Abstract and Applied Analysis

Volume 2013 (2013), Article ID 918569, 10 pages

http://dx.doi.org/10.1155/2013/918569

Research Article

Stochastic Delay Population Dynamics under Regime Switching: Global Solutions and Extinction

Zheng Wu, Hao Huang, and Lianglong Wang

School of Mathematical Sciences, Anhui University, Hefei 230039, China

Received 30 January 2013; Accepted 26 March 2013

Academic Editor: Zhiming Guo

Copyright © 2013 Zheng Wu 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.

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Finally, an example is given to illustrate the main results.

1. Introduction

The delay differential equation

has been used to model the population growth of certain species and is known as the delay Lotka-Volterra model or the delay logistic equation. The delay Lotka-Volterra model for interacting species is described by the -dimensional delay differential equation

where , , , and . There is an extensive literature concerned with the dynamics of this delay model and have had lots of nice results. We here only mention Ahmad and Rao [1], Bereketoglu and Győri [2], Freedman and Ruan [3], and in particular, the books by Gopalsamy [4], Kolmanovskiĭ and Myshkis [5], and Kuang [6], among many others.

In the equations above, the state denotes the population sizes of the species. Naturally, we focus on the positive solutions and also require the solutions not to explode at a finite time. To guarantee the positive solutions without explosion (i.e., the global positive solutions), some conditions are in general needed to impose on the system parameters. For example, it is generally assumed that , and for (1) while much more complicated conditions are required on matrices and for (2) [7] (and the references cited therein).

On the other hand, population systems are often subject to environmental noise, and the system will change significantly, which may change the dynamics of solutions significantly [8, 9]. It is therefore necessary to reveal how the noise affects the dynamics of solutions for the delay population systems. In fact, many authors have discussed population systems subject to white noise [7–18]. Recall that the parameter in (2) represents the intrinsic growth rate of species . In practice we usually estimate it by an average value plus an error term. According to the well-known central limit theorem, the error term follows a normal distribution. In term of mathematics, we can therefore replace the rate by , where is a white noise (i.e., is a Brownian motion) and represents the intensity of noise. As a result, (2) becomes a stochastic differential equation (SDE, in short)

where . We refer to [7] for more details.

To our knowledge, much of the attention paid to environmental noise is focused on white noise. But another type of environmental noise, namely, color noise, say telegraph noise, has been studied by many authors ([19–25] and the references cited therein). In this context, telegraph noise can be described as a random switching between two or more environmental regimes, which differ in terms of factors such as nutrition or rain falls [23, 24]. Usually, the switching between different environments is memoryless and the waiting time for the next switch has an exponential distribution. This indicates that we may model the random environments and other random factors in the system by a continuous-time Markov chain with a finite state space . Therefore stochastic delay population system (3) in random environments can be described by the following stochastic model with regime switching:

The mechanism of ecosystem described by (4) can be explained as follows. Assume that initially, the Markov chain . Then the ecosystem (4) obeys the SDE

until the Markov chain jumps to another state, say, . Therefore, the ecosystem (4) satisfies the SDE

for a random amount of time until the Markov chain jumps to a new state again.

It should be pointed out that the stochastic population systems under regime switching have received much attention lately. For instance, the stochastic permanence and extinction of a logistic model under regime switching were considered in [20, 24], asymptotic results of a competitive Lotka-Volterra model in random environment are obtain in [25], a new single-species model disturbed by both white noise and colored noise in a polluted environment was developed and analyzed in [26], and a general stochastic logistic system under regime switching was proposed and was treated in [27].

Equation (4) describes the dynamics of populations. This paper is concerned with the positive global solutions, ultimate boundedness and extinction. The stochastic permanence and asymptotic estimations of solutions will be investigated in the next note [28].

This paper is organized as follows. In the next section, some sufficient conditions for global positive solutions for any initial positive value are given by using generalized Itô formula, Gronwall inequality, and -function techniques. In Section 3, the stochastically ultimate boundedness of solutions is obtained by virtue of Young's inequality. Section 4 is devoted to the extinction of solutions. Finally, an example and its numerical simulation are given to illustrate our main results.

