Two types of permanence of a stochastic mutualism model
© Qiu et al.; licensee Springer 2013
Received: 3 February 2012
Accepted: 30 January 2013
Published: 18 February 2013
A stochastic mutualism model is proposed and investigated in this paper. We show that there is a unique solution to the model for any positive initial value. Moreover, we show that the solution is stochastically bounded, uniformly continuous and globally attractive. Under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean. Finally, we introduce some figures to illustrate our main results.
where , , , , are all positive, continuous and bounded functions on , and , are independent Brownian motions, and represent the intensities of the white noises.
To the best of our knowledge, a very little amount of work has been done on the stochastic system (4). Therefore, we aim to consider dynamical properties of the stochastic model (4) in this paper.
Since stochastic differential equation (4) describes population dynamics, it is necessary for the solution of the system to be positive and not to explode to infinity in a finite time. In this paper, we firstly show that the stochastic system (4) has a unique global (no explosion in a finite time) solution for any positive initial value in Section 2.1. To a population system, the stochastic boundedness is one of most important topics. Section 2.2 tells us that the stochastic model (4) is stochastically ultimately bounded. Furthermore, we will show that the solution of (4) is uniformly continuous and globally attractive in Section 2.3 and Section 2.4 respectively. Moreover, we obtain that the stochastic system is stochastically permanent (cf. [6, 8]) in Section 3. Section 4 shows that the stochastic system is persistent in mean (cf. [7, 12]). And under some conditions, we discuss the stochastic extinction of the system (4) in Section 5. We work out some figures to illustrate the various theorems obtained before in Section 6. Finally, we close the paper with conclusions in Section 7. The important contributions of this paper are therefore clear.
2 Basic properties of the solution
2.1 Positive and global solution
Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions. We denote by the positive cone in , and . And we use K to denote a positive constant whose exact value may be different in different appearances.
Theorem 1 For any given initial value , there is a unique solution to stochastic differential equation (4) on and the solution will remain in with probability 1, that is, for all almost surely.
Letting leads to the contradiction . Hence, we have a.s. The proof is complete. □
2.2 Stochastic boundedness
Stochastic boundedness is one of most important topics because boundedness of a system guarantees its validity in a population system. We first present the definition of stochastically ultimate boundedness.
Definition 1 (see )
Theorem 2 The solution of the system (4) is stochastically ultimately bounded for any initial value .
Thus . So, we have .
Applying the Chebyshev inequality yields the required assertion. □
2.3 Uniform continuity
In other words, almost every sample path of is locally but uniformly Hölder-continuous with exponent γ.
Theorem 3 For any initial value , almost every sample path of to (4) is uniformly continuous on .
where dropping from .
where dropping from and . Consequently, it follows from Lemma 1 that almost every sample path of is locally but uniformly Hölder continuous with an exponent , and therefore almost every sample path of is uniformly continuous on .
Similarly, by virtue of Lemma 1, almost every sample path of is uniformly continuous on . In a word, almost every sample path of to (4) is uniformly continuous on . □
2.4 Global attractivity
Here we show that the solution of (4) is globally attractive.
Lemma 2 (Barbalat )
Let be a non-negative function defined on such that is integrable on and is uniformly continuous on . Then .
then we say the system is globally attractive.
Theorem 4 Let , on hold. Then, for any initial value , the solution is globally attractive.
So, we complete the proof. □
3 Stochastic permanence
Let us now impose a hypothesis.
Assumption 1 .
Theorem 6 Let Assumption 1 hold. Then the system (4) is stochastically permanent.
The proof is an application of the well-known Chebyshev inequality and Theorems 2 and 5. Here it is omitted.
4 Persistence in mean
In view of ecology, a good situation occurs when all species co-exist. In this section, we will consider another stochastic persistence, that is, stochastic persistence in mean. Now, we present the definition of persistence in mean.
Firstly, we introduce a fundamental lemma which will be used.
So, the proof is complete. □
To continue our analysis, let us impose the following hypothesis.
Assumption 2 , .
Lemma 3, (10), (11) and (14) can straightforward imply the assertion.
Lemma 3, (12), (13) and (14) prove the result.
The proof is complete. □
In Sections 3 and 4, we showed that under certain conditions, the system was stochastically permanent and persistent in mean respectively. In view of ecology, a bad thing happens when a species disappears. Here, we will show that if the noise is sufficiently large, the solution to the associated stochastic model will become extinct with probability one.
So, we complete the proof. □
6 Numerical simulations
In this section we use the Milstein method mentioned in Higham  to substantiate the analytical findings.
where and are Gaussian random variables that follow .
By comparing Figure 1a,b with Figure 1c, we can observe that small environmental noise can retain the stochastic system permanent; however, sufficiently large environmental noise makes the stochastic system extinct.
In this paper, we consider the stochastic mutualism system (4). We show that there is a unique positive solution to the model for any positive initial value. Moreover, we show that the positive solutions are uniformly continuous, globally attractive. Especially, we conclude the following: under Assumption 1, the stochastic model (4) is stochastically permanent; under Assumption 2, the stochastic model (4) is persistent in mean. It is interesting and surprising to obtain the results. It is easy to see that Assumptions 1 and 2 have almost the same meaning. To a great extent, when the intensity of environmental noise is not too big, some nice properties such as non-explosion, boundedness, permanence are desired. However, Theorem 8 reveals that a large white noise will force the population to become extinct.
The authors are grateful to the associate editor and the referees for their helpful suggestions. This work was supported by the National Natural Science Foundation of P.R. China (No. 11171081, 11171056), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (No. HIT.NSRIF.2011094), the Scientific Research Foundation of Harbin Institute of Technology at Weihai (No. HIT(WH)ZB201103).
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