Asymptotic Behavior of Positive Solutions of a Competitive System Subject to Environmental Noise

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.

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   (0) > 0,   () is independent white noise with   (0) = 0,  ≥ 0, and  2  represents the intensity of the noise,  = 1, 2.   () is standard Brownian motion defined on the complete probability space (Ω, F, {F  } ≥0 , ) with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 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.
Lemma 6 (see [27,28]).Suppose that a stochastic process () on  ≥ 0 satisfies the condition for some positive constants , , and .Then there exists a continuous modification X() of (), which has the property that, for every  ∈ (0, /), there is a positive random variable ℎ() such that  { : sup In other words, almost every sample path of X is locally but uniformly Hölder continuous with exponent .

Existence, Uniqueness, Stochastic Boundedness, and Extinction
We first present the existence and uniqueness of positive solution of system (2).
Next, we investigate the stochastic boundedness of the positive solutions of system (2).To this end, we first give the following Lemma 8. 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   (0) < (  + (1/2)( − 1) 2  )/  ,  = 1, 2, we have Furthermore, using the standard comparison principle, one can show that Then we can obtain that where This completes the proof.
Proof.Define, respectively, Lyapunov functions ln  1 () and ln  2 ().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 lim sup This completes the proof.

Global Attraction
In this section, we first introduce Lemma 11 before we show the global attraction of system (2).
Proof.It follows from system (2) that where Applying Lemmas 4 and 8, for any  > 1, one derives that Without loss of generality, we assume that  > 2. Using the moment inequality (see [10]) to stochastic integral (45), we can obtain that where 0 ⩽  1 <  2 < +∞ and  > 2. We further let then by ( 47), (48), and Lemma 4, one yields that It follows from Lemma 6 that almost every sample path of  1 () is uniformly continuous on  ≥ 0. Similarly, we can show that almost every sample path of  2 () is uniformly continuous on  ≥ 0. Therefore, ( 1 (),  2 ()) is uniformly continuous on  ≥ 0, a.s.This completes the proof.
We can now present the result on global attraction of system (2).
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
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.