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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 401031, 24 pages
http://dx.doi.org/10.1155/2012/401031
Research Article

The Complex Dynamics of a Stochastic Predator-Prey Model

1College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
2School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China

Received 19 May 2012; Accepted 29 July 2012

Academic Editor: Sergey V. Zelik

Copyright © 2012 Xixi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A modified stochastic ratio-dependent Leslie-Gower predator-prey model is formulated and analyzed. For the deterministic model, we focus on the existence of equilibria, local, and global stability; for the stochastic model, by applying Itô formula and constructing Lyapunov functions, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. Based on these results, we perform a series of numerical simulations and make a comparative analysis of the stability of the model system within deterministic and stochastic environments.

1. Introduction

The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1]. In recent years, one of important predator-prey models is Leslie-Gower model [2, 3], which has been extensively studied [4, 5]. And, more and more obvious evidences of biology and physiology show that in many conditions, especially when the predators have to search for food (consequently, have to share or compete for food), a more realistic and general predator-prey system should rely on the theory of ratio-dependence, this is confirmed by lots of experimental results [4, 6]. A ratio-dependent Leslie-Gower predator-prey model [7], takes the form where , stand for the population (the density) of the prey and the predator at time , respectively, and is the predator functional response to prey. And we assumed that the prey grows logistically with growth rate and carrying capacity in the absence of predation. The predator consumes the prey according to the functional response and grow logistically with growth rate and carrying capacity proportional to the population size of prey (or prey abundance). The parameter is a measure of the food quality that the prey provides for conversion into predator birth. The term of this equation is called the Leslie-Gower term.

On the other hand, the predator can switch over to other population when the prey population is severely scarce, but its growth will be limited, because we cannot forget the fact that its most favorite food, the prey , is not in abundance. In this situation, a positive constant can be added to the denominator, measures the extent to which the environment provides protection to the predator [8, 9], and the second equation of model (1.1) becomes

Based on the above discussions, in the paper, we will focus on the following ratio-dependent Leslie-Gower model:

The rest of the paper is organized as follows. In Section 2, we give some theorems about the stability property of the equilibria of model (1.3). In Section 3, we establish a stochastic model based on model (1.3) and focus on the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability of the stochastic model. In Section 4, we give some numerical examples and make a comparative analysis of the stability of the model system within deterministic and stochastic environments.

2. The Dynamics of Model (1.3)

2.1. Dissipativeness

By standard simple arguments, one can show that the solution of model (1.3) always exists and stays positive. In fact, from the first equation of model (1.3), we can get Hence, there exists a such that, for , .

From the second equation of model (1.3), we see that, for , we have . A standard comparison argument shows that

Thus, we have the following conclusion.

Lemma 2.1. Model (1.3) is dissipative.

Lemma 2.2. If , then model (1.3) is permanent.

Proof. If , from the first equation of model (1.3), we have . Therefore, by standard comparison argument, we have
Hence, for any and large , , and
From the arbitrariness of , we can get that

2.2. Stability Analysis of the Equilibria

In this section, we will focus on the existence of equilibria and their stabilities of model (1.3).

It is easy to find that model (1.3) always has three boundary equilibria , , . And the positive equilibria satisfies the equations which yields

For simplicity, we define then (2.7) can be rewritten as

Lemma 2.3 (existence of equilibria). Suppose and , in (2.9), then one has(a1) If , then it has two positive roots given by Therefore, model (1.3) has two positive equilibria and .(a2) If , (2.9) has a unique positive root of multiplicity-2 given by . Thus, model (1.3) has a unique positive equilibrium (a3) If , there are no positive roots that exist and then, model (1.3) has no positive equilibrium.
, (2.9) has a unique positive root , model (1.3) has a unique positive equilibrium .
, (2.9) has a unique positive root , model (1.3) has a unique positive equilibrium .

Next, we discuss the local stabilities of these equilibria. Easy to obtain the following results:

Lemma 2.4.    is a saddle point;(ii) if , is a stable node point. If , is a saddle point.

Following, let be an arbitrary positive equilibrium. The Jacobian matrix for is given by

Then we can get the following:

So the sign of is determined by

Theorem 2.5. For model (1.3), the stabilities of and are as follows.(a) The positive equilibrium is(a1) stable if and only if (a2) unstable if and only if (b) The positive equilibrium is a saddle point.

