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Mathematical Problems in Engineering
Volume 2014, Article ID 637862, 10 pages
http://dx.doi.org/10.1155/2014/637862
Research Article

Analysis of Stochastic Gilpin-Ayala Competition System

1College of Science, Hohai University, Nanjing 210098, China
2School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, China

Received 4 August 2014; Accepted 4 September 2014; Published 5 November 2014

Academic Editor: Ramachandran Raja

Copyright © 2014 Lei Liu and Quanxin Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the asymptotic behavior for stochastic Gilpin-Ayala competition system. The sufficient conditions for existence of stationary distribution and extinction are established. And a certain asymptotic property of the solution is also obtained. A nontrivial example is provided to illustrate our results.

1. Introduction

One of the most common phenomena considering ecological population is that many species which grow in the same environment compete for the limited resources or in some way inhibit others’ growth. It is therefore very important to study the competition models for multispecies. It is well known that one of the famous models is the following classical Lotka-Volterra competition system: where represents the population size of species at time , the constant is the growth rate of species , and represents the effect of interspecific () or intraspecific () interaction. The Lotka-Volterra models have often been severely criticized. One disadvantage of Lotka-Volterra models is that in such a model, the rate of change in the density of each species is a linear function of densities of the interacting species. In order to yield significantly more accurate results, Gilpin and Ayala proposed the the following Gilpin-Ayala models; detailed studies related to the model may be found in [1]: where are the parameters to modify the classical Lotka-Volterra model.

On the other hand, population systems are inevitably affected by environmental noise. It is therefore useful to reveal how the noise affects the population systems. Recall that the parameter in (2) represents the intrinsic growth rate of the population. In practice we usually estimate it by an average value plus an error which follows a normal distribution; then the intrinsic growth rate becomes where    are Brown motions with and represent the intensities of the noise. As a result, system (2) becomes the stochastic Gilpin-Ayala system as follows: and we impose the following condition:

The stochastic Lotka-Volterra model has been extensively studied due to its universal existence and importance; see [210]. More recently, the existence of stationary distribution and extinction of stochastic Lotka-Volterra system have received a lot of attention, which can give a good explanation of the recurring phenomena in population system. Under what conditions can a stochastic Lotka-Volterra system has a stationary distribution? It is an open topic until very recently Mao [11] gave a positive answer. Since then, this topic has received a lot of attention; the readers are referred to [1114]. In addition, the asymptotic behavior of ,   for various stochastic Lotka-Volterra systems has been considered by many authors [4, 5, 10, 12], which is an important and useful property on asymptotic estimation for corresponding population systems.

However, these properties for stochastic Gilpin-Ayala system (4) have not been investigated, which remain an interesting research topic. We aim to establish new results on these properties for system (4). It is well known that the stochastic Gilpin-Ayala system (4) is a highly nonlinear system; the method for classic Lotka-Volterra system cannot be directly applied to system (4). By the Lyapunov methods, and some techniques to deal with the nonquadratic item, sufficient criteria are established which ensure the existence of a stationary distribution and extinction. By using some stochastic analysis techniques, an asymptotic property for system (4) is obtained.

2. Notation

Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contains all -null sets). Let be a n-dimensional Brownian motion defined on the probability space. If and are real numbers, then denotes the maximum of and , and stands for the minimum of and . If is symmetric, its largest and smallest eigenvalues are denoted by and . Let be the positive equilibrium of the deterministic Gilpin-Ayala competition system (2), that is, the solution of the following equation: In the same way as Mao et al. [8] did, we can also show the following result on the existence of global positive solution.

Lemma 1. Assume that condition (5) holds. Then, for any given initial value , there is a unique solution to system (4) and the solution will remain in with probability 1; namely, for any .

Lemma 2. Let condition (5) hold. Then, for any and any given initial value , there exists a constant such that

The proof of the lemma is rather standard so it is omitted.

3. An Asymptotic Property

The main aim of this section is to consider the large time behavior of ,  . To this end, we consider two auxiliary stochastic differential equations as follows: Then it follows from comparison principle (see [15]) that

Lemma 3. Let condition (5) hold. Then the solution to system (9) has the following property:

The proof is similar to Li et al. [5] and is omitted here.

Theorem 4. Let condition (5) hold and be the global solution to system (4) with any positive initial value . Assume moreover that Then the solution of system (4) has the following property:

Proof. Let be for simplicity. By virtue of Lemma 3 and (11), we have ,  ,  . Thus it remains to show that ,  . It is sufficient to show By Ito’s formula, satisfies A simple computation shows that The well-known Hölder inequality yields For , it follows from the inequality that
For , set Substituting these inequalities into (16) yields Similarly, we get Substituting (21) and (22) into (16) yields where is the solution of the following system: A simple computation shows that Using the property of Brownian motion, we conclude that It is easy to see that if , then we have Besides, it follows from Lemma 3 that The required assertion (15) follows by letting on both sides of (25) and using conditions (26)–(28). The proof is therefore completed.

4. Stationary Distribution

The main aim of this section is to study the existence of a unique stationary distribution of the system (4). Let us prepare a known lemma (see Hasminskii [16, pp. 106–125]). Let be a homogeneous Markov process in described by the following stochastic differential equation: The diffusion matrix is To be more precise, let denote the probability measure induced by , that is where is the -algebra of all the Borel sets .

