Abstract
A competitive system subject to environmental noise is established. By using the theory of stochastic differential equations and Lyapunov function, sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solution of the above system are established, respectively. An example together with its corresponding numerical simulations is presented to confirm our analytical results.
1. Introduction
Mathematical modelling plays an important role in the mathematical ecology. In the past several years, ecological models based on determinate systems emerged in large numbers (see [1–9]). While the disturbance of environmental noise is unavoidable in the real world, more and more researchers start to pay attention to the study on nonlinear dynamic systems with environmental noise and many valuable results have been obtained (see [10–24]).
In [25], Gopalsamy introduced the following competitive system: where may represent the densities of species. The coefficients , , , and are all positive constants. In the absence of interspecific interactions, each species is governed by the logistic equation; however, in the presence of interspecific interactions, each species retains the average growth rate of the other. In this contribution, we consider the influence of environmental noise and obtain the following form: where , is independent white noise with , , and represents the intensity of the noise, . is standard Brownian motion defined on the complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets).
In this paper, we focus on the asymptotic behavior of positive solution of system (2). To the best of our knowledge, there are few published papers concerning system (2). The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. The existence, uniqueness, and stochastic boundedness of positive solution of system (2) are discussed in Section 3. The global attraction of system (2) is studied in Section 4. As an application of our main results, we present an example and its numerical simulations to support our theoretical results in Section 5.
2. Preliminaries
In this section, we introduce some definitions and lemmas which are useful for establishing our main results.
Definition 1. The solution of system (2) is stochastically bounded if, for any , there exist positive constants such that
Definition 2. Let be a positive solution of system (2). If another positive solution of system (2) satisfies then is global attractive.
Definition 3. Let be a positive solution of system (2). The solution to system (2) is said to be exponentially extinct with probability one if
Lemma 4 ( inequality). Suppose that are all real numbers; then for any positive real number we have where
Lemma 5 (see [26]). Let be a nonnegative integrable and uniformly continuous function defined on such that is integrable and uniformly continuous on . Then .
Lemma 6 (see [27, 28]). Suppose that a stochastic process on satisfies the condition for some positive constants , , and . Then there exists a continuous modification 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 .
3. Existence, Uniqueness, Stochastic Boundedness, and Extinction
We first present the existence and uniqueness of positive solution of system (2).
Theorem 7. System (2) has a unique positive solution, say , on . Furthermore, the solution will remain in with probability one.
Proof. The proof of this lemma is rather standard. It is obvious that the coefficients of system (2) are local Lipschitz continuous. Then, for any initial value with , there exists a unique local solution , where is the explosion time (see [10, 19]). Therefore, to prove that the local solution is also global, we only need to show that a.s. Let be sufficiently large so that every component of lies in . For each integer , we define the stopping time as follows: Here we set ( denotes the empty set). Obviously, is increasing as . Denote , whence a.s. We need to show that a.s. Otherwise, there exist constants and such that . Then, by denoting , there exists an integer such that for all , We now define a -function as where and . It is obvious that is nonnegative. By Itô’s formula, one has where A calculation can show that is upper bounded, denoted by . Thus (13) can be rewritten as Integrating both sides from 0 to , we acquire that As a consequence, one has Since , taking expectations one shows that Thus On the other hand, for every , either or equals to either or . Then is not less than either or . Consequently, from (19) we have where is the indicator function of . Let , one can show the following contradiction: Hence, a.s. and there exists a unique positive solution of system (2) on . This completes the proof.
Next, we investigate the stochastic boundedness of the positive solutions of system (2). To this end, we first give the following Lemma 8.
Lemma 8. If , , then for any real number the solution of system (2) satisfies where
Proof. By Itô’s formula, one can show that Integrating from 0 to , we have Taking expectations, we obtain that So Let we have As a consequence Noting that , , we have Furthermore, using the standard comparison principle, one can show that Then we can obtain that where This completes the proof.
Finally, we discuss the stochastic boundedness of the positive solutions of system (2).
Theorem 9. If , , then the solution of system (2) is stochastically bounded.
Proof. On one hand, for any positive number , , one derives that On the other hand, by the Chebyshev inequality and Lemma 8, we obtain that Let . Then we have It follows from Lemma 8 that which, together with (37), leads to Let , and noting that , one shows that Therefore, which implies that the solution of system (2) is stochastically bounded. The proof is complete.
Theorem 10. Suppose that all coefficients of system (2) are positive and , . Then the solution of system (2) is exponentially extinct with probability one.
Proof. Define, respectively, Lyapunov functions and . Then the following conclusions can be obtained by Itô’s formula Integrating from 0 to , one concludes that Dividing on both sides of (43), sending , and employing the strong law of large numbers for local martingales, one acquires that This completes the proof.
4. Global Attraction
In this section, we first introduce Lemma 11 before we show the global attraction of system (2).
Lemma 11. If , , , then almost every sample path of the solution of system (2) is uniformly continuous on .
Proof. It follows from system (2) that where Applying Lemmas 4 and 8, for any , one derives that Without loss of generality, we assume that . Using the moment inequality (see [10]) to stochastic integral (45), we can obtain that where and . We further let then by (47), (48), and Lemma 4, one yields that It follows from Lemma 6 that almost every sample path of is uniformly continuous on . Similarly, we can show that almost every sample path of is uniformly continuous on . Therefore, is uniformly continuous on , a.s. This completes the proof.
We can now present the result on global attraction of system (2).
Theorem 12. If , , , then system (2) has a unique global attractive positive solution, denoted by , on .
Proof. It follows from Theorem 7 that system (2) has a unique positive solution . Assume that is another positive solution of system (2). Consider a Lyapunov function defined by Applying Itô’s formula, a calculation of the right differential of along the solution, one yields that Integrating from 0 to and taking expectations one can show that Thus and hence integrating from 0 to one derives that which implies that So which, together with Lemmas 5 and 11, leads to and hence is global attractive on .
5. An Example
In this section, we first give an example to verify the feasibilities of Theorems 9 and 12. Using the Milsten method mentioned in Higham [29], we can derive the following discrete version of system (2): where and are Gaussian random variables which follow . Let us choose , , , , , , , , , and , . A calculation shows that the conditions of Theorems 9 and 12 are satisfied. Figures 1 and 2 show that the positive solution of system (59) is stochastically bounded and global attractive on .
Recalling the whole paper, we have derived sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solutions of system (2). However, there are still some limitations in our work which need to be improved. We only especially consider the white noise which is an idealized situation. In fact, the effect of colorful noise on system (2) is more general in line with the actual situation, and we leave it for our future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (no. 212111), the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (PKLHB1323), and the Key Subject of Hubei Province (Mathematics, Forestry), and the Project of Education Department of Hubei Province (B20111909).