Proof. (a) At the point , we have the following: thus, , and the nature of singularity depends on the trace given by Clearly, (a1) If , then , and is stable.(a2) If , then , and is unstable.(b) At the point , we have the following: then , and is a saddle point.

Figure 1 shows the dynamics of model (1.3). In this case, is a nodal source, is a saddle point, is a nodal sink, is a saddle point, and is locally asymptotically stable. There exists a separatrix curve determined by the stable manifold of , which divides the behavior of trajectories, that is, the stable manifold of saddle split the feasible region into two parts such that orbits initiating inside tend to the positive equilibrium , while orbits initiating outside tend to except for the stable manifolds of .

401031.fig.001
Figure 1: The phase portraits of model (1.3). The parameters are taken as . In this case, is a source, and are saddle points, and is a nodal sink, which is stable. The positive equilibrium is locally asymptotically stable. The dashed curve is the -nullcline , and the dotted curve is the -nullcline .

Theorem 2.6. The singularity is(a) stable if and only if (b) unstable if and only if

Proof. At the point , we have as , then , and the nature of singularity depends on the trace given by
Clearly,(a) If , then , and is stable.(b) If , then , and is unstable.

Theorem 2.7. Singularity is(a) stable if and only if (b) unstable if and only if

Proof. At the point , as , so as , then , and the nature of singularity depends on the trace given by Clearly,(a) if , then , is stable;(b) if , then , is unstable.

Theorem 2.8. Singularity is(a) a non-hyperbolic attractor node if and only if (b) a non-hyperbolic repellor node if and only if

Proof. At the point , we have then, , and .
Hence we can conclude (2.28) and (2.29).

Theorem 2.9. If and hold the boundary equilibria of model (1.3) is globally asymptotically stable.

Proof. Since and , from the first equation of model (1.3), for any , there exists a , for all , we have From the arbitrariness of , we can get that
As , by standard comparison arguments, it follows that thus,
As a result, using the second equation of model (1.3), one can easily know that . The proof is complete.

Figure 2 shows the dynamics of model (1.3). In this case, is a nodal source, is a saddle point, and is globally asymptotically stable, that is, all orbits tend to the equilibrium for any initial values.

401031.fig.002
Figure 2: The phase portraits of model (1.3). The parameters are taken as . In this case, is a nodal source, is a saddle point. The boundary equilibria is globally asymptotically stable. The dashed curve is the -nullcline , and the dotted curve is the -nullcline .

Theorem 2.10. Assume , is globally asymptotically stable, if the following conditions hold(i); (ii), where .

Proof. Define a Lyapunov function: so, let If Hence, according to Lemma 2.2, we can obtain that so,
Considering , we obtain . This ends the proof.

Figure 3 shows the dynamics of model (1.3) for the case of . In this case, is a nodal source, is a saddle point, is a saddle point, and is globally asymptotically stable, that is, all orbits tend to the equilibrium for any initial values.

401031.fig.003
Figure 3: The phase portraits of model (1.3). The parameters are taken as ,  . In this case, and are saddle points, the positive interior equilibrium point is globally asymptotically stable. The dashed curve is the -nullcline , and the dotted curve is the -nullcline .

3. The Stochastic Model

Those important and useful works on deterministic models provide a great insight into the effect of the pollution. In the real world, population dynamics is inevitably subjected to environmental noise (see e.g., [10, 11]), which is an important component in an ecosystem. May [12] pointed out the fact that due to environmental noise, the birth rates, carrying capacity, competition coefficients, and other parameters involved in the system exhibit random fluctuation to a greater or lesser extent.

In this part, we focus on the stochastic stability analysis of model (1.3).

Taking into account the effect of randomly fluctuating environment, we incorporate white noise in each equations of model (1.3). We assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the growth rates of the prey population and the predator population, set then the stochastic version of model (1.3) is given by the following Itô type where are the 1-dimensional standard Brownian motion defined on a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and increasing while contains all -null sets) and , are, respectively, white noises with possible intensity , .

3.1. Existence of Global Positive Solutions

Lemma 3.1. There is a unique local positive solution for to model (3.2) almost surely (a.s.) for the initial value , , where is the explosion time.

Proof. Consider the equations on with initial value , .
It is easy to see that the coefficients of model (3.3) satisfy the local Lipschitz condition, then there is a unique local solution on [13]. Therefore, by Itô formula, are the unique positive local solutions to model (3.3) with initial value .

