Stochastic Delay Lotka-Volterra Model

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.

The most simple stochastic model is given in the form of a stochastic delay differential equation (also called a diffusion process); we call it a delay Lotka-Volterra model with diffusion. The model will be

(1.1)

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

This model of the stochastic delay Lotka-Volterra is different from Mao et al. [3–10], which paid more attention to the mathematical properties of the model than the real background of the model. However, our model has the following three characteristics. First, it is another stochastic delay subclass of the Lotka-Volterra model which is different from Mao et al. Then we can obtain more comprehensive properties in Theorem 2.1. Second, in this field no paper gives more attention to it so far, especially for the stochastic delay model which is the focus in our model. Third, this model has many real applications, for example, in economic growth model it is different from the old delay Lotka-Volterra model which only palys a role in predator-prey system, for example, the stochastic R&D model [9, 10] is the best application of this model. We hope our model can have new applications of the Lotka-Volterra model. Throughout this paper, we impose the condition

(1.2)

where .

Of course, it is important for us to point that the condition (1.2) may be not real in predator-prey interactions, but in the stochastic R&D model in economic growth model, it has a special meaning

(1.3)

If or 1, the inequality (1.3) can be deduced to

(1.4)

(1.5)

If condition (1.2) is satisfied, then

(1.6)

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 .

Condition (1.4) will be used in a further topic in the paper; the condition (1.4) is complicated, we can find many matrixes that have a property like this. For example,

(1.7)

satisfy the condition (1.4). Furthermore, if , or are proper small enough positive numbers, condition (1.4) holds too. Particularly, if , the condition can be induced into

(1.8)

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

Let

(1.9)

The homogeneous function of degree has the following key property.

Lemma 1.1.

Suppose the matrix satisfies condition (1.2). Let

(1.10)

then

(1.11)

where is given in condition (1.3).

Proof.

Fix , so . We will show . We have

(1.12)

with satisfying condition (1.2), where . Since, from condition (1.2), , , and

(1.13)

then

(1.14)

Since , we have , , and there exits at least , such that and at least for some . Thus

(1.15)

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.

Let be a homogeneous function of degree , , and ; then the function as follows has an upper bound for some constant .

(1.16)

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.

Let us assume that satisfy

(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.

Since the coefficients of the equation are locally Lipschitz continuous, for any given initial data , there is a unique maximal local solution on , where is the explosion time [3–10]. To show this solution is global, we need to show that a.s. Let be sufficiently large for

(2.2)

For each integer , define the stopping time

(2.3)

where throughout this paper we set (as usual denotes the empty set). Clearly, is increasing as . Set , whence 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 show is that a.s. Or for all , we have . To show this statement, let us define a -functions by

(2.4)

The nonnegativity of this function can be seen from

(2.5)

Let and be arbitrary. For , we apply the Itô formula to to obtain that

(2.6)

where is a homogeneous function of a degree not above 1, , and by (1.9), , and let , for all; then . By Lemma 1.1, we obtain

(2.7)

where we use the fact and , and

(2.8)

where , if , and .

Thus

(2.9)

Put

(2.10)

Then, if , by Lemma 1.2, we obtain

(2.11)

with a constant .

Consequently,

(2.12)

On the other hand, if , then for some ; therefore,

(2.13)

so ; Theorem 2.1 is proved.

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.

Suppose (2.1) and the following condition:

(3.1)

hold. Then for all and any initial data , there is a positive constant , which is independent of the initial data, such that the solution of (1.1) has the property that

(3.2)

Proof.

If condition (1.2) is satisfied, then

(3.3)

By Liapunov inequality,

(3.4)

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

Define the Lyapunov functions.

(3.5)

It is sufficient to prove

(3.6)

with a constant , independent of initial data .

We have

(3.7)

where is constant and is given in (1.9). Let , for all ; by Lemma 1.1, we have

(3.8)

Then,

(3.9)

Thus we obtain

(3.10)

and from (1.3)

(3.11)

if is big enough, then

(3.12)

By Lemma 1.2 and inequality (3.12),

(3.13)

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.

Suppose (2.1) holds and the following condition:

(4.1)

is satisfied, where is given by (1.5) and . Then for any initial data , the solution of (1.1) has the property that

(4.2)

where .

Proof.

Define the Lyapunov functions

(4.3)

By Itô's formula, we have

(4.4)

Therefore,

(4.5)

where

(4.6)

where is a real-valued continuous local martingale vanishing at and its quadratic form is given by

(4.7)

and then

(4.8)

Now, let be arbitrary. By the exponential martingale inequality [3–10], we can show that for every integer ,

(4.9)

Since the series converges, the well-known Borel-Cantelli lemma yields that there is with such that for every there exists a random integer such that for all ,

(4.10)

which implies

(4.11)

Substituting this into (4.6) and making use of the upper inequality, we derive that

(4.12)

Therefore,

(4.13)

Puting , we can get inequality (4.2).

5. Further Topic

In this section, we introduce an economic model named stochastic R&D model in economic growth [10]; for the notion of the model, see details in reference [9, 10]. The equation is

(5.1)

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.

Let the following conditions be satisfied

(5.2)

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).