Abstract and Applied Analysis
Volume 2013, Issue 1 129072
Research Article
Open Access

Stochastic Delay Population Dynamics under Regime Switching: Permanence and Asymptotic Estimation

Zheng Wu, 

Zheng Wu

School of Mathematical Sciences, Anhui University, Hefei 230039, China ahu.edu.cn

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Hao Huang, 

Hao Huang

School of Mathematical Sciences, Anhui University, Hefei 230039, China ahu.edu.cn

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Lianglong Wang, 

Corresponding Author

Lianglong Wang

School of Mathematical Sciences, Anhui University, Hefei 230039, China ahu.edu.cn

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First published: 28 May 2013
https://doi.org/10.1155/2013/129072
Citations: 3
Academic Editor: Yuming Chen

Abstract

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. Permanence and asymptotic estimations of solutions are investigated by virtue of V-function technique, M-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 and is known as the delay Lotka-Volterra model or the delay logistic equation. The delay Lotka-Volterra model for n interacting species is described by the n-dimensional delay differential equation
()
where , , , . There is an extensive literature concerned with the dynamics of this delay model and have had lots of nice results and we here only mention the work of Ahmad and Rao [1], Bereketoglu and Győri [2], and 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 x(t) 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 a > 0, b > 0, and c < b for (1) while much more complicated conditions are required on matrices A and B 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 behavior 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 bi in (2) represents the intrinsic growth rate of species i. 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 terms of mathematics, we can therefore replace the rate bi by , where is a white noise (i.e., w(t) is a Brownian motion) and σi ≥ 0 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 attention 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 (see [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 r(t), t ≥ 0 with a finite state space S = {1,2, …, N}. 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 r(0) = ι ∈ S. Then the ecosystem (4) obeys the SDE
()
until the Markov chain r(t) jumps to another state, say ς. Therefore, the ecosystem (4) satisfies the SDE
()
for a random amount of time until the Markov chain r(t) 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 obtained 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].

In [28], some results have been obtained for (4), such as existence of global positive solutions, stochastically ultimate boundedness, and extinction. In contrast to the existing results, our new contributions in this paper are as follows.
  • (i)

    The stochastic permanence of solutions is derived.

  • (ii)

    The asymptotic estimations of the solutions are obtained, which is related to the stationary probability distribution of the Markov chain.

The rest of the paper is arranged as follows. For convenience of the reader, we briefly recall the main result of [28] in Section 2. The main results of this paper are arranged in Sections 3 and 4. Section 3 is devoted to the stochastic permanence. The asymptotic estimations of the solutions are obtained in Section 4. Finally, an example is given to illustrate our main results.

2. Properties of the 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 ℱ0 contains all P-null sets). Let w(t), t ≥ 0, be a scalar standard Brownian motion defined on this probability space. We also denote by the positive cone in Rn, that is, for all 1 ≤ i ≤ n} and denote by the nonnegative cone in Rn, that is, for all 1 ≤ i ≤ n}. If A is a vector or matrix, its transpose is denoted be AT. If A is a matrix, its trace norm is denoted by , whilst it operator norm is denoted by ∥A∥ = sup {|Ax| : |x| = 1}. Moreover, let τ > 0 and denote by C([−τ, 0]; R+) the family of continuous functions from [−τ, 0] to R+.

In this paper we will use a lot of quadratic functions of the form xTAx for the state only. Therefore, for a symmetric n × n matrix A, we naturally introduce the following definition:
()
For more properties of , please see the appendix in [7].
Let r(t) be a right-continuous Markov chain on the probability space, taking values in a finite state space S = {1,2, …, N}, with the generator Γ = (γuv) given by
()
where δ > 0, γuv is the transition rate from u to v and γuv ≥ 0 if u ≠ v, while γuu = −∑v≠u γuv. We assume that the Markov chain r(·) is independent of the Brownian motion w(·). It is well known that almost every sample path of r(·) 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 r(t) 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 u, v ∈ S, one can find finite numbers i1, i2, …, ik ∈ S such that . Under this condition, the Markov chain has a unique stationary (probability) distribution π = (π1, π2, …, πN) ∈ R1×N which can be determined by solving the following linear equation:
()
subject to
()
For the fundamental theory of stochastic differential equations, one can refer to [12, 29].
For convenience and simplicity in the following discussion, for any constant sequence fi(k), (1 ≤ i ≤ n,  k ∈ S), let
()
To proceed, we first state a result, whose proof can be found in [28].

Assumption 1. Assume that there exist positive numbers c1, …, cn such that

()
where .

Assumption 2. Assume that there exist positive numbers c1, …, cn such that

()
where .