Lemma 3.1 only tells us that there has a unique local positive solution to model (3.2). Next, we show this solution is global, that is, .

Theorem 3.2. Consider model (3.2), for any given initial value , there is a unique solution on and the solution will remain in with probability 1, where .

Proof. For convenience of statement, we introduce some notations. Define Let be sufficiently large for and lying within the interval . For each integer , define the stopping times
Assume (as usual the empty set). Clearly, is increasing as . Let , then a.s. If we can show that a.s., then a.s., and a.s., for all . In other words, to complete the proof all we need to prove that a.s. If this statement is false, then there is a pair of constants and such that . There is an integer , such that Define a function by Obviously, is nonnegative. And where are positive numbers. Integrating both sides of the above inequality from to and then taking expectations, yields Set for and by (3.6), we have . Note that for every , there is some such that equals either or for , hence is no less than . It then follows from (3.9) that where is the indicator function of . Let , then
This completes the proof.

3.2. Stochastic Boundedness

Let us now recall the definition of stochastically ultimate boundedness [1416].

Definition 3.3. The solution of model (3.2) is said to be stochastically ultimately bounded if for any , there is a positive constant , such that for any initial data , the solution of model (3.2) has the property that

Lemma 3.4. For any , there is a positive constant , which is independent of the initial data , such that the solution of model (3.2) has the property that

Proof. Define For the sake of discussion, we rewrite the above as
By using the Itô formula, we have as , we have here is an integer. So
Now, by using the Itô formula again, we have
According to the above, we can easily get
On the other hand, so, we can obtain
Set , this completes the proof.

Theorem 3.5. The solutions of model (3.2) is stochastically ultimately bounded.

Proof. By Lemma 3.4, let , there is a such that
Now, for any , let . Then by Chebyshevs inequality
Hence,
This ends the proof.

Next, we study the asymptotic properties of the moment solutions of model (3.2).

Theorem 3.6. For any given , there is a , let be the solution of model (3.2) with any initial value , then

Proof. Set , from Lemma 3.4, we have so, we get here is a positive number, so
Let , we can get
That is,
Set , so

3.3. Stochastic Asymptotic Stability

Note that a solution of model (1.3) is also a solution of model (3.2), so, in the following, we will focus on stochastic asymptotic stability of the positive equilibria of model (3.2). As an example, we only give the proof of the unique positive equilibrium of model (3.2).

Theorem 3.7. Let if when , then the equilibrium position of model (3.2) is stochastically asymptotically stable in the large, that is, for any initial data , the solution of model (3.2) has the property that

Proof. From the theory of stability of stochastic differential equations [13], we only need to establish a Lyapunov function satisfying and the identity holds if and only if , where is the solution of the stochastic differential equation is the equilibrium position of model (3.36), and Define Lyapunov functions the nonnegativity of this function can be observed from on . Applying Itô formula leads to when , according to Lemma 2.2, we have , so,
Similarly, by using the Itô formula, we can obtain when , according to Lemma 2.2, we have , so hence,
So,
Let then we get
Clearly, if (3.34) hold, then the above inequality implies along all trajectories in the first quadrant except . Then the desired assertion (3.35) follows immediately. This completes the proof.

4. Conclusions and Remarks

In this paper, we consider a modified stochastic Leslie-Gower predator-prey model. The value of this study lies in two aspects. First, it presents the analysis of stability for the equilibria of model (1.3). Second, it verifies some relevant properties of the stochastic model (3.2) with white noise, which shows that the existence of global positive solutions, stochastic boundedness, and stochastic asymptotic stability.

Next, we give some numerical examples to illustrate the dynamical behaviors of model (3.2) by using the method mentioned in [17].

In Figure 4(a), we choose , that is, without noise, we observe that the positive equilibrium is globally stable. In Figure 4(b), with noise densities , starting with a homogeneous state , the random white noise leads to a slight oscillations, and the later random noise makes the oscillations decay, ending with the time-independent stability. Comparing Figure 4(a) with Figure 4(b), one can realize that, if the white noise is not strong, the stochastic perturbation does not cause sharp changes of the dynamics of the system. However, in Figure 5, we choose that and , which violates condition (3.34), we find that the stochastic model (3.2) is not permanent. This shows that strong white noise might make a permanent system be nonpersistent.

fig4
Figure 4: The solution of the stochastic model (3.2). The parameters are taken as