Abstract and Applied Analysis

Volume 2012 (2012), Article ID 241702, 26 pages

doi:10.1155/2012/241702

Research Article

Stochastic Delay Logistic Model under Regime Switching

Zheng Wu, Hao Huang, and Lianglong Wang

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

Received 2 April 2012; Accepted 14 June 2012

Academic Editor: Elena Braverman

Copyright © 2012 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 logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue of -function technique, -matrix method, and Chebyshev's inequality. 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, known as the delay logistic equation. There is an extensive literature concerned with the dynamics of this delay model. We here only mention Gopalsamy [1], Kolmanovskiĭ, and Myshkis [2], Kuang [3] among many others.

In (1.1), the state denotes the population size of the species. Naturally, we focus on the positive solutions and also require the solutions not to explode at a finite time. To guarantee positive solutions without explosion (i.e., there exists global positive solutions), it is generally assumed that ,  , and [4] (and the references cited therein).

On the other hand, the population growth is often subject to environmental noise, and the system will change significantly, which may change the dynamical behavior of solutions significantly [5, 6]. It is therefore necessary to reveal how the noise affects on the dynamics of solutions for the delay population model. First of all, let us consider one type of environmental noise, namely, white noise. In fact, recently, many authors have discussed population systems subject to white noise [7–9]. Recall that the parameter in (1.1) represents the intrinsic growth rate of the population. 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 is a positive number representing the intensity of noise. As a result, (1.1) becomes a stochastic differential equation (SDE, in short)

We refer to [4] for more details.

To our knowledge, much attention to environmental noise is paid on white noise ([10–14] and the references cited therein). But another type of environmental noise, namely, color noise or say telegraph noise, has been studied by many authors (see, [15–19]). In this context, telegraph noise can be described as a random switching between two or more environmental regimes, which are different in terms of factors such as nutrition or as rain falls [20, 21]. 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, the stochastic delay logistic (1.3) in random environments can be described by the following stochastic model with regime switching:

The mechanism of ecosystem described by (1.4) can be explained as follows. Assume that initially, the Markov chain , then the ecosystem (1.4) obeys the SDE

until the Markov chain jumps to another state, say, . Then the ecosystem satisfies the SDE

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

It should be pointed out that the stochastic logistic systems under regime switching have received much attention lately. For instance, the study of stochastic permanence and extinction of a logistic model under regime switching was considered in [18], a new single-species model disturbed by both white noise and colored noise in a polluted environment was developed and analyzed in [22], a general stochastic logistic system under regime switching was proposed and was treated in [23].

Since (1.4) describes a stochastic population dynamics, it is critical to find out whether or not the solutions will remain positive or never become negative, will not explode to infinity in a finite time, will be ultimately bounded, will be stochastically permanent, will become extinct, or have good asymptotic properties.

This paper is organized as follows. In the next section, we will show that there exists a positive global solution with any initial positive value under some conditions. In Sections 3 and 4, we give the sufficient conditions for stochastic permanence or extinction, which show that both have closed relations with the stationary probability distribution of the Markov chain. If (1.4) is stochastically permanent, we estimate the limit of the average in time of the sample path of its solution in Section 5. Finally, an example is 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 interval and denote by the interval . Moreover, let and denote by the family of continuous functions from to .

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. 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 [9, 24] for the fundamental theory of stochastic differential equations.

For convenience and simplicity in the following discussion, define

where is a constant vector.

As in model (1.4) denotes population size at time , it should be nonnegative. Thus, for further study, we must give some condition under which (1.4) has a unique global positive solution.

Theorem 2.1. Assume that there are positive numbers such that

Then, for any given initial data , there is a unique solution to (1.4) 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 this solution is global, we need to prove a.s.

Let be sufficiently large for

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 we can show that a.s., then a.s. and a.s. for all . In other words, we need to show a.s. Define a -function by

which is not negative on . Let and be arbitrary. For , it is not difficult to show by the generalized Itô formula that

where is defined by

Using condition (2.5), we compute

Moreover, there is clearly a constant such that

Substituting these into (2.10) yields

Noticing that , we obtain that

where is a positive constant. Substituting these into (2.9) yields

Now, for any , we can integrate both sides of (2.15) from 0 to and then take the expectations to get

Compute

and, similarly

Substituting these into (2.16) gives

where .

By the Gronwall inequality, we obtain that

Note that for every equals either or , thus

It then follows from (2.20) that

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

Corollary 2.2. Assume that there is a positive number such that

Then the conclusions of Theorem 2.1 hold.