2. Global Positive Solution

Throughout this paper, unless otherwise specified, let , be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let , be a scalar standard Brownian motion defined on this probability space. We also denote by the positive cone in , that is , and denote by the nonnegative cone in , that is . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by , and its operator norm is denoted by . Moreover, let and denote by the family of continuous functions from to .

In this paper we will use a lot of quadratic functions of the form for the state only. Therefore, for a symmetric matrix , we naturally introduce the following definition

For more properties of , refer to the appendix in [7].

Let be a right-continuous Markov chain on the probability space, taking values in a finite state space , with the generator given by

where is the transition rate from to , and if , while . We assume that the Markov chain is independent of the Brownian motion . It is well known that almost every sample path of is a right-continuous step function with a finite number of jumps in any finite subinterval of . As a standing hypothesis we assume in this paper that the Markov chain is irreducible. This is a very reasonable assumption as it means that the system can switch from any regime to any other regime. This is equivalent to the condition that for any , one can find finite numbers such that . Under this condition, the Markov chain has a unique stationary (probability) distribution which can be determined by solving the following linear equation:

subject to

We refer to [12, 29] for the fundamental theory of stochastic differential equations.

For convenience and simplicity in the following discussion, for any constant sequence let

As in system (4) denotes populations size at time , it should be nonnegative. Thus for further study, we must give some condition under which (4) has a unique global positive solution.

Theorem 1. Assume that there are positive numbers and such that

where . Then for any given initial data , there is a unique solution to (4) on and the solution will remain in with probability 1, namely, for all a.s.

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. To show that the solution is global, we need to show that a.s.

Let be sufficiently lager such that

For each integer , define the stopping time

where throughout this paper we set (as usual denotes the empty set). Clearly, is increasing as . Set , where a.s. If a.s., then a.s. and a.s. for all . In other words, to complete the proof, one should show that a.s. Define by

The nonnegativity of this function can be seen from . Let and be arbitrary. For , it is easy to see by the generalized Itô formula that

where is defined by

and . Using condition (12) we compute

Moreover, there is a constant such that

Substituting these inequalities into (17) yields

Noticing that , we compute

It follows from (20) and (21) that

where is a positive constant. Substituting this inequality into (16) yields

Compute

and, similarly

Substituting these inequalities into (23) gives

where .

By the Gronwall inequality, we obtain that

Noting that for every ,

one has by (27) that

where is the indicator function of . Letting gives and hence . Since is arbitrary, we must have , so as required.

Assumption 1. Assume that there exist positive numbers such that

where .

The following theorem is easy to verify in applications, which will be used in the sections below.

Theorem 2. Under Assumption 1, for any given initial data , there is a unique solution to (4) on and the solution will remain in with probability 1, namely, for all a.s.

Proof. Define by . The non-negativity of this function can be seen from , and then we have (16) and (17).

If , then . Consequently

Otherwise for . In this case, we also have that

Thus,

Denote . By (33) and Assumption 1, one has

The rest of the proof is similar to that of Theorem 1 and omitted.

3. Ultimate Boundness

Theorem 2 shows that solutions of the SDE (4) will remain in the positive cone . This nice property provides us with a great opportunity to discuss how solutions vary in in detail. In this section, we give the definitions of stochastically ultimate boundedness of the SDE (4) and some sufficient conditions under which solutions of SDE (4) are stochastically ultimate bounded.

Definition 3. The solutions of (4) are called stochastically ultimately bounded, if for any , there exists a positive constant , such that the solutions of (4) with any positive initial value have the property that

Assumption 2. Assume that there exist positive numbers such that

where .

Theorem 4. Under Assumption 2, for any given initial data and any given positive constant , there are two positive constants and , such that the solution of (4) has the properties that

Proof. By Theorem 2, the solution will remain in for all with probability 1. If , we let and . Define . It has by the generalized Itô formula that

where is defined by

Meanwhile, by Assumption 2 and Young’s inequality, one gets

where

On the other hand,

by (41) and (43), we obtain that

which yields

Since , it has and the desired assertion (37) follows by setting . It is easy to verify this result, if . We omit its proof here.

