Abstract
We consider a predator-prey model in which the preys disperse among patches () with stochastic perturbation. We show that there is a unique positive solution and find out the sufficient conditions for the extinction to the system with any given positive initial value. In addition, we investigate that there exists a stationary distribution for the system and it has ergodic property. Finally, we illustrate the dynamic behavior of the system with via numerical simulation.
1. Introduction
Interest has been growing in the study of the dynamic relationship between predators and their preys due to its universal existence and importance. However, due to the spatial heterogeneity and the increasing spread of human activities, the habitats of many biological species have been separated into isolated patches. In some of these patches, without the contribution from other patches, the species will go to extinction. Recently, the effect of dispersion on the species survival has been an important subject in population biology (see [1–10] and the references cited therein). Particularly, two species predator-prey systems with dispersal have received great attention from both theoretical and mathematical biologists and many good results have been achieved (see [1, 2, 7, 9–11]). The analysis of these papers has been centered around the coexistence of populations, stability (local and global), and permanence of equilibria. Zhang and Teng [11] established the sufficient conditions on the boundedness, permanence, and existence of positive periodic solution for two species predator-prey model. Kuang and Takeuchi [1] studied a predator-prey system with prey dispersal in a two-patch environment; they obtained the existence, local and global stability of the positive steady state and analyzed both the stabilizing and destabilizing effects of dispersion by introducing examples.
Li and Shuai [2] considered the model where , denote the densities of preys and predators on the patch , respectively. The parameters in the model are nonnegative constants and , are positive. The constants are the dispersal rate from patch to , and the constants can be selected to represent different boundary conditions in the continuous diffusion case [12]. Let denote dispersal matrix. By constructing a Lyapunov function and using graph theory, Li and Shuai proved the uniqueness and globally asymptotically stability of a positive equilibrium, whenever it exists, if is irreducible and there exists such that or .
The model mentioned above is a deterministic model which assumes that the parameters in the model are all deterministic irrespective environmental fluctuations. In fact, population dynamics is inevitably affected by environmental white noise, such as weather and epidemic disease. Therefore, the deterministic models are often subject to stochastic perturbation, and it is useful to reveal how the noise affects the population system. There are some authors who have studied the dynamics of predator-prey models with stochastic perturbations (see [13–15]). Ji et al. [13] studied a predator-prey with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation; they got some good results about existence, uniqueness, and extinction of positive solution. Cai and Lin [15] investigated a predator-prey stochastic system with competition among predators and obtained the probability distribution of the system state variables at the state of statistical stationarity. But, until now, few people study the dynamical behavior of the predator-prey system with diffusion under the influence of white noise. However, the diffusion phenomenon and environmental white noise are universal existence in nature. Therefore, we want to study the effect of random perturbations on the predator-prey system on the basis of the existing diffusion model and the contents of this paper are of great significance.
In this paper, we take into account the effect of randomly fluctuating and stochastically perturb intrinsic growth rate in each equations of (1.1): where are mutually independent Brownian motions and are positive constants representing the intensity of the white noises, respectively. Then the stochastic system takes the following form: Throughout this paper, we assume are nonnegative constants, is irreducible, and the parameters , , , , , are positive constants.
In order to obtain better dynamic properties of the SDE (1.3), we will show that there exists a unique positive global solution with any initial positive value, and its th moment is bounded in Section 2. In the study of a population dynamics, permanence is a very important and interesting topic regarding the survival of populations in ecological system. In a deterministic system, it is usually solved by showing the global attractivity of the positive equilibrium of the system. But, as mentioned above, it is impossible to expect stochastic system (1.3) to tend to a steady state. So we attempt to investigate the stationary distribution of this system by Lyapunov functional technique. The stationary can be considered a weak stability, which appears as the solution is fluctuating in a neighborhood of the equilibrium point of the corresponding deterministic model. In Section 3, we will show if the white noise is small, there is a stationary distribution of SDE (1.3) and it has ergodic property. Existing results on dynamics in a patchy environment have largely been restricted to extinction analysis which means that the population system will survive or die out in the future due to the increased complexity of global analysis. In Section 4, we give the sufficient conditions for extinction. In Section 5, we make numerical simulation to conform our analytical results. Finally, for the completeness of the paper, we give an appendix containing some theories which will be used in previous sections.
The key method used in this paper is the analysis of Lyapunov functions [6, 13, 14, 16]. We will also use the graph theory in Section 3 and some graph definitions can be found in the appendix.
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, . For convenience and simplicity in the following discussion, denote . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by whilst its operator norm is denoted by .
2. Positive and Global Solutions
In order for a stochastic differential equation to have a unique global (i.e., no explosion at any finite time) solution, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (see [17]). However, the coefficients of SDE (1.3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of SDE (1.3) may explode at a finite time. In this section, we will prove the solution of stochastic system (1.3) with any positive initial value is not only positive but also not exploive in infinity at any finite time.