Assumption 3. Assume that there exist positive numbers c1, …, cn such that

()
where and .

Theorem 1. (1) Under Assumption 1, for any given initial data , there is a unique solution x(t) to (4) on t ≥ −τ and the solution will remain in with probability 1, namely, for all t ≥ −τ almost surely.

(2) Under Assumption 2, for any given initial data and any given positive constant p, there are two positive constant K1(p) and K2(p), such that the solution x(t) of (4) has the properties that

()
()

(3) Solutions of (4) are stochastically ultimately bounded under Assumption 2; that is, for any ε ∈ (0,1), there exists a positive constants H = H(ε), such that the solutions of (4) with any positive initial value have the property that

()

(4) Under Assumption 3, for any given initial data {x(t) : −τ ≤ t ≤ 0} ∈ C([−τ, 0]; R+), the solution x(t) of (4) has the properties that

()
where . Particularly, if  , then
()

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

3. Stochastic Permanence

Definition 2. Equation (4) is said to be stochastically permanent if, for any ε ∈ (0,1), there exist positive constants H = H(ε), δ = δ(ε) such that

()
where x(t) is the solution of (4) with any positive initial value.

It is obvious that if a stochastic equation is stochastically permanent, its solutions must be stochastically ultimately bounded. For convenience, let

()
and we impose the following assumptions.

Assumption 4. For some u ∈ S, γiu > 0 (for all i ≠ u).

Assumption 5. .

Assumption 6. For each k ∈ S, α(k) > 0.

Let G be a vector or matrix. By G ≫ 0, we mean all elements of G are positive, and by G ≥ 0, we mean all elements of G are nonnegative. We also adopt here the traditional notation by letting
()

Lemma 3 (see [29].)If has all of its row sums positive, that is,

()
then A > 0.

Lemma 4 (see [29].)If A ∈ ZN×N, then the following statements are equivalent:

  • (1)

    A  is a nonsingular M-matrix.

  • (2)

    All of the principal minors of A are positive; that is,

    ()

  • (3)

    A is semipositive; that is, there exists x ≫ 0 in RN such that Ax ≫ 0.

Lemma 5 (see [23].)Assumptions 4 and 5 imply that there exists a constant θ > 0 such that the matrix

()
is a nonsingular M-matrix, where
()

Lemma 6 (see [23].)Assumption 6 implies that there exists a constant θ > 0 such that the matrix A(θ) is a nonsingular M-matrix.

Lemma 7. If there exists a constant θ > 0 such that A(θ) is a nonsingular M-matrix and B(k) ≥ 0  (k = 1,2, …, N), then the global positive solution x(t) of (4) has the property that

()
where H is a fixed positive constant (defined by (42) in the proof).

Proof. Define on . Then

()

Define also

()
Let y(t) = x(t − τ). Applying the generalized It formula, we derive from (28) that
()
dropping x(t) from U(x(t)), V(x(t)) and t from x(t), y(t), respectively. By Lemma 4, for given θ, there is a vector such that
()
namely,
()

Define the function by . It follows from the generalized It formula that

()
where
()

It is easy to see that, for all ,

()
where K is a positive constant, while
()

Consequently,

()

Substituting (37) into (34) yields

()

Now, choose a constant κ > 0 sufficiently small such that it satisfies , that is,

()

Then, by the generalized Itô formula again,

()

It is computed that

()
where
()

This implies

()

For , note that . Consequently,

()

The required assertion (27) is obtained.

Assumption 7. Assume that there exist positive numbers c1, …, cn such that

()
where . Moreover for each k ∈ S, B(k) ≥ 0.

Theorem 8. Under Assumptions 4, 5, and 7, (4) is stochastically permanent.

The proof is a simple application of the Chebyshev’s inequality, Lemmas 5 and 7, and Theorem 1(3). Similarly, we have the following result.

Theorem 9. Under Assumptions 6 and 7, (4) is stochastically permanent.