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

Theorem 2.3. Assume that

Then for any given initial data , there is a unique solution to (1.4) on and the solution will remain in with probability 1, namely, for all a.s.

Proof. The proof of this theorem is the same as that of the theorem above. Let

then we have (2.9) and (2.10). By (2.24), we get

where is a positive constant. The rest of the proof is similar to that of Theorem 2.1 and omitted.

Note that condition (2.5) is used to derive (2.13) from (2.10). In fact, there are several different ways to estimate (2.10), which will lead to different alternative conditions for the positive global solution. For example, we know

Therefore, if we assume that

then

hence

from which we can show in the same way as in the proof of Theorem 2.1 that the solution of (1.4) is positive and global. In other words, the arguments above can give an alternative result which we describe as a theorem as below.

Theorem 2.4. Assume that there are positive numbers such that

Then for any given initial data , there is a unique solution to (1.4) on and the solution will remain in with probability 1, namely, for all a.s.

Similarly, we can establish a corollary as follows.

Corollary 2.5. Assume that there is a positive number such that

Then the conclusions of Theorem 2.4 hold.

3. Asymptotic Bounded Properties

For convenience and simplicity in the following discussion, we list the following assumptions.

(A1) For each .() For each .() For each .(A2) For some .

(A3).().(A4) For each .() For each .

Definition 3.1. Equation (1.4) is said to be stochastically permanent if for any , there exist positive constants such that

where is the solution of (1.4) with any positive initial value.

Definition 3.2. The solutions of (1.4) are called stochastically ultimately bounded, if for any , there exists a positive constant , such that the solutions of (1.4) with any positive initial value have the property that

It is obvious that if a stochastic equation is stochastically permanent, its solutions must be stochastically ultimately bounded. So we will begin with the following theorem and make use of it to obtain the stochastically ultimate boundedness of (1.4).

Theorem 3.3. Let hold and is an arbitrary given positive constant. Then for any given initial data , the solution of (1.4) has the properties that

where both and are positive constants defined in the proof.

Proof. By Theorem 2.3, the solution will remain in for all with probability 1. Let

Define the function by

By the generalized Itô formula, we have

where is defined by

By (3.5) and Young's inequality, we obtain that

where . Moreover,

By (3.10) and (3.11), one has

which yields

where

By the generalized Itô formula, Young's inequality and (3.5) again, it follows

where . This implies

The inequality above implies

where and the desired assertion (3.4) follows by setting .

Remark 3.4. From (3.3) of Theorem 3.3, there is a such that

Since is continuous, there is a such that

Taking , we have

This means that the th moment of any positive solution of (1.4) is bounded.

Remark 3.5. Equation (3.4) of Theorem 3.3 shows that the average in time of the th () moment of solutions of (1.4) is bounded.

Theorem 3.6. Solutions of (1.4) are stochastically ultimately bounded under .

Proof. This can be easily verified by Chebyshev's inequality and Theorem 3.3.

Based on the results above, we will prove the other inequality in the definition of stochastic permanence. For convenience, define

Under (A3), it has . Moreover, let be a vector or matrix. By we mean all elements of are positive. We also adopt here the traditional notation by letting

We will also need some useful results.

Lemma 3.7 (see [24]). If , then the following statements are equivalent.

(1) is a nonsingular -matrix (see [24] for definition of -matrix).(2)All of the principal minors of are positive; that is,

(3) is semipositive, that is, there exists in such that .

Lemma 3.8 (see [18]). (i) Assumptions and imply that there exists a constant such that the matrix

is a nonsingular -matrix, where .

(ii) Assumption (A4) implies that there exists a constant such that the matrix is a nonsingular -matrix.

Lemma 3.9. If there exists a constant such that is a nonsingular -matrix and , then the global positive solution of (1.4) has the property that

where is a fixed positive constant (defined by (3.35) in the proof).

Proof. Let   on  . Applying the generalized Itô formula, we have

By Lemma 3.7, for the given , there is a vector such that

namely,

Define the function by . Applying the generalized Itô formula again, we have

where is defined by

Now, choose a constant sufficiently small such that

that is,

Then, by the generalized Itô formula again,

It is computed that

where

which implies

Then

Recalling the definition of , we obtain the required assertion.

Theorem 3.10. Under (A1′′), (A2), and (A3), (1.4) is stochastically permanent.

The proof is a simple application of the Chebyshev inequality, Lemmas 3.8 and 3.9, and Theorem 3.6. Similarly, it is easy to obtain the following result.

Theorem 3.11. Under (A1′′) and (A4), (1.4) is stochastically permanent.

