Research Article | Open Access
Chunyan Ji, Daqing Jiang, Hong Liu, Qingshan Yang, "Existence, Uniqueness and Ergodicity of Positive Solution of Mutualism System with Stochastic Perturbation", Mathematical Problems in Engineering, vol. 2010, Article ID 684926, 18 pages, 2010. https://doi.org/10.1155/2010/684926
Existence, Uniqueness and Ergodicity of Positive Solution of Mutualism System with Stochastic Perturbation
We discuss a two-species Lotka-Volterra mutualism system with stochastic perturbation. We show that there is a unique nonnegative solution of this system. Furthermore, we investigate that there exists a stationary distribution for this system, and it has ergodic property.
It is well known that the differential equation denotes the population growth of mutualism system for the two species. and represent the densities of the two species at time , respectively, and the parameters are all positive. Goh  showed that the asymptotic stability equilibrium state of (1.1) in local must be asymptotic stability in global. That is, if , and , then where is the unique positive equilibrium of system (1.1) and While if , then the population of both species increase to infinite. There are extensive literature concerned with mutualism system; see [2–7].
The papers mentioned above are all deterministic models, which do not incorporate the effect of fluctuating environment. In fact, environmental fluctuations are important components in the population system. Most of natural phenomena do not follow strictly deterministic laws, but rather oscillate randomly around some average values, hence the deterministic equilibrium is no longer an absolutely fixed state . Therefore stochastic differential equation models play a significant role in various branches of applied sciences including the population system, as they provide some additional degree of realism compared to their deterministic counterpart [9–13]. Recently, many authors have paid attention to how population systems are affected by random fluctuations from environment (see, e.g., [14–18]). However, as far as we known, there is few paper consider how environmental noises affect the dynamical behaviors of the mutualism system, Zeng et al.  discussed the effects of noise and time delay on (the normalized correlation function) and (the associated relaxation time) of a mutualism system, in which they considered the intraspecies interaction parameters were stochastically perturbed. Motivated by this, the main aim of this paper is to study the dynamical behaviors of the mutualism system with stochastic perturbation.
In this paper, considering the effect of randomly fluctuating environment, we incorporate white noise in each equation of system (1.1). Here we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the natural growth rates . Suppose , where are mutually independent one dimensional standard Brownian motions with , and are the intensities of white noises. The stochastic version corresponding to the deterministic system (1.1) takes the following form:
This paper is organized as follows. In Section 2, we show there is a unique positive solution of (1.4) if , and give out the estimation of the solution. The stability of system (1.4) is investigated in Section 3. Since (1.4) does not have interior equilibrium, we cannot discuss the stability as the deterministic system. First, we show there is a stationary distribution of (1.4) and it has ergodic property. Next, by estimating the moment, we explore some properties of the solution.
Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let denote the positive cone of , namely . If is a vector, its transpose is denoted by . For .
2. The Existence and Estimation of the Solution
To investigate the dynamical behavior, the first concern thing is whether the solution is global existence. Moreover, for a population model, whether the value is nonnegative is also considered. Hence in this section we first show the solution of (1.4) is global and nonnegative. As we have known, for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (cf. Arnold , Mao ). However, the coefficients of (1.4) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of (1.4) may explode at a finite time. In this section, following the way developed by Mao et al. , we show there is a unique positive solution of (1.4).
Theorem 2.1. There is a unique positive solution of system (1.4) for any given initial value provided .
Proof. The proof is similar to Theorem in . Here we define a -function : where satisfies
Remark 2.2. Theorem 2.1 shows stochastic equation (1.4) also has a global positive solution under the same condition of the corresponding deterministic system (1.1). That is to say, the white noise does not affect the existence of the unique global positive solution.
In the remaining of this section, we give the estimation of the solution of system (1.4).
Jiang and Shi  discussed a randomized nonautonomous logistic equation: where is 1-dimensional standard Brownian motion, and is independent of . They showed the following.
Lemma 2.3 (see ). Assume that and are bounded continuous functions defined on , and . Then there exists a unique continuous positive solution of (2.3) for any initial value , which is global and represented by
Theorem 2.4. Assume that and is the solution of system (1.4) with initial value . Then has the property that where and are the solutions of equations:
3. Stationary Distribution and Ergodicity for System (1.4)
In the introduction, we have mentioned that if , and , then the unique positive equilibrium of (1.1) is globally stable. But there is none positive equilibrium for (1.4). We investigate there is a stationary distribution for system (1.4) instead of asymptotically stable equilibria . Before giving the main theorem, we first give a lemma (see ).
Assumption B. There exists a bounded domain with regular boundary , having the following properties.(B.1)In the domain and some neighbourhood thereof, the smallest eigenvalue of the diffusion matrix is bounded away from zero.(B.2)If , the mean time at which a path issuing from reaches the set is finite, and for every compact subset .
Lemma 3.1 (see ). If (B) holds, then the Markov process has a stationary distribution . Let be a function integrable with respect to the measure . Then for all .
Proof. Define : Then where according to the equality (2.2). By Young inequality, we have Then Note that ; then the ellipsoid lies entirely in . We can take to be any neighborhood of the ellipsoid with , so for , , which implies that condition (B.2) in Lemma 3.1 is satisfied. Besides, there is such that which implies condition (B.1) is also satisfied. Therefore, the stochastic system (1.4) has a stable a stationary distribution and it is ergodic.
Remark 3.3. If and , we can choose , then and .
Since system (1.4) is ergodic, next we explore some properties of the solution.
Consider the equation with initial value . It is well known, when , (3.9) has a unique positive solution
Lemma 3.4. Suppose that and is the solution of (2.6), then one has where is the solution of
Proof. From the representation of the solution , we have where the last inequality is based on the property of Brownian motion that . Similarly, we have Therefore which is as required.
Lemma 3.5. Suppose that and is the solution of (2.6), then one has
Proof. It is easy to drive from Lemma 3.4 that Note that the distribution of is the same as , and that has the same distribution as—, then by (3.11) and the strong law of large numbers, we get This completes the proof of Lemma 3.5.
Lemma 3.6. Suppose that and is the solution of (2.7), then one has
Lemma 3.7. Let , where is 1-dimensional standard Brownian motion, then
Proof. The proof can be found on [21, page 70].
Based on these lemmas, now we show the main result in this section.
Let , ; then by Itô's formula we obtain If and , then the equation has a unique positive solution:
Lemma 3.8. Assume and . Then for any initial value , the solution of system (1.4) has the following property: where
Proof. It follows from (3.22) that
Obviously, to prove the result, it is an easy consequence of
We first show that
In fact, the results of Theorem 2.4 and Lemmas 3.5 and 3.6 imply that (3.28) is true.
Next, we will prove If , then there exist positive constants such that From (3.22) we get Note that the function has its maximum value at , and the function has its maximum value at ; then where . Integrating both sides of it from to , yields where . It is easy to drive from Lemma 3.7 that which implies Moreover (3.35) together with (3.28) shows that that is, Similarly, we have which is as required.
Lemma 3.9. Assume and . Then for any initial value , there exists a positive constant such that the solution of system (1.4) has the following property:
Proof. By Itô's formula and Young inequality, we compute
Hence, for positive constants and , we have
Next, we claim that there are such that
if and . In fact, we only need which can be simplified to It is obviously true, if and .
Let , . Then and Hence, By comparison theorem, we can get which implies that there is a , such that Besides, note that is continuous, then there is a such that Let , then
By the ergodic property, for , we have On the other hand, by dominated convergence theorem, we can get which together with (3.49) implies Letting , we get That is to say, functions and are integrable with respect to the measure . Therefore one has the following.
Theorem 3.10. Assume , , and , where , and . Then for any initial value , the solution of system (1.4) has the following property:
Moreover, we can get the following.
Theorem 3.11. Assume , and . Then for any initial value , the solution has following property where is defined as in (3.24).
Proof. By (3.39), for , we have In view of the well-known Borel-Cantelli lemma, we see that for almost all , holds for all but finitely many . Hence there exists a , for all excluding a -null set, for which (3.56) holds whenever . Consequently, letting , we have, for almost all Similarly, we can obtain Besides, by (3.39) and its ergodic property, we get On the other hand, we have Then Therefore, (3.25), (3.57), (3.58), and (3.59) imply
At the end of this section, to conform the results above, we numerically simulate the solution of (1.4). By the method mentioned in , we consider the discretized equation: given the values of and parameters in the system, by Matlab software we get Figure 1.
Figure 1 gives the solutions of (1.1) and (1.4), and the real lines and the imaginary lines represent the deterministic and the stochastic, respectively. In this figure, we choose parameters such that the conditions said in theorems are satisfied. Hence, although there is no equilibrium of the stochastic system (1.4) as the deterministic system (1.1), but the solution of (1.4) is ergodicity. From the figure, we can see that the solution of system (1.4) is fluctuating around a constant.
The work was supported by the Ministry of Education of China (no. 109051), the Ph.D. Programs Foundation of Ministry of China (no. 200918), Young Teachers of Northeast Normal University (no. 20090104), and the Fundamental Research Funds for the Central Universities (no. 09SSXT117), the Fundamental Research Funds for the Central Universities.
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