Define . By the generalized Itô formula, it follows

where is defined by

By Assumption 2 and Young’s inequality again,

It is easy to compute

Moreover,

hence, we get

where . This implies immediately that

and the desired assertion (38) follows by setting .

Remark 5. From (37) of Theorem 4, there is a such that

Since is continuous, there is a such that

Let , we have

which implies that the th moment of any positive solution of (4) is bounded.

Remark 6. Conclusion (38) of Theorem 4 shows that the average in time of the th () moment of solutions of (4) will be bounded.

Theorem 7. Solutions of (4) are stochastically ultimately bounded under Assumption 2.

Proof. This can be easily verified by Chebyshev’s inequality and Theorem 4.

4. Extinction

Assumption 3. Assume that there exist positive numbers such that

where and .

Theorem 8. Under Assumption 3, for any given initial data , the solution of (4) has the properties that

where . Particularly, , then

That is, the population will become extinct exponentially with probability 1.

Proof. By Theorem 2, the solution will remain in for all with probability 1. Define

where . Then

By the generalized Itô formula,

It is computed

Substituting these two inequalities into (61) yields

where is a martingale defined by

The quadratic variation of this martingale is

hence

Applying the strong law of large numbers for martingales [29], we therefore have

It finally follows from (64) by dividing on the both sides and then letting that

which is the required assertion (57).

Similarly, we can prove the following conclusions.

Theorem 9. Assume that Assumption 3 holds. Assume moreover that the noise intensities are sufficiently large in the sense that

then for any given initial data , the solution of (4) has the properties that

where

That is, the population will become extinct exponentially with probability 1.

Proof. Let be the same as defined in the proof of Theorem 8, so we have (60), (61), and (62). It is also computed

where and . Substituting (62) and (73) into (61) yields

Note that , the th element of the matrix is positive by (70). It is therefore easy to verify

where has been defined in the statement of the theorem. Substituting this inequality into (74) yields

The rest of the proof is similar to that of Theorem 8 and omitted.

5. Examples

In this section, an example and corresponding numerical simulations are given to illustrate our main results.

Example 10. Consider the two-species Lotka-Volterra system with regime switching described by

where , , ,

and is a right-contiuous Markov chain taking values in , and and are independent. Here

Let , the identity matrix. It is easy to compute

Then

By Theorems 2 and 7, the solutions of (77) will remain in for all with probability 1 and are stochastically ultimately bounded.

Let the generator of the Markov chain be

By solving the linear equation , we obtain the unique stationary (probability) distribution . Then . Therefore, by Theorems 8, (77) is extinctive, shown in Figure 1.

(a)

(b)

Figure 1

In Figure 1, for numerical solutions of (77), step size , delay . Initial datum of are random numbers in . Initial datum are not shown in Figure 1.

6. Conclusion

This work is concerned with delay Lotka-Volterra model under regime switching diffusion in random environment. It should be pointed out that (77) is more difficult to handle than (3) in [23]. Fortunately, the difficulties caused by delay term are overcome by using Young's inequality. The model in [7] is similar to (4), while the coefficients in (4) are varied with regime switching. Similar results are technically obtained by making use of comparison principle.

Acknowledgments

The authors are grateful to the editor Prof. Zhiming Guo and anonymous referees for their helpful comments and suggestions which have improved the quality of this paper. The authors are indebted to Professor Tao Wu, Anhui University, for giving ideas on simulations of the example. This work is supported by Research Fund for Doctor Station of Ministry of Education of China (no. 20113401110001, no. 20103401120002), TIAN YUAN Series of Natural Science Foundation of China (no. 11126177, No. 11226247), Key Natural Science Foundation (no. KJ2009A49, no. KJ20 12A19), 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 1308085MA01, no. 1308085QA15, and no. 1208085QA15), Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).

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