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.
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 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:where are the densities of the prey and the th predator () population, respectively. is the intrinsic rate of growth of the prey, and is the carrying capacity of the prey, which describes the richness of resources for prey. are the effects of the th predation on the prey, are the natural death rates of the th predator in the absence of prey, and 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 [11–14]. 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 are independent Brownian motions defined on a complete probability space with a filtration satisfying the usual conditions, and 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 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 , system (2) has a unique positive solution 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
Lemma 2. Assume . Then for any initial value , the solution of system (2) has the property where is a positive constant that satisfies .
Proof. Let By Itô’s formula and (2), we have Define the function by . Thenwhere here .
Now, choose a constant sufficiently small such that it satisfies Applying Itô’s formula, we have where here and . This implies Then There exists a positive constant such that For sufficiently small , , (7) implies that where here + + + − − , and according to the Burkholder-Davis-Gundy inequality. Therefore, In particular, choose such that , and so Let be arbitrary. Then, by the Chebyshev’s inequality, we have In view of the Borel-Cantelli lemma, we see that, for almost all ,holds for all but finitely many . Hence there exists an integer , for almost all , for which (21) holds whenever . Consequently, for almost all , if and , Therefore, Letting , we obtain the desired assertion and Hence the proof of this lemma is complete.
Theorem 3. Let be the solution of system (2) with initial value . Assume . The principle of competitive exclusion holds for system (2) if one of the following conditions holds.
(i) If , , and , then where and is the boundary equilibrium of system (1).
(ii) If , , and , then where and is the boundary equilibrium of system (1).
Proof. (i) If , system (1) has the boundary equilibrium . Then Define By Itô’s formula and (2), we have where according to . Integrating both sides of it from to yieldsLet which is a real-valued continuous local martingale, and .
By Lemma 2, for arbitrary small , there exists a constant and a set such that and, for , , Then, Let ; by the strong law of large numbers [16], It then follows from (32) thatBy Itô’s formula, we have and then which together with and (37) implies From the condition , it is easy to see that namely, tends to zero exponentially almost surely.
We derive from (2) thatandFrom Lemma 2, we know that which impliesSubstituting (45) into (43) yields thatIt then follows from (42), (46), and (47), using and the condition as well, that (ii) The proof is similar to the former part of the proof.
Theorem 4. Let be the solution of system (2) with initial value . The principle of competitive exclusion does not hold for system (2) if , , and . In this case In addition, if .
Proof. Since the solution of system (2) is positive, by a classical comparison theorem of stochastic differential equations, it is clear that , where is the solution of the stochastic logistic equation: From the result in [17], we knowprovided .
By Itô’s formula, we have and then From (52) and the condition , it is easy to see thatnamely, tends to zero exponentially almost surely.
By Itô’s formula, we have so Together with (52) and the condition , we havenamely, tends to zero exponentially almost surely.
We derive from (42) that This together with (52), (55), and (58) implies that From we obtain If , it is easy to see that 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 parameters . These values imply that . For deterministic system (1), the predator species survives, and the predator species goes to extinction.
Next, we observe the role of competition in the stochastic system (2) under environmental disturbance.
(i) We take . In this case, and We conclude from Theorem 3, for the initial value , that the solution of system (2) obeys 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 ( and ), 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.
(ii) We choose . The conditions in Theorem 4 and are satisfied. In this case, The numerical simulations, in Figure 4, support these results clearly, illustrating extinction of the competing predator species and .
(iii) We choose . In this case, . Then 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 . These values imply that . For deterministic system (1), two competing predators coexist at a positive equilibrium.
However, we choose . The conditions in Theorem 3 and 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 .
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 , the principle of competitive exclusion holds for system (2) under certain conditions. In this case, the predator species (or ) goes to extinction, and the predator species (or ) survives. Theorem 4 shows that if , the principle of competitive exclusion does not hold for system (2) under certain conditions. In this case, both competing predators go to extinction. If , it is easy to see that . Furthermore, from the numerical simulation, if conditions about the principle of competitive exclusion hold, we can see that there exists a stationary distribution of 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.
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).