Abstract
We investigate a stochastic Food-Chain System , , , where , = is a standard Brownian motion. Firstly, the existence, the uniqueness, and the positivity of the solution are proved. Secondly, the stochastically ultimate boundedness of the system is investigated. Thirdly, the boundedness of moments and upper-growth rate of the solution are obtained. Then the global attractivity of the system is discussed. Finally, the main results are illustrated by several examples.
1. Introduction
In recent decades, a number of authors have studied the predator-prey type systems [1–8], especially the simple Food-Chain Systems [9–14]. These deterministic systems assume that all parameters in the models are deterministic, and some beautiful and valuable conclusions are obtained. Then it is well known that biology systems are often subject to environmental noise [15]. It is therefore useful to reveal how the noise affects these systems. Though there are various types of environmental noise, May [16] have claimed that the birth rates in the system should be stochastic. At the same time, the natural growth of many species vary with time, for example, due to the seasonality.
On the other hand, basing on some natural phenomenon and statistics in laboratory, the predator-dependent functional response should be considered in the predator-prey type systems. As we know, there are three classical predator-dependent functional responses: Hassell and Varley [17], Beddington et al. [18, 19], and Crowley and Martin [20]. Recently, Skalski and Gilliam [21] pointed out that these functional responses can provide better descriptions, and the Beddington-DeAngelis (B-D) type functional response performed even better in some cases.
Based on this background, in this paper we will discuss the following stochastic system: where , , is a Brownian motion defined on a complete probability space . All parametric functions are continuous and bounded functions on . , and denote the population density of resource, consumer, and predator at time , respectively. For convenience, the model, consisting of (1), together with , , , will be referred to as model SDE. On the other hand, for easy expression, if is a continuous bounded function on , we define and . Throughout this paper, we assume that , , and the others are similar. The notation used in this paper expresses the positive constant, for convenience, it denotes different constant in different places, and they only differ in positive constant times.
The rest of the paper is organized as follows. In Section 2, we show that the system has a global positive solution. In Section 3, the stochastically ultimate boundedness of the system is acquired. In Section 4, we obtain the asymptotic moment estimation and the upper-growth rate of the solution. In Section 5, the global attractivity of SDE is investigated. In Section 6, we illustrate our main results through several numerical examples. Last but not least, conclusions are drawn in Section 7.
The key method used in this paper is the analysis of Lyapunov functions. This Lyapunov function analysis for stochastic differential equations was developed by Khasminskii [22] and has been used by many authors [23–28].
2. Positive and Global Solutions
As the state of the SDE is the population density of species in the system at time , it should be nonnegative. Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial data, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [25]. However, the coefficients of each equation in SDE neither obey the linear growth condition nor local Lipschitz continuous. In this section, we show existence and uniqueness of the positive solution.
Lemma 1. For any initial value , , , system (SDE) has a unique positive local solution for almost surely (a.s.), where is the explosion time.
Proof. To begin with, consider the following equations: on with initial value , , , where , , . Notice that the last equations’ coefficients satisfy the local Lipschitz condition; thus, there is a unique solution on . Therefore, it follows from Itô’s formula that , , is the unique positive local solution of SDE with initial value , , .
Theorem 2. Consider system SDE, for any given initial value , and there is a unique solution on and the solution will remain in with probability 1, where .
Proof. From Lemma 1, to complete the proof we only need to show that . Hence, let be sufficiently large for , , and lying within the interval . For each integer , define the stopping times:
Clearly, is increasing as . Set , from which . Now, we only need to prove . If this statement is false, there is a pair of constants and such that . Thus there exists an integer such that
Define . If , we have
Integrating both sides of the above inequality from 0 to and then taking the expectations lead to
Set ; then it follows from inequality (4) that . Clearly, for every , there is , , or which equals either or ; therefore, is no less than
Then an application of (6) results in
where is the indicator function of . Letting leads to the contradiction
3. Stochastically Ultimate Boundedness
In the previous section, we have showed that the solutions of SDE will remain in the positive cone forever. However, this nonexplosion property in a population dynamical system is often not good enough while the property of ultimate boundedness is more desired. Let us discuss it in this section. We begin with giving the definition of stochastically ultimate boundedness.
Definition 3. The SDE is said to be stochastically ultimate bounded if for any , there is a positive constant such that for any initial value, the solution of SDE has the property that
Theorem 4. For any there is a positive constant such that for any initial value, the solution of SDE has the property that In particular, the SDE is stochastically ultimate bounded.
