Abstract and Applied Analysis

Volume 2011 (2011), Article ID 518719, 16 pages

http://dx.doi.org/10.1155/2011/518719

## Analysis on a Stochastic Predator-Prey Model with Modified Leslie-Gower Response

^{1}Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China^{2}School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received 30 October 2010; Accepted 30 March 2011

Academic Editor: Wing-Sum Cheung

Copyright © 2011 Jingliang Lv and Ke Wang. 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

This paper presents an investigation of asymptotic properties of a stochastic predator-prey model with modified Leslie-Gower response. We obtain the global existence of positive unique solution of the stochastic model. That is, the solution of the system is positive and not to explode to infinity in a finite time. And we show some asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained. Finally we work out some figures to illustrate our main results.

#### 1. Introduction

The dynamic interaction between predators and their prey has been one of the dominant themes in mathematical biology due to its universal existence and importance. Much literature exists on the general problem of food chains in the classical Lotka-Volterra model. In [1, 2], Leslie introduced a predator-prey model where the capacity of the predators environment is proportional to the number of preys. Leslie stresses the fact that there are upper limits to the rates of increase of both prey and predator, which are not recognized in the Lotka-Volterra model. Broer et al. [3] studied the dynamical properties of a predator-prey model with nonmonotonic response function. Reference [4] considered two-species autonomous system which incorporated a modified Leslie-Gower functional response as well as that of the Holling II as follows: where , and are all positive constants and , represent the population densities at time .

Hsu and Huang [5] studied the global stability property of the following predator-prey system:

Recently, [6] discussed the following model with modified Leslie-Gower response: where are all positive constants and , are the growth rates of prey and predator , respectively. Here, we change the form of the predator-prey model above which reads

As a matter of fact, population systems are often subject to environmental noise. Recently, more and more interest is focused on stochastic systems. Reference [7] investigated the predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation: By virtue of comparison theorem, [7] obtained some interesting results, including globally positive solutions, persistence in mean and extinction. Moreover, [8] continued to consider the stochastic ratio-dependent predator-prey system: where , , are independent standard Brownian motions. And [8] also obtained some nice conclusions on the stochastic model.

According to (1.4), taking into account the effect of randomly fluctuating environment, we will consider the corresponding autonomous stochastic system described by the Itô equation where , , are independent standard Brownian motions and , and are all positive.

When white noise is taken into account in our model (1.7), we obtain the global existence of positive unique solution of the stochastic model, that is, the solution of the system is positive and not to explode to infinity in a finite time in Section 2. Section 3 shows some fundamental asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained in Section 3. The main contributions of this paper are therefore clear.

Throughout the paper, we use to denote a positive constant whose exact value may be different in different appearances.

#### 2. Positive and Global Solution

As , of the SDE (1.7) are sizes of the species in the system at time , it is obvious that the positive solutions are of interest. The coefficients of (1.7) are locally Lipschitz continuous and do not satisfy the linear growth condition, so the solution of (1.7) may explode at a finite time. The following Theorem shows that the solution will not explode at a finite time.

Theorem 2.1. *For a given initial value , there is a unique positive solution to (1.7) on , and the solution will remain in with probability one, namely, for all almost surely.*

*Proof. *The proof is similar to [9, 10]. Since the coefficients of the equation are locally Lipschitz continuous, for a given initial value , there is a unique local solution on , where is the explosion time. To show that this solution is global, we need to show that a.s. Let be sufficiently large for every component of and all lying within the interval . For each integer , define the stopping time
where throughout this paper we set . Obviously, is increasing as . Let , whence a.s. If we can show that a.s., then a.s. and a.s. for all . So we just prove that a.s. If not, there is and such that
Hence, there is an integer such that for all . Define a function by . The nonnegativity of this function can be seen from
If , by virtue of on , we obtain
dropping from and . Making use of the Itô formula yields
The Gronwall inequality yields
Set for ; then . Note that, for every , there is or equal to either or , and hence is no less than either
or
Therefore
So
where is the indicator function of . Letting implies the contradiction
So we have that a.s. The proof is complete.

Theorem 2.1 shows that the solution of the SDE (1.7) will remain in the positive cone for any initial value . The conclusion is fundamental which will be used later.

#### 3. Asymptotic Behavior

##### 3.1. Limit Results

To begin our discussion, we impose the following assumption:(H).

And we list the interesting lemma as follows.

Lemma 3.1 (see [7, 8]). * Consider one-dimensional stochastic differential equation
**
where are positive and is standard Brownian motion. Under condition , for any initial value , the solution has the properties
*

To demonstrate asymptotic properties of the stochastic system (1.7), we firstly discuss the long time behavior of and .