Theorem 2.1. For any given initial value , there is a unique positive solution of SDE (1.3), and the solution will remain in with probability 1.
Proof. We define a -function : Applying Itô’s formula, we have It is clear that the coefficient of quadratic term is negative, so we must be able to find a positive constant number that satisfies and is independent of , , and . By the similar proof of Li and Mao [18, Theorem 2.1], we can obtain the desired assertion.
Theorem 2.1 shows that the solution of SDE (1.3) will remain in the positive cone with probability 1. This nice property provides us with a great opportunity to discuss the th moment and stochastically ultimately boundedness of the solution.
Definition 2.2 (see [18]). The solution of SDE (1.3) is said to be stochastically ultimately bounded, if for any , there exists a positive constant , such that for any initial value , the solution to (1.3) has the property that
Lemma 2.3. For any given initial value , there exist positive constants , , and , such that the solution of SDE (1.3) has the following property:
Proof. By Itô’s formula and Young inequality, we compute where are positive constants to be determined. Hence, for positive constants , we have Next, we claim that there exist , and such that In fact, we only need and , where is a sufficiently large positive integer. Let It is clear that . Then Integrating it from 0 to and taking expectations of both sides, we obtain that Therefore, letting , we have Notice that the solution of equation obeys By comparison theorem, we can get which implies that there is a , such that In addition is continuous, so we have Let , then This completes the proof.
Theorem 2.4. The solutions of system (1.3) are stochastically ultimately bounded for any initial value .
Proof. From Theorem 2.1, the solution will remain in with probability 1. Let . Note that and . Therefore, we get and by (2.5) we have Applying the Chebyshev inequality yields the required assertion.
3. Stationary Distribution
In this section, we investigate there is a stationary distribution for SDE (1.3) instead of asymptotically stable equilibria. Before giving the main theorem, we first give a lemma (see [2]).
Lemma 3.1 (see [2]). Assume is irreducible. If there exists such that or , then, whenever a positive equilibrium exists for system (1.1), it is unique and globally asymptotically stable in the positive cone .
In the section, we assume system (1.1) exists and the positive equilibrium satisfies the equation where .
We now state a theorem in which the graph theory will be used. Assume defined as in Lemma A.1 are nonnegative constants and .
Theorem 3.2. Assume . Then there is a stationary distribution for SDE (1.3) and it has ergodic property. Here is the solution of (3.1), and .
Proof. Define : Then By (3.1), we have Substituting this into (3.3) gives Here we use the fact: , for with equality holding if and only if . Since is irreducible, , , we know matrix ) is irreducible. Let , and by Lemma A.1, we have which implies that The following proof of ergodicity is similar to Theorem 3.2 in [19]. Assume , then Note that , then the ellipsoid lies entirely in . We can take to be a neighborhood of the ellipsoid with , so for , ( is a positive constant), which implies the condition in Lemma A.2 is satisfied. Therefore, the solution is recurrent in the domain , which together with Remark A.3 and Lemma A.4 implies is recurrent in any bounded domain . Thus, for any , there is such that for all , , which implies that condition is also satisfied. Therefore, the stochastic system (1.3) has a stable stationary distribution and it is ergodic.
4. Extinction
In this section, we will show that if the noise is sufficiently large, the solution to the associated SDE (1.3) will become extinct with probability 1, although the solution to the original equation (1.1) may be persistent. For example, recall a simple case, namely, the scalar logistic equation with initial value . It is well known that, when , the solution is persistent, because However, consider its associated stochastic equation where , then the solution to this stochastic equation will become extinct with probability 1, that is to say, if , The following theorem reveals the important fact that the environmental noise may make the population extinct.
Theorem 4.1. For any given initial value , the solution of the SDE (1.3) has the property that Here Particularly, if , then .
Proof. Define By Itô’s formula, we have Thus we compute Letting , we compute By Cauchy inequality, we compute also where . Substituting these two inequalities into (4.9) yields This implies where is a martingale defined by with . The quadratic variation of this martingale is where . By the strong law of large numbers for martingale (see [17, 20]), we have It finally follows from (4.13) by dividing on the both sides and then letting that which implies the required assertion.
Remark 4.2. Theorem 4.1 shows that if the condition holds, that is, when the prey population is disturbed by large white noise, the species of prey will extinct.
Now we give the following theorem which describes the entire extinction.
Theorem 4.3. For any given initial value , the solution of the SDE (1.3) has the property that Here Particularly, if , then .
Proof. The proof of the theorem is similar to Theorem 4.1, we only give the main proof procedure. Define
Let and . By Itô’s formula, we compute
The rest of the proof is similar to Theorem 4.1.
Remark 4.4. Theorem 4.3 states that when the prey and predator population are all disturbed by large white noise and the condition holds, the two species will be extinct.