Lemma 5 (see [16]). We assume that there is a bounded open subset with a regular (i.e., smooth) boundary such that its closure , and consider the following:(i)in the domain and some neighborhood, therefore, the smallest eigenvalue of the diffusion matrix is bounded away from zero;(ii)if , the mean time at which a path issuing from reaches the set is finite, and for every compact subset . And throughout this paper one sets .
We then have the following assertions.(1)The Markov process has a stationary distribution with density in , such that, for any borel set , (2)(ergodic property) Let be a function integrable with respect to the measure . Then

Remark 6. The proof is given by [16] in detail. Exactly, the existence of stationary distribution with density is referred to Theorem 4.3 on page 117 while ergodic property (33) is referred to Theorem 4.2, page 110.

Theorem 7. Let condition (5) hold and be the global solution to system (4) with any positive initial value . Assume that there exists such that Then there is a stationary distribution for system (4) and it has the ergodic property.

Proof. By Lemma 5, it suffices to prove that there exists some neighborhood and a nonnegative -function such that the diffusion matrix is uniformly elliptical in and, for any , is negative (for details refer to [11]).
Applying Itô’s formula to yields If , since , for , then If , then Substituting (37) and (38) into (36) yields By the inequality , we have Note that ,  , and Then the ellipsoid lies entirely in . Let be a neighborhood of the ellipsoid such that, for any , . We therefore have verified condition (ii) in Lemma 5.
Now we begin to verify condition (i) in Lemma 5. It is easy to see that . If , then there exists such that . This implies that . Then we have , which contradicts the fact that . Noting that is a continuous function of , we therefore have This immediately implies condition (i) in Lemma 5. The proof is completed.

Now we denote by the stationary distribution. The mean vector of is important and useful information on population systems, from which we can infer asymptotically the mean of and the size of each species. If we can show that , then the mean vector is well defined. In this case, the ergodic theory stated above implies that

Theorem 8. Let assumptions in Theorems 4 and 7 hold. Then

Proof. The proof is composed of two parts. The first part is to show the well-definition of by dominated control convergence theorem. The second part is to prove assertion (45). Let for simplicity.
By the ergodic property of stationary distribution, for ,  , we have The dominated convergence theorem yields that It follows from Lemma 2 that Letting yields That is to say, for any , the functions are integrable with respect to the measure . The well-definition of follows by letting in (49) straightforward.
Now we process to show assertion (45). For , simple computation shows that The well-known Hölder inequality yields where . This implies The well-known Hölder inequality yields This implies By the law of strong large numbers for martingales and Theorem 4, letting on both sides of (54) yields which is the required assertion (45).

5. Extinction

One of the most basic questions one can ask in population dynamics is extinction, which means a species will be doomed. The interesting question is can the exponential extinction rate be estimated precisely? In many cases, we need to know the extinction rate of the species in order to have a suitable policy in investment and to have timely measures to protect them from the extinct disaster.

Theorem 9. Let condition (5) and ,  , hold and be the global solution to system (4) with any positive initial value . Then the solution to system (4) has the property that That is, the population will become extinct exponentially with probability one and the exponential extinction rate of the th species is .

Proof. Let for simplicity. It follows from Itô’s formula that where is the real-valued continuous local martingale vanishing at , with the quadratic variation . Dividing both sides by yields Using the law of strong large numbers for martingales (see [17]), we can claim that Letting yields This shows that, for any and , there is a positive random variable such that, with probability one, It follows that which means The required assertion (58) follows by letting on both sides of (54).

Remark 10. Theorem 9 showed that when the perturbation is large in the sense that ,  , the population will be forced to expire. And the exponential extinction rate is given precisely in terms of system’s coefficients.

6. Numerical Simulations

In this section, to illustrate the usefulness and flexibility of the theorem developed in previous section, we present a numerical example.

Example 11. Consider a 2-dimensional stochastic Gilpin-Ayala system as follows: System (64) is exactly system (4) with , , , , , , and ,  . We compute that and . The existence and uniqueness of the solution follows from Lemma 1. We consider the solution with initial data and . Let for simplicity.
(i) : simple computation shows that By Theorem 4, the solution to system (64) has the following property Figures 1 and 2 show the stochastic trajectories of and generated by the Heun scheme for time step for system (64) on , respectively.
Choosing and  , we further compute that By virtue of Theorem 7, system (64) has a unique stationary distribution. Figures 3 and 4 show the stochastic trajectories of and generated by the Heun scheme for time step for system (64) on , respectively.
(ii) Consider .
Note that , , by virtue of Theorem 9, system (64) is exponentially extinctive. Figures 5 and 6 show the stochastic trajectories of and generated by the Heun scheme for time step for system (64) on , respectively.

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Figure 1
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Figure 2
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Figure 3
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Figure 4
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Figure 5
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Figure 6

7. Conclusion

In this paper, we have investigated the asymptotic behavior for the stochastic Gilpin-Ayala competition system. Firstly, by utilizing stochastic analysis techniques and the stochastic comparison principle, the larger time behavior ,  . has been researched. Secondly, by applying some techniques to deal with the nonquadratic item, sufficient conditions are obtained under which there is a stationary distribution to the system. Based on the condition, the estimation on the mean of the stationary distribution is presented. Finally, the sufficient criteria for extinction are established.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was jointly supported by the National Natural Science Foundation of China (Grant nos. 61304070, 11271146, and 61374080), the National Key Basic Research Program of China (973 Program) (2013CB228204), the Natural Science Foundation of Zhejiang Province (LY12F03010), the Fundamental Research Funds for the Central Universities of China (Grant no. 2013B00614), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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