4. Asymptotic Properties

Lemma 10. Under Assumption 2, for any given initial data {x(t) : −τ ≤ t ≤ 0} ∈ C([−τ, 0]; R+), the solution x(t) of (4) with any positive initial value has the property

()

Proof. By Theorem 1 (1), the solution x(t) will remain in for all t ≥ −τ with probability 1. Denote , on . It is known that

()
We can also derive from this that
()

From (15), we know that limsup t→∞E|x(t)| ≤ K1(1) and limsup t→∞E|x(t)|2 ≤ K1(2). By the well-known BDG’s inequality [29] and the Hölder’s inequality, we derive that

()

Combining the inequality above with

()
we get that
()

Recalling the following inequality for any  , we obtain

()

It is following from (52) that there is a positive constant M such that

()

Let ε > 0 be arbitrary. Then, by Chebyshev’s inequality, we have

()

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

()
holds for all but finitely many k. Hence, there exists a k0(ω), for almost all ω ∈ Ω, for which (55) holds whenever k ≥ k0. Consequently, for almost all ω ∈ Ω, if k ≥ k0 and k ≤ t ≤ k + 1,
()

Therefore, lim  sup t→∞ (log  (|x(t)|)/log  t) ≤ 1 + ε  a.s. Letting ε → 0, we obtain the desired assertion (46).

Lemma 11. If there exists a constant θ > 0 such that A(θ) is a nonsingular M-matrix and for each k ∈ S, B(k) ≥ 0, then the global positive solution x(t) of SDE (4) has the property that

()

Proof. Let be the same as defined by (29); for convenience, we write U(x(t)) = U(t). Applying the generalized It formula, for the fixed constant θ > 0, we derive from (37) that

()

By (43), there exists a positive constant M such that

()

Let δ > 0 be sufficiently small such that

()

Then (58) implies that

()
It is computed that
()

On the other hand, by the BDG’s inequality, we derive that

()

Substituting this and (62) into (61) gives

()
Making use of (59) and (60), we obtain that
()

Let ε > 0 be arbitrary. Then, by Chebyshev inequality, we have

()
Applying the Borel-Cantelli lemma, we obtain that for almost all ω ∈ Ω
()
holds for all but finitely many k. Hence, there exists an integer k0(ω) > 1/δ + 2, for almost all ω ∈ Ω, for which (67) holds whenever k ≥ k0. Consequently, for almost all ω ∈ Ω, if k ≥ k0 and (k − 1)δ ≤ t ≤ kδ,
()
Therefore limsup t→∞ (log (1+U(t))θ/log t) ≤ 1 + ε  a.s. Let ε → 0, we obtain the desired assertion
()
Recalling the definition of U(t), this yields limsup t→∞(log |x(t)|−θ/log t) ≤ 1  a.s., which further implies
()
This is our required assertion (57).

Assumption 8. Assume that there exist positive numbers c1, …, cn such that

()
where and . Moreover for each k ∈ S, B(k) ≥ 0.

Theorem 12. Under Assumptions 4, 5, and 8, for any given initial data {x(t) : −τ ≤ t ≤ 0} ∈ C([−τ, 0]; R+), the solution x(t) of (4) obeys

()

where .

Proof. By Theorem 1(1), the solution x(t) will remain in R+ for all t ≥ −τ with probability 1. Define , for . By generalized It formula, one has

()
From Lemmas 5, 10, and 11, it follows that
()
By (73), it has
()
Meanwhile,
()
Substituting (76) into (75) yields
()
Hence,
()
Applying the strong law of large numbers for martingales, we have
()
Dividing both sides of (78) by t and letting t → ∞, we obtain that
()
which implies the required assertion (72).

On the other hand, it is observed from (75)-(76) that

()
Hence,
()
Consequently, one gets that
()
which implies the other required assertion (4.12).

Similarly, using Lemmas 6, 10, and 11, we can show the following

Theorem 13. Under Assumptions 5 and 8, for any given initial data {x(t) : −τ ≤ t ≤ 0} ∈ C([−τ, 0]; R+), the solution x(t) of (4) obeys

()
where .

5. Examples

In this section, an example is given to illustrate our main results.

Example 1. Consider the two-species Lotka-Volterra system with regime switching described by

()
where , b(r(t)) = (b1r(t)), b2(r(t)))T, σ(r(t)) = (σ1(r(t)), σ2(r(t)))T,
()
and r(t) is a right-continuous Markov chain taking values in S = {1,2}, and r(t) and w(t) are independent. Here
()
Let . It is easy to compute that
()
Then
()
Moreover, α(1) = 3, α(2) = −4, β(1) = 7, and β(2) = −2.

By Theorem 1(1), the solution x(t) of (85) will remain in R+ for all t ≥ −τ with probability 1. Let the generator of the Markov chain r(t) be

()
By solving the linear equation πΓ = 0, we obtain the unique stationary (probability) distribution π = (π1, π2) = (2/3, 1/3). Then
()
Therefore, by Theorems 8 and 12, (85) is stochastically permanent and the solutions have the following properties:
()

Acknowledgments

The authors are grateful to Editor Proffessor Yuming Chen for the diligent work. 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), 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 1308085MA01, no. 1308085QA15, and no. 1208085QA15), and Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).

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