5. Numerical Simulation
In this section, in order to better study the effect of white noise in diffusion system, we assume , are nonnegative constants, is irreducible , and . Consider the predator-prey system with , that is,
In order to better study the previous results, we will numerically simulate the solution of (5.1). By the method mentioned in [21], we consider the following discretized equation:
Using the discretized equation and the help of Matlab software, we choose the appropriate parameters , the initial value , and time step ; then . In order to better investigate the white noise, we divide its intensity into small, medium, and large three cases to study.
In Figures 1, 2, and 3, we choose and ( satisfy Lemma A.1 and Theorem 3.2); then , and so the condition is also satisfied. Therefore, by Theorem 3.2, there is a stationary distribution (see the histogram on the right in Figure 1). From the left picture in Figure 1, we can see that the solution of system (5.1) is fluctuating in a small neighborhood. Moreover, from Figure 2, we find that almost all population distribution lies in the neighborhood, which can be imagined by a circular or elliptic region centered at (see the scatter picture in Figure 2). Figure 3 shows that when the white noise is small, stochastic system imitates deterministic system and their curves are almost coincident. Hence, the solution of (5.1) is ergodicity, although there is no equilibrium of the stochastic system as the deterministic system. All of these imply system (5.1) is a stochastic stability.
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Comparing with small white noise as in Figures 1, 2, and 3, we select the relatively large white noise in Figures 4, 5, and 6. We find that , so the conditions of Theorem 3.2 are not satisfied; therefore, there is not a stationary distribution although the deterministic system is global asymptotic stability. The condition , that is, all extinction condition does not hold by Theorem 4.3 and the predator will die. From Figure 4, we see that the fluctuations on the left figures are more intense and histogram distribution is not concentrated comparing with Figure 1, close to the point .
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In Figure 7, we assume both the predator and the prey population suffered large white noise; we choose , which satisfy the cases said in Theorem 4.1, that is . As the case in Theorem 4.1 expected, the large white noise leads to the extinction of the prey which also leads to the extinction of the predator, so the solution of system (5.1) tends to zero.
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In Figure 8, we choose the same parameters as in Figure 1, but change the value of to , which do not meet the conditions for extinction of the two species as in Theorem 4.3. The predator population suffer the large white noises and then die out, but the prey will survive.
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From these figures, we can get the following conclusions: when the white noise is small, system (5.1) imitates its deterministic system (see Figures 1, 2, and 3). But when the white noise is relatively large, it will bring more big deviation (see Figures 4, 5, and 6) even the extinction of the species (see Figures 6, 7, and 8), which will not happen in the deterministic system. However, when the intensity of the white noise on prey species is large, the predator and prey population will be extinct (see Figure 7). In real world, the large white noise may be bad weather, serious epidemic, which can be considered as the decisive factors responsible for the extinction of populations. Therefore, the study of stochastic model has great practical significance.
Appendix
In this section, we list some theories used in the previous sections.
Some Graph Theories [2, 22]
A directed graph or digraph = contains a set of vertices and a set of arcs leading from initial vertex to terminal vertex . In our convention, if and only if there exists an arc from vertex to vertex in .
A digraph is weighted if each arc is assigned a positive weight . Given a weighted digraph (, ) with vertices, where is weight matrix, whose entry equals the weight of arc if it exists, and 0 otherwise.
A digraph (, ) is strongly connected if for any pair of distinct vertices, there exists a directed path from one to the other. A weighted digraph (, ) is strongly connected iff the weight matrix is irreducible.
The Laplacian matrix of (, ) is defined as
Let denote the cofactor of the th diagonal element of , which has the following property.
Lemma A.1 (see [2]). Assume . If (, A) is strongly connected. Then(1) for ,(2)the following identity holds: where () are arbitrary functions.
Some Theories about the Stationary Distribution [23]
Let be a homogeneous Markov Process in ( denotes -space) described by the stochastic equation
The diffusion matrix is
Assumption B. There exists a bounded domain with regular boundary , having the following properties: In the domain and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix is bounded away from zero. If , the mean time at which a path issuing from reaches the set is finite and for every compact subset .
Lemma A.2 (see [23]). If Assumption B holds, then the Markov process has a stationary distribution . Let be a function integrable with respect to the measure . Then for all .
Remark A.3. Theorem 2.1 shows that there exists a unique positive solution of SDE (1.3). Besides, from the proof of Theorem 2.1, we obtain Now define ; then and we can get where . By [23] (Remark 2 of Theorem 4.1), we obtain the solution is a homogeneous Markov process in .
Lemma A.4 (see [23]). Let be a regular temporally homogeneous Markov process in . If is recurrent relative to some bounded domain , then it is recurrent relative to any nonempty domain in .
Acknowledgments
The work was supported by the NSFC of China (no. 10971021), the NNSF of China (no. 11001032), the Ministry of Education of China (no. 109051), the Ph.D. Programs Foundation of inistry of China (no. 200918) and Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT).