Abstract
A stochastic competitive system is investigated. We first show that the positive solution of the above system does not explode to infinity in a finite time, and the existence and uniqueness of positive solution are discussed. Later, sufficient conditions for the stochastically ultimate boundedness of positive solution are derived. Also, with the help of Lyapunov function, sufficient conditions for the global attraction of positive solution are established. Finally, numerical simulations are presented to justify our theoretical results.
1. Introduction
In recent years, many researches have been done on the dynamics of many types of Lotka-Volterra competitive systems. Owing to their theoretical and practical significance, these competitive systems have been investigated extensively and there exists a large volume of literature relevant to many good results (see [1–7]). Particularly, in [8], Gopalsamy introduced the following autonomous two-species competitive system: where and can be interpreted as the population size of two competing species at time , respectively. All parameters involved with the above model are positive constants and can be interpreted in more detail; and are the intrinsic growth rates of two species; , , , and represent the effects of intraspecific competition; and are the effects of interspecific competition. Recently, Tan et al. [9] have considered the effect of impulsive perturbations and discussed the uniformly asymptotic stability of almost periodic solutions for a corresponding nonautonomous impulsive version of (1). It has also been noticed that the ecological systems, in the real world, are often perturbed by various types of environmental noise. May [10] also pointed out that, due to environmental fluctuation, the birth rate, the death rate, and other parameters usually show random fluctuation to a certain extent. To accurately describe such systems, it is necessary to use stochastic differential equations. Recently, various stochastic dynamical models have been introduced extensively in [11–17] and many interesting and valuable results including extinction, persistence, and stability can be found in [18–20].
Motivated by the above works, in this contribution, we assume that the environmental noise affects mainly the intrinsic growth rate with where are independent white noises, are standard Brownian motions defined on the complete probability space with a filtration satisfying the usual conditions, and represent the intensities of the white noises. Then the stochastically perturbed sytem (1) can be Itô’s equations with the initial values .
In this paper, we will focus on the stochastically ultimate boundedness and global attraction of positive solutions of system (3). To the best of our knowledge, there are few published papers concerning system (3). The rest of this paper is organized as follows. In Section 2, we present some assumptions, definitions, and lemmas. In Section 3, we investigate the existence and uniqueness of positive solution, and then, we discuss the stochastically ultimate boundedness of positive solutions. In Section 4, we discuss the global attraction of positive solutions. Finally, we conclude and present a specific example to justify the analytical results.
2. Preliminaries
Throughout this paper, we give the notation and assumptions. , . For any initial value , there exists such that , .
In the following, let us briefly review several basic definitions and lemmas which will be useful for establishing our main results.
Definition 1. The solution of system (3) is stochastically ultimately bounded a.s. if for arbitrary , there exists a positive constant such that
Definition 2. Let be a positive solution of system (3). Then is said to be globally attractive provided that any other solution of system (3) satisfies
Lemma 3 (see [21]). Let such that where is the family of processes such that Then
Lemma 4 (see [22]). Suppose that are real numbers; then where and
Lemma 5 (see [23]). Assume that an n-dimensional stochastic process on satisfies the condition for positive constants , , and . Then there exists a continuous version of which has the property that, for every , there is a positive random variable such that In other words, almost every sample path of is locally but uniformly Hölder continuous with exponent .
Lemma 6 (see [24]). Let be a nonnegative function on such that is integrable on and is uniformly continuous on . Then .
3. Stochastically Ultimate Boundedness
In this section, we first show, under the assumption , that the positive solution of system (3) will not explode to infinity at any finite time.
Lemma 7. Let hold and the initial value . Then system (3) has a unique solution on , which will remain in with probability one.
Proof. It is easy to see that the coefficients of system (3) satisfy the local Lipschitz condition. Then for any given initial value , there exists a unique local solution on , where is the explosion time. To show that the positive solution is global, we only need to show that . Let be sufficiently large for every component of which remains in the interval . For each integer , one can define the stopping time
Clearly, is increasing as . Assign , whence . If we can show that ., then . and . for all . In other words, to complete the proof, we just need to show that .
By reduction to absurdity, we suppose that ; then there exists a pair of constants and such that
As a result, there exists an integer such that for all
Define a -function as
Obviously, is a nonnegative function. If , then using Itô’s formula, one can derive that
where
A simple calculation shows that
It then follows from that the upper bound of , noted by , exists. We therefore have
Integrating both sides from to yields that
whence taking expectations leads to
Set for , and then . Note that arbitrary , there exist some such that equals either or . Then is not less than
As a consequence,
where is the indicator function of . Let lead to the contradiction
So we must have . This completes the proof of Lemma 7.
Lemma 7 is fundamental in this paper. In the following, we will show that the th moment of the positive solution of system (3) is upper bounded and then discuss the stochastically ultimate boundedness.
Theorem 8. Let and hold; then the positive solution of system (3) with initial value is stochastically ultimately bounded.
Proof. From Lemma 7, we can see that system (3) has a unique positive solution under assumption . Assign arbitrarily; then by Itô’s formula we can show that
Integrating and taking expectations on both sides yield that
We then derive that
By Hölder inequality one has
and moreover
Denote , ; then (31) can be rewritten as
It follows from that
that is,
Meanwhile, it is easy to see that
By the standard comparison theorem, we therefore derive that
which implies that the th moment of positive solution is upper bounded.
Let us now proceed to discuss the stochastically ultimate boundedness of system (3). Setting , then by the Chebyshev inequality, we obtain that
This gives that
The proof of Theorem 8 is complete.
4. Global Attraction
In this section, we will establish sufficient conditions for global attraction of system (3).
Lemma 9. Let hold and let be a solution of (3) with initial value ; then almost every sample path of is uniformly continuous for .
Proof. We first prove . Let us consider the following integral equation:
where
Recalling , (32), and the standard comparison theorem, we can know that
Then from Lemma 4 and (41) one derives that
Meanwhile, by Lemma 3, we obtain that, for and ,
Let
and then from (42), (43), and Lemma 4, one can derive that
Thus, it follows from Lemma 5 that almost every sample path of is locally but uniformly Hölder-continuous with exponent for and therefore almost every sample path of is uniformly continuous on .
By a similar procedure as above, can be proven. Thus, is uniformly continuous on . The proof of Lemma 9 is complete.
Theorem 10. Let , , and hold; then the unique positive solution of system (3) is globally attractive for initial data .
Proof. It follows from that, for initial data , system (3) has a unique solution (see Lemma 7). Assume that is another solution of system (3) with initial values .
Define a Lyapunov function as
Using Itô’s formula, a calculation of the right differential along (3) shows that
Integrating both sides yields that
that is,
In view of and , then it follows from (49) that
So recalling Lemmas 9 and 6, we can show that
This completes the proof of Theorem 10.
5. Numerical Simulations
In this paper, we derived the sufficient conditions for the existence, uniqueness, stochastically ultimate boundness, and global attraction of positive solutions of system (3). In order to illustrate the above theoretical results, we will perform a specific example. Motivated by the Milsten method mentioned in Higham [25], we can obtain the following discrete version of system (3): where and are Gaussian random variables that follow . Let , , , , and ; , , , , and ; ; and the initial value , . After a calculation, we can see that the assumptions of Theorems 8 and 10 hold. Figures 1 and 2 show that the positive solution of system (52) is stochastically ultimately bounded and globally attractive.
It follows from Theorem 8 that a preliminary result on the stochastically ultimate boundness of system (3) is obtained. We would like to mention here that an interesting but challenging problem associated with the investigation of system (3) should be the stochastic permanence; we leave this for future work.
Conflict of Interests
The authors declare that they have no conflict of interests.
Acknowledgments
The authors thank the anonymous referees for their valuable comments. The work is supported by the National Natural Science Foundation of China (no. 11261017), the Foundation of Hubei University for Nationalities (PY201401), the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (PKLHB1329), and the Key Subject of Hubei Province (Forestry).