Proof. Define
Applying Itôs formula gives
Using Itôs formula again we have
Integrating the above inequality and then taking expectations on both sides, we can see that
This implies immediately that
On the other hand,
and then we have
which yields the required assertion by setting . Especially, let ; then
Now, for any , let , by Chebyshev’s inequality,
Hence,
In other word, SDE is stochastically ultimate bounded.
4. Boundedness of Moments and Upper-Growth Rate Estimation
Theorems 2 and 4 show that the solutions of SDE will remain in a subset of . Now let us construct different types of Lyapunov functions to investigate how the solutions behave in .
Theorem 5. Consider system SDE for any positive numbers , there exists a positive number such that, for any initial value , the solution of SDE obeys
Proof. Define
Applying Itôs formula gives
Integrating the above inequality and then taking expectations on both sides, we can see that
Together with , this inequality shows that
From this inequality, we obtain the desired assertion.
This theorem, together with Theorem 2, denotes , in other word, the average value of on time and “space” is bounded. It is similar to and .
Theorem 6. For any positive numbers , , and , solutions of SDE satisfy
Proof. Define
Applying Itôs formula gives
Let
where , and by the Itôs formula, we have
Therefore,
Define
Notice that the quadratic variation of is
Let and . In view of the exponential martingale inequality, for every and , we have
An application of Borel-Cantelli lemma yields that, for almost any , there is a random integer such that, for ,
So
for , a.s. Substituting the above inequality into (32), we have
In other words, we have shown that
Thus, for all , , we have
Then
Letting , , and , we can obtain the required assertion.
Based on this theorem, obviously, the upper-growth rate is linear.
5. Global Attractivity
In this section, we turn to establishing sufficient criteria for the global attractivity of system SDE. Firstly, we prepare some useful lemmas.
Lemma 7. Let be a solution to SDE with initial value . Then for all , there exist three constants related to : for , such that
Proof. Define for . Making use of Itôs formula we have Applying Itôs formula again to and then taking expectations on both sides, we can see that which is the desired assertion of , and the other two inequalities can be proved similarly.
Lemma 7 tells us that there is a , such that , , and for all . At the same time, it follows from the continuity of , , and that there exist , , and such that , , and for all . Denote , , and . Then for all ,
Lemma 8 (see [29]). Suppose that an -dimensional 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 .
Lemma 9. Let be a solution of SDE on with initial value ; then almost every sample path of is uniformly continuous on .
Proof. The second equation of SDE is equivalent to the following stochastic integral equation:
We estimate
On the other hand, as an application of the moment inequality for stochastic integrals [30], we see that for and ,
Then for , , , we obtain that
Then it follows from Lemma 8 that almost every sample path of is locally but uniformly Hölder-continuous with exponent for every and therefore almost every sample path of is uniformly continuous on . Similarly, we can show that almost every sample path of and is uniformly continuous on .
Lemma 10 (see [31]). Let be a nonnegative function defined on such that is integrable on and is uniformly continuous on . Then .
Now, we give our main result of this section.
Definition 11 (see [32]). Let and be two arbitrary solutions of system SDE with initial values , . If then we say that the system is globally attractive.
Theorem 12. If , , , then system SDE is globally attractive.
Proof. Define
then it is a continuous and positive function on . A direct calculation of the right differential of , and then applying Itö’s formula, we have
Integrating both sides gives
Therefore,
Making use of and the known conditions result in that
Then the required assertion follows from Lemmas 9 and 10 immediately.
6. Numerical Results
We present numerical experiments in this section to show how the proposed model works in the constructive examples. The results enhance the readers to understand the theoretical conclusions from the practical applications.
Here, we use the Milstein method [33] to construct the discretization equation of SDE. That is, where , , and , , are the Gaussian random variables. We obtain the following results.
As we pointed out in Theorem 6, a.s., and Figure 1 confirms this. Here we choose , , , , , , , , , , , and .
Figure 2 confirms that the system is globally attractive (Theorem 12). Here , , , , , , , , , . The initial values are and . Then , , , and the assumptions of Theorem 12 are satisfied.
7. Conclusions
This paper studies a stochastic nonautonomous food-chain system with Beddington-DeAngelis functional response. This system retains some conventional properties of stochastic systems. That is, it has a unique global positive solution for any initial conditions, the moments of the solutions are bounded, and the solutions are global attractive and stochastically ultimate bounded.
Nevertheless, there are rooms to continue working on this issue. It is interesting to study other parameters; for example, are stochastic. The permanence and extinction of the system and the stability in distribution deserve to be investigated too.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research is supported by the National Natural Science Foundation of China under Grant no. 11302126.