# Optimal Harvest of a Stochastic Predator-Prey Model

- Jingliang Lv
^{1}Email author and - Ke Wang
^{1, 2}

**2011**:312465

https://doi.org/10.1155/2011/312465

© Jingliang Lv and Ke Wang. 2011

**Received: **12 January 2011

**Accepted: **20 February 2011

**Published: **17 March 2011

## Abstract

We firstly show the permanence of hybrid prey-predator system. Then, when both white and color noises are taken into account, we examine the asymptotic properties of stochastic prey-predator model with Markovian switching. Finally, the optimal harvest policy of stochastic prey-predator model perturbed by white noise is considered.

## 1. Introduction

where , represent the prey and the predator populations at time , respectively, and , , are all positive constants.

when a population model is discussed, one of most important and interesting themes is its permanence, which means that the population system will survive forever. In this paper, we show that the hybrid model (1.3) is permanent.

where is a Markov chain. Therefore, we aim to obtain its dynamical properties in more detail.

In this paper, based on the arguments on model (1.4), we will obtain the the optimal harvest policy of stochastic predator-prey system (1.6).

The organization of the paper is as follows: we recall the fundamental theory about stochastic differential equation with Markovian switching in Section 2. We show that the hybrid system (1.3) is permanent in Section 3. Since stochastic predator-prey system (1.4) describes population dynamics, it is necessary for the solution of the system to be positive and not to explode to infinity in a finite time. Section 4 is devoted to the existence, uniqueness of global solution by comparison theorem, and its asymptotic properties. Based on the arguments of Section 4, in Section 5 predator-prey model perturbed by white noise (1.5) is considered, and the limit of the average in time of the sample path of the solution is obtain, moreover, optimal harvest policy of population model is derived. Finally, we close the paper with conclusions in Section 6. The important contributions of this paper are therefore clear.

## 2. Stochastic Differential Equation with Markovian Switching

where . Here, is the transition rate from to if , while . We assume that the Markov chain is independent of the Brownian motion. And almost every sample path of is a right-continuous step function with a finite number of simple jumps in any finite subinterval of .

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

## 3. Hybrid Predator-Prey Model

In this section, we mainly consider the permanence of the hybrid prey-predator system (1.3).

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Theorem 3.3.

Suppose that , , , , and hold. Then, the solution to (1.3) is permanent.

Proof.

## 4. Stochastic Predator-Prey Model With Markovian Switching

In this section, we consider the stochastic differential equation with regime switching (1.4). If stochastic differential equation has a unique global (i.e., no explosion in a finite time) solution for any initial value, the coefficients of the equation are required to obey the linear growth condition and local Lipschitz condition. It is easy to see that the coefficients of (1.4) satisfy the local Lipschitz condition; therefore, there is a unique local solution on with initial value , , where is the explosion time.

And since our purpose is to reveal the effect of environmental noises, we impose the following hypothesis on intensities of environmental noises.

It can be easily verified that , , , all exist on , hence

Theorem 4.2.

where , , , are defined as (4.1),(4.4),(4.6), and (4.9).

Theorem 4.2 tells us that (1.4) has a unique global solution, which makes us to further discuss its properties.

Then, we give the following essential theorems which will be used.

Theorem 4.3.

Proof.

So, we complete the proof.

Theorem 4.4.

Proof.

Theorem 4.5.

Proof.

The proof is complete.

## 5. Optimal Harvest Policy

When the harvesting problems of population resources is discussed, we aim to obtain the optimal harvesting effort and the corresponding maximum sustainable yield.

In the same way of Theorems 4.2–4.5, we can conclude the following results. Here, we do not list the corresponding proofs in detail, only show the main results.

Theorem 5.1.

When the harvesting problems are considered, the corresponding average population level is derived below.

Theorem 5.2.

When the two species are both subjected to exploitation, it is important and necessary to discuss the corresponding maximum sustainable revenue.

Theorem 5.3.

where , are the price of and .

Proof.

That is, under the condition , we obtain (5.4) and (5.5). So, we obtain the optimal harvesting efforts of and .

as desired. Therefore, we complete the proof.

## 6. Conclusions

The optimal management of renewable resources has a direct relationship to sustainable development. When population system is subject to exploitation, it is important and necessary to discuss the optimal harvesting effort and the corresponding maximum sustainable yield. Meanwhile, population systems are often subject to environmental noise. It is also necessary to reveal how the noise affects the population systems. Our work is an attempt to carry out the study of optimal harvest policy of population system in a stochastic setting. When both white noise and color noise are taken into account, we consider the limit of the average in time of the sample path of the stochastic model (1.4). Based on the arguments of (1.4), we discuss the corresponding stochastic system perturbed by white noise (1.5). We obtain the the optimal harvesting effort and the corresponding maximum sustainable yield.

## Declarations

### Acknowledgment

This research is supported by the national natural science foundation of China (no. 10701020)

## Authors’ Affiliations

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