub.1] = 0.005, 3 = [b.sub.3]/[b.sub.31] [greater than or equal to] [b.sub.2]/[b.sub.21] = 3, 3.45 = [b.sub.3] + (1/2) [[sigma].sup.2.sub.3]/[b.sub.31] > max {[b.sub.2]/[b.sub.21] + [(K([sigma])/[a.sub.11]).sup.1/2]} = 3.28 (71)

and

4.983 = [b.sub.1] - (1/2) [[sigma].sup.2.sub.1]/[a.sub.11] > [b.sub.2] + (1/2) [[sigma].sup.2.sub.2]/[b.sub.21] = 3.017 (72)

are 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).

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 [[sigma].sup.2.sub.1] < 2[b.sub.1], 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 [[sigma].sup.2.sub.1] < 2[b.sub.1], the principle of competitive exclusion does not hold for system (2) under certain conditions. In this case, both competing predators go to extinction. If [[sigma].sup.2.sub.1] > 2[b.sub.1], it is easy to see that [lim.sub.t[right arrow][infinity]]S(t) = 0, [lim.sub.t[right arrow][infinity]]X(t) = 0 and [lim.sub.t[right arrow][infinity]]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.

https://doi.org/10.1155/2018/7312581

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by Program for Natural Science Foundation of China (nos. 11701209 and 11601038), the natural science foundation of Jilin Province (no. 20170101146JC), Science and Technology Research Project of Jilin Provincial Department of Education (no. JJKH20170487KJ), Youth Science Foundation of Jilin Province (no. 20160520110JH), Education Department of Jilin Province (no. JJKH20180462KJ), and Youth Teacher Development Program of Changchun University (no. 2018JBC08L13).

References

 A. L. Koch, "Competitive coexistence of two predators utilizing the same prey under constant environmental conditions," Journal of Theoretical Biology, vol. 44, no. 2, pp. 387-395, 1974.

 S. B. Hsu, S. P. Hubbell, and P. Waltman, "Competing predators," SIAM Journal on Applied Mathematics, vol. 35, no. 4, pp. 617-625, 1978.

 S. Hsu, S. Hubbell, and P. Waltman, "A contribution to the theory of competing predators," Ecological Monographs, vol. 48, pp. 337-349, 1978.

 A. Korobeinikov and G. Wake, "Global properties of the three-dimensional predator-prey Lotka-Volterra systems," Journal of Applied Mathematics and Decision Sciences, vol. 3, pp. 155-162, 1999.

 B. Li and H. L. Smith, "Global dynamics of microbial competition for two resources with internal storage," Journal of Mathematical Biology, vol. 55, no. 4, pp. 481-515, 2007.

 J. Llibre and D. Xiao, "Global dynamics of a Lotka-Volterra model with two predators competing for one prey," SIAM Journal on Applied Mathematics, vol. 74, no. 2, pp. 434-453, 2014.

 W. Liu, D. Xiao, and Y. Yi, "Relaxation oscillations in a class of predator-prey systems," Journal of Differential Equations, vol. 188, no. 1, pp. 306-331, 2003.

 G. S. K. Wolkowicz and Z. Lu, "Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates," SIAM Journal on Applied Mathematics, vol. 52, no. 1, pp. 222-233, 1992.

 X. Li, D. Jiang, and X. Mao, "Population dynamical behavior of Lotka-Volterra system under regime switching," Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 427-448, 2009.

 Y. Zhao, D. Jiang, and D. O'Regan, "The extinction and persistence of the stochastic SIS epidemic model with vaccination," Physica A: Statistical Mechanics and its Applications, vol. 392, no. 20, pp. 4916-4927, 2013.

 Y. Zhao and D. Jiang, "The threshold of a stochastic SIS epidemic model with vaccination," Applied Mathematics and Computation, vol. 243, pp. 718-727, 2014.

 M. Liu and P. S. Mandal, "Dynamical behavior of a one-prey two-predator model with random perturbations," Communications in Nonlinear Science and Numerical Simulation, vol. 28, no. 1-3, pp. 123-137, 2015.

 Q. Zhang, D. Jiang, Z. Liu, and D. O'Regan, "The long time behavior of a predator-prey model with disease in the prey by stochastic perturbation," Applied Mathematics and Computation, vol. 245, pp. 305-320, 2014.

 Q. Zhang and D. Jiang, "Competitive exclusion in a stochastic chemostat model with Holling type II functional response," Journal of Mathematical Chemistry, vol. 54, no. 3, pp. 777-791, 2016.

 Q. Zhang and D. Jiang, "The coexistence of a stochastic Lotka-Volterra model with two predators competing for one prey," Applied Mathematics and Computation, vol. 269, Article ID 21456, pp. 288-300, 2015.

 X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 2nd edition, 1997.

 C. Ji, D. Jiang, and N. Shi, "Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation," Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 482-498, 2009.

 D. J. Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations," SIAM Review, vol. 43, no. 3, pp. 525-546, 2001.

 D. Li, S. Liu, and J. Cui, "Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching," Journal of Differential Equations, vol. 263, no. 12, pp. 8873-8915, 2017.

 M. Liu, X. He, and J. Yu, "Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays," Nonlinear Analysis: Hybrid Systems, vol. 28, pp. 87-104, 2018.

 M. Liu and Y. Zhu, "Stability of a budworm growth model with random perturbations," Applied Mathematics Letters, vol. 79, pp. 13-19, 2018.

 L. Wang and D. Jiang, "Ergodic property of the chemostat: a stochastic model under regime switching and with general response function," Nonlinear Analysis: Hybrid Systems, vol. 27, pp. 341-352, 2018.

Zhongwei Cao (iD), (1) Qiumei Zhang (iD), (2) and Yanan Zhao (2)

(1) Department of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China

(2) School of Science, Changchun University, Changchun 130022, China

Correspondence should be addressed to Qiumei Zhang; zhangqm1110@163.com

Received 12 March 2018; Revised 29 June 2018; Accepted 9 July 2018; Published 17 July 2018

Caption: Figure 1: Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with [[sigma].sub.1] = 0.1, [[sigma].sub.2] = 0.1, [[sigma].sub.3] = 0.1, with initial value S(0) = 1, X(0) = 1, Y(0) = 1.

Caption: 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).

Caption: 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.

Caption: Figure 4: Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with [[sigma].sub.1] = 0.1, [[sigma].sub.2] = 0.6, [[sigma].sub.3] = 0.3, with initial value S(0) = 1, X(0) = 1, Y(0) = 1.

Caption: Figure 5: Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with [[sigma].sub.1] = 1.5, [[sigma].sub.2] = 0.8, [[sigma].sub.3] = 1.0, with initial value S(0) = 1, X(0) = 1, Y(0) = 1.

Caption: Figure 6: Numerical simulation of the solution of system (2) and its corresponding deterministic system (1) with [[sigma].sub.1] = 0.1, [[sigma].sub.2] = 0.1, [[sigma].sub.3] = 0.6, with initial value S(0) = 1, X(0) = 1, Y(0) = 1.