On the one hand, by the comparison theorem of stochastic equations, it is obvious that

Denote by the solution of the following stochastic equation: We have that On the other hand, by the comparison theorem of stochastic equations, it is obvious that we denote by the solution of stochastic differential equation Consequently To sum up, we have that So we have the explicit solutions of and as follows:

Theorem 3.2. *Under assumption (H), for any initial value , the solutions and satisfy
*

*Proof. *By assumption (H) and Lemma 3.1, the assertion is straightforward.

Theorem 3.3. *Under assumption (H), for any initial value , the solution satisfies
*

*Proof. *By virtue of (3.8) and Theorem 3.2, we can imply the desired assertion.

Now let us continue to consider the asymptotic behavior of the species . By the comparison theorem of stochastic equations, we have that

Denote by the solution of the stochastic equation as follows: We have that On the other hand, applying the comparison theorem again, denote by the solution of stochastic equation Consequently, To sum up, we have that Moreover, and have the explicit solutions, respectively,

Lemma 3.4. *Under assumption (H), for any initial value , the solutions and satisfy
*

*Proof. *The proof is motivated by [7]. Obviously, Lemma 3.1 and assumption (H) yield

On the other hand, it follows from (3.20) that
Choose satisfying for . Thus we have that for . Then for , from (3.10), we obtain
Thus,
where So we derive
Dividing on both sides yields
The distributions of and are that same as and , respectively, and and have the same distributions as and , respectively.

From the representation of , we can simplify it as follows:
By assumption (H), constants satisfy . Then,
It follows from , that
Hence, letting and by the strong law of large numbers, we have that
Then,
as desired.

Theorem 3.5. *Under assumption (H), for any initial value , the solution of (1.7) has the property
*

*Proof. *It follows from (3.18) and Lemma 3.4 that
Consequently,
The proof is complete.

##### 3.2. Persistent in Mean and Extinction

As we know, the property of persistence is more desirable since it represents the long-term survival to a population dynamics. Now we present the definition of persistence in mean proposed in [7, 11].

*Definition 3.6. *System (1.7) is said to be persistent in mean if

Theorem 3.7. *Assume that condition (H) holds. Then system (1.7) is persistent in mean.*

*Proof. *Define the function ; by the Itô formula, we get
That is,
Dividing on both sides and using the strong law of large numbers, it follows from Theorem 3.3 that

Moreover, define the function ; using the Itô formula again, we have that
Thus,
Dividing both sides by and letting and also by the strong law of large numbers and Theorem 3.5, we have that
So the system is persistent in mean and we complete the proof.

Under condition (H), we show that the system is persistent in mean. To a large extent, (H) is the condition that stands for small environmental noises. That is, small stochastic perturbation does not change the persistence of the system. Here, we will consider that large noises may make the system extinct.

Theorem 3.8. *Assume that condition holds. Then system (1.7) will become extinct exponentially with probability one.*

*Proof. *Define the function ; by the Itô formula, we get
Then,
By the strong law of large numbers of martingales, we have that
Therefore,

On the other hand, by the Itô formula again, we derive
Applying the strong law of large numbers of martingales, we obtain
The proof is complete.

We continue to discuss the asymptotic behaviors of the stochastic system (1.7).

Theorem 3.9. *Assume that condition holds. Then the prey of system (1.7) is persistent in mean; however, the predator will become extinct exponentially with probability one.*

*Proof. *Define the function ; by the Itô formula, we get
Thus,
Under condition , it follows from the proof of Theorem 3.3 that
So
That is, the prey is persistent in mean. However, under condition , from the proof of Theorem 3.8, we have that
That is, the predator will become extinct exponentially with probability one.

#### 4. Numerical Simulations

In this section, some simulation figures are introduced to support the main results in our paper.

For model (1.7), we consider the discretization equations where and are Gaussian random variables that follow .

In Figure 1, we choose , , , , , and . By virtue of Theorem 3.7, the system will be persistent in mean. What we mentioned above can be seen from Figure 1. The difference between the conditions of Figure 1 and Figure 2 is that the values of and are different. In Figure 1, we choose . In Figure 2, we choose . In view of Theorem 3.8, both species and will go to extinction. Figure 2 confirms this.

In Figure 3, we choose , , , , ,, and . Then the prey is persistent in mean; however, the predator will become extinct. Figure 3 confirms the assertion (Theorem 3.9).

By comparing Figures 1 and 2, with Figure 3, we can observe that small environmental noise can retain the stochastic system permanent; however, sufficiently large environmental noise makes the stochastic system extinct.

*Remark 4.1. *White noise is taken into account in our model in this paper. It tells us that, when the intensities of environmental noises are not too big, some nice properties such as nonexplosion and permanence are desired. However, Theorem 3.8 reveals that a large white noise will force the population to become extinct while the population may be persistent under a relatively small white noise. To some extent, Theorem 3.9 shows that, though the predator has plenty of food , they may be extinct because of large environmental noise.

#### Acknowledgment

This research is supported by China Postdoctoral Science Foundation (no. 20100481000).

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