# Stochastic Delay Lotka-Volterra Model

- Lian Baosheng
^{1}Email author, - Hu Shigeng
^{2}and - Fen Yang
^{1}

**2011**:914270

https://doi.org/10.1155/2011/914270

© Lian Baosheng et al. 2011

**Received: **15 October 2010

**Accepted: **20 January 2011

**Published: **7 February 2011

## Abstract

This paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process, namely the delay Lotka-Volterra model. The stochastic version of this process appears to have some intriguing properties such as pathwise estimation and asymptotic moment estimation. Indeed, their solutions will be stochastically ultimately bounded.

## 1. Introduction

As is well known, Lotka-Volterra Model is nonlinear and tractable models of predator-prey system. The predator-prey system is also studied in many papers. In the last few years, Mao et al. change the deterministic model in this field into the stochastic delay model. and give it more important properties [1–8].

Fluctuations play an important role for the self-organization of nonlinear systems; we will study their influence on a simple nonlinear model of interacting populations, that is, the Lotka-Volterra model. A simple analysis shows the result that the system allows extreme behaviour, leading to the extinction of both of their species or to the extinction of the predator and explosion of the prey. For example, in Mao et al. [1–8], we can see that once the population dynamics are corporate into the deterministic subclasses of the delay Lotka-Voterra model, the stochastic model will bear more attractive properties: the solutions will be be stochastically ultimately bounded, and their pathwise estimation and asymptotic moment estimation will be well done.

where , (where denotes the transpose of a vector or matrix , , , , and is the -dimensional Brownian motion, diag is the diag matrix.

Therefore, if is big enough, condition (1.2) implies condition (1.3).

It is obvious the conditions (1.3)–(1.5) are dependent on the matrix , independent on .

It is clear that the upper inequalities are the key conditions in the stochastic R&D model in economic growth model.

The homogeneous function of degree has the following key property.

Lemma 1.1.

where is given in condition (1.3).

Proof.

Now, from condition (1.3), the right hand of the upper equation is just , so ; Lemma 1.1 is proved.

We use the ordinary result of the polynomial functions.

Lemma 1.2.

## 2. Positive and Global Solutions

Let be a complete probability space with filtration satisfying the usual conditions, that is, it is increasing and right continuous while contains all -null sets [8]. Moreover, let be an -dimensional Brownian motion defined on the filtered space and . Finally, denote the trace norm of a matrix by (where denotes the transpose of a vector or matrix ) and its operator norm by . Moreover, let and denote by the family of continuous functions from to .

The coefficients of (1.1) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of (1.1) may explode at a finite time.

let us emphasize the important feature of this theorem. It is well known that a deterministic equation may explode to infinity at a finite time for some system parameters and . However, the explosion will no longer happen as long as conditions (1.2) and (1.3) hold. In other words, this result reveals the important property that conditions (1.2) and (1.3) suppress the explosion for the equation. The following theorem shows that this solution is positive and global.

Theorem 2.1.

Then for any given initial data , there exists a unique global solution to (1.1) on . Moreover, this solution remains in with probability 1, namely, for all almost surely.

Proof.

## 3. Stochastically Ultimate Boundedness

Theorem 2.1 shows that under simple hypothesis conditions (1.2), (1.3), and (2.1), the solutions of (1.1) will remain in the positive cone . This nice positive property provides us with a great opportunity to construct other types of Lyapunov functions to discuss how the solutions vary in in more detail.

As mentioned in Section 2, the nonexplosion property in a population dynamical system is often not good enough but the property of ultimate boundedness is more desired. Let us now give the definition of stochastically ultimate boundedness.

Theorem 3.1.

Proof.

So in the proof, we suppose is big enough, and these hypotheses will not effect the conclusion of the theorem.

with a constant , independent of initial data .

where is a constant. Then (3.2) follows from the above inequality and Theorem 3.1 is proved.

## 4. Asymptotic Pathwise Estimation

In the previous sections, we have discussed how the solutions vary in in probability or in moment. In this section, we will discuss the solutions pathwisely.

Theorem 4.1.

Proof.

## 5. Further Topic

where we put the delay ; it is clear that the property of the model can be done by the example of condition (1.4). So we have the following theorem.

Theorem 5.1.

Then for any given initial data , there exists a unique global solution to (5.1) on . Moreover, this solution remains in with probability 1.

Remark 5.2.

The explanations in population dynamic of the conditions (1.2), (1.3), and (2.1) for (1.1) are worth pointing out. Each species has a special ability to inhibit the fast growth; the relationship of the species is the role of either species competition ( ), or a low level of cooperation ( , but they are small enough).

## Declarations

### Acknowledgment

This paper is supported by the National Natural Science Foundation of China (10901126), research direction: Theory and Applications of Stochastic Differential Equations.

## Authors’ Affiliations

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