Remark 3.12. It is well-known that if ,   and , then the solution of (1.1) is persistent, namely,

Furthermore, we consider its associated stochastic delay equation (1.4), that is,

where ,  for  ,  and  . Thus, applying Theorem 3.10 or Theorem 3.11, we can see that (1.4) is stochastically permanent, if the noise intensities are sufficiently small in the sense that

Corollary 3.13. Assume that for some ,, and . Then the subsystem

is stochastically permanent.

4. Extinction

In the previous sections we have shown that under certain conditions, the original (1.1) and the associated SDE (1.4) behave similarly in the sense that both have positive solutions which will not explode to infinity in a finite time and, in fact, will be ultimately bounded. In other words, we show that under certain condition the noise will not spoil these nice properties. However, we will show in this section that if the noise is sufficiently large, the solution to (1.4) will become extinct with probability 1.

Theorem 4.1. Assume that (A1) holds. Then for any given initial data , the solution of (1.4) has the property that

Proof. By Theorem 2.3, the solution will remain in for all with probability 1. We have by the generalized Itô formula and that

where is used in the last step. Then,

where . The quadratic variation of is given by

Therefore, applying the strong law of large numbers for martingales [24], we obtain

It finally follows from (4.3) by dividing on the both sides and then letting that

which is the required assertion (4.1).

Similarly, it is easy to prove the following conclusions.

Theorem 4.2. Assume that (A1) and (A3′) hold. Then for any given initial data , the solution of (1.4) has the property that

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

Theorem 4.3. Assume that (Al) and (A4′) hold. Then for any given initial data , the solution of (1.4) has the property that

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

Remark 4.4. If the noise intensities are sufficiently large in the sense that

then the population represented by (1.4) will become extinct exponentially with probability 1. However, the original delay equation (1.1) may be persistent without environmental noise.

Remark 4.5. Let A and A2 hold, . Then, SDE (1.4) is either stochastically permanent or extinctive. That is, it is stochastically permanent if and only if , while it is extinctive if and only if .

Corollary 4.6. Assume that for some ,

Then for any given initial data , the solution of subsystem

tend to zero a.s.

5. Asymptotic Properties

Lemma 5.1. Assume that (A1′) holds. Then for any given initial data , the solution of (1.4) has the property

Proof. By Theorem 2.3, the solution will remain in for all with probability 1. It is known that

From (3.3) of Theorem 3.3, it has

By the well-known BDG's inequality [24] and the Hölder's inequality, we obtain

Note that

Therefore,

This, together with (5.3), yields

From (5.7), there exists a positive constant such that

Let be arbitrary. Then, by Chebyshev's inequality,

Applying the well-known Borel-Cantelli lemma [24], we obtain that for almost all

for all but finitely many . Hence, there exists a , for almost all , for which (5.10) holds whenever . Consequently, for almost all , if and , then

Therefore,

Letting , we obtain the desired assertion (5.1).

Lemma 5.2. If there exists a constant such that is a nonsingular -matrix and , then the global positive solution of SDE (1.4) has the property that

Proof. Applying the generalized Itô formula, for the fixed constant , we derive from (3.26) that

where on . By (3.37), there exists a positive constant such that

Let be sufficiently small for

Then (5.14) implies that

By directly computing, we have

By the BDG's inequality, it follows

Substituting this and (5.18) into (5.17) gives

Making use of (5.15) and (5.16), we obtain

Let be arbitrary. Then, we have by Chebyshev's inequality that

Applying the Borel-Cantelli lemma, we obtain that for almost all ,

holds for all but finitely many . Hence, there exists an integer , for almost all , for which (5.23) holds whenever . Consequently, for almost all , if and ,

Therefore,

Let , we obtain

Recalling the definition of , this yields

which further implies

This is our required assertion (5.13).

Theorem 5.3. Assume that (A1′′), (A2), and (A3) hold. Then for any given initial data , the solution of (1.4) obeys

Proof. By Theorem 2.3, the solution will remain in for all with probability 1. From Lemmas 3.8, 5.1, and 5.2, it follows

By generalized Itô formula, one has

Dividing by on both sides, then we have

Let , by the strong law of large numbers for martingales and (5.31), we therefore have

which is the required assertions (5.29). And we also have

Let , by the strong law of large numbers for martingales and (5.21), we therefore have

which is the required assertions (5.30).

Similarly, by using Lemmas 3.8, 5.1, and 5.2, it is easy to show the following conclusion.

Theorem 5.4. Assume that (A1′′) and (A4) hold. Then for any given initial data , the solution of (1.4) obeys

Corollary 5.5. If that for some ,

then the solution with positive initial value to subsystem

has the property that

Remark 5.6. If , (1.4) will be written by

which is investigated in [18]. It should be pointed out that (1.4) is more difficult to handle than (5.41). Fortunately, it overcomes the difficulties caused by delay term with the help of Young's inequality. Meanwhile, we get the similar results for .

6. Examples

Example 6.1. Consider a 2-dimensional stochastic differential equation with Markovian switching of the form

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

It can be computed that

By Theorem 2.3, the solution of (6.1) will remain in for all with probability 1.

Case 1. Let the generator of the Markov chain be

By solving the linear equation , we obtain the unique stationary (probability) distribution

Therefore,

By Theorems 3.10 and 5.3, (6.1) is stochastically permanent and its solution with any positive initial value has the following properties:

Case 2. Let the generator of the Markov chain be

By solving the linear equation , we obtain the unique stationary (probability) distribution

So,

Applying Theorems 4.2, (6.1) is extinctive.

Acknowledgments

The authors are grateful to Editor Professor Elena Braverman and anonymous referees for their helpful comments and suggestions which have improved the quality of this paper. This work is supported by the National Natural Science Foundation of China (no. 10771001), Research Fund for Doctor Station of The Ministry of Education of China (no. 20113401110001 and no. 20103401120002), TIAN YUAN Series of Natural Science Foundation of China (no. 11126177), Key Natural Science Foundation (no. KJ2009A49), and Talent Foundation (no. 05025104) of Anhui Province Education Department, 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 090416237, no. 1208085QA15), Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).

References

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.

V. Kolmanovskiĭ and A. Myshkis, Applied Theory of Functional-Differential Equations, vol. 85, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, Boston, Mass, USA, 1993.

A. Bahar and X. Mao, “Stochastic delay population dynamics,” International Journal of Pure and Applied Mathematics, vol. 11, no. 4, pp. 377–400, 2004.

X. Mao, S. Sabanis, and E. Renshaw, “Asymptotic behaviour of the stochastic Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 141–156, 2003.

X. Mao, G. Marion, and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stochastic Processes and their Applications, vol. 97, no. 1, pp. 95–110, 2002.

D. Jiang and N. Shi, “A note on nonautonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 164–172, 2005.

D. Jiang, N. Shi, and X. Li, “Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 588–597, 2008.

T. C. Gard, Introduction to Stochastic Differential Equations, vol. 114, Marcel Dekker, New York, NY, USA, 1988.

A. Bahar and X. Mao, “Stochastic delay Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 364–380, 2004.

X. Mao, “Delay population dynamics and environmental noise,” Stochastics and Dynamics, vol. 5, no. 2, pp. 149–162, 2005.

S. Pang, F. Deng, and X. Mao, “Asymptotic properties of stochastic population dynamics,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 15, no. 5, pp. 603–620, 2008.

I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,” Abstract and Applied Analysis, vol. 2011, Article ID 920412, 14 pages, 2011.

Z. Yu, “Almost surely asymptotic stability of exact and numerical solutions for neutral stochastic pantograph equations,” Abstract and Applied Analysis, vol. 2011, Article ID 143079, 14 pages, 2011.

Y. Takeuchi, N. H. Du, N. T. Hieu, and K. Sato, “Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 938–957, 2006.

Q. Luo and X. Mao, “Stochastic population dynamics under regime switching,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 69–84, 2007.

X. Li, D. Jiang, and X. Mao, “Population dynamical behavior of Lotka-Volterra system under regime switching,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 427–448, 2009.

X. Li, A. Gray, D. Jiang, and X. Mao, “Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 11–28, 2011.

C. Zhu and G. Yin, “On hybrid competitive Lotka-Volterra ecosystems,” Nonlinear Analysis A, vol. 71, no. 12, pp. e1370–e1379, 2009.

N. H. Du, R. Kon, K. Sato, and Y. Takeuchi, “Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise,” Journal of Computational and Applied Mathematics, vol. 170, no. 2, pp. 399–422, 2004.

M. Slatkin, “The dynamics of a population in a Markovian environment,” Ecology, vol. 59, no. 2, pp. 249–256, 1978.

M. Liu and K. Wang, “Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment,” Journal of Theoretical Biology, vol. 264, no. 3, pp. 934–944, 2010.

M. Liu and K. Wang, “Asymptotic properties and simulations of a stochastic logistic model under regime switching,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2139–2154, 2011.

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.