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

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: dS (t) dt = S (t) (b1 − a11S (t) − a12X (t) − a13Y (t)) , dX (t) dt = X (t) (−b2 + b21S (t)) , dY (t) dt = Y (t) (−b3 + b31S (t)) , (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  3 rather than an equilibrium.The similar themes were discussed in [2][3][4][5][6][7][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.
where (), (), () are the densities of the prey and the th predator (i = 1, 2) population, respectively. 1 is the intrinsic rate of growth of the prey, and 1/ 11 is the carrying capacity of the prey, which describes the richness of resources for prey. 12 ,  13 are the effects of the th predation on the prey,  2 ,  3 are the natural death rates of the th predator in the absence of prey, and  21 ,  31 are the efficiency and propagation rates of the th 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][12][13][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: where  1 (),  2 (),  3 () are independent Brownian motions defined on a complete probability space (Ω, F, {F  } ≥0 , P) with a filtration {F  } ≥0 satisfying the usual conditions, and  2 1 ,  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  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.

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.

𝐸 [ sup
In particular, choose  > 0 such that 3 ) 1/2  1/2 ≤ 1/2, and so Let  > 0 be arbitrary.Then, by the Chebyshev's inequality, we have  { : sup In view of the Borel-Cantelli lemma, we see that, for almost all  ∈ Ω, holds for all but finitely many .Hence there exists an integer  0 () > Hence the proof of this lemma is complete.
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].

Numerical Simulations
We present some examples to confirm and visualize the observed by using Milstein's Higher Order Method [18].
Example Choose parameters  1 = 1, The numerical simulations in Figure 1 support these results clearly, illustrating survival of the predator species () and extinction of the predator species ().Furthermore, we choose the same parameters as in Figure 1 but change the intensities of the white noise ( 2 1 =  2 2 =  2 3 = 0.1 2 and  2 1 =  2 2 =  2 3 = 0.06 2 ), 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 ((), ()) has a stationary distribution.
The numerical simulations in Figure 5 support these results clearly, illustrating extinction of the prey species () and the competing predator species (), ().
Example 2. Choose parameters  1 = 1.5,  2 = 0.9,      are satisfied.For stochastic system (2), the numerical simulations in Figure 6, support these results clearly, illustrating survival of the predator species () and extinction of the predator species ().

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.

Figure 2 :
Figure2: ((), ()) 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 :Figure 4 :
Figure3: There also exists a stationary distribution of ((), ()), and the fluctuation is reduced with the decreasing of the white noise.The lines have the same meaning as Figure2.