Stochastic Delay Logistic Model under Regime Switching

and Applied Analysis 3 extinction of a logistic model under regime switching was considered in 18 , a new singlespeciesmodel disturbed by bothwhite 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 Ω,F, {Ft}t≥0, P be a complete probability space with a filtration {Ft}t≥0 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 R the interval 0,∞ and denote by R the interval 0,∞ . Moreover, let τ > 0 and denote by C −τ, 0 ;R the family of continuous functions from −τ, 0 to R . 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 P{r t δ v | r t u} ⎧ ⎨ ⎩ γuvδ o δ , if u/ v, 1 γuvδ o δ , if u v, 2.1 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 R . 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. 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: πΓ 0, 2.2 subject to n ∑ i 1 πi 1, πi > 0, ∀ i ∈ S. 2.3 We refer to 9, 24 for the fundamental theory of stochastic differential equations. 4 Abstract and Applied Analysis For convenience and simplicity in the following discussion, define f̂ min i∈S f i , f̌ max i∈S f i , f max i∈S ∣ ∣f i ∣ ∣, 2.4 where {f i }i∈S is a constant vector. As x t in model 1.4 denotes population size at time t, 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 θ i i 1, 2, . . . , n such that max i∈S ( −b i 1 4θ i c2 i θ̌ ) ≤ 0. 2.5 Then, for any given initial data {x t : −τ ≤ t ≤ 0} ∈ C −τ, 0 ;R , there is a unique solution x t to 1.4 on t ≥ −τ and the solution will remain in R with probability 1, namely, x t ∈ R for all t ≥ −τ a.s. Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial data {x t : −τ ≤ t ≤ 0} ∈ C −τ, 0 ;R , there is a unique maximal local solution x t on t ∈ −τ, τe , where τe is the explosion time. To show that this solution is global, we need to prove τe ∞ a.s. Let k0 > 0 be sufficiently large for 1 k0 < min −τ≤t≤0 x t ≤ max −τ≤t≤0 x t < k0. 2.6 For each integer k ≥ k0, define the stopping time τk inf { t ∈ 0, τe : x t / ∈ ( 1 k , k )} , 2.7 where throughout this paper we set inf ∅ ∞ as usual ∅ denotes the empty set . Clearly, τk is increasing as k → ∞. Set τ∞ limk→∞τk, where τ∞ ≤ τe a.s. If we can show that τ∞ ∞ a.s., then τe ∞ a.s. and x t ∈ R a.s. for all t ≥ 0. In other words, we need to show τ∞ ∞ a.s. Define a C2-function V : R → R by V x x − 1 − logx, 2.8 which is not negative on x > 0. Let k ≥ k0 and T > 0 be arbitrary. For 0 ≤ t ≤ τk ∧ T , it is not difficult to show by the generalized Itô formula that dV x t LV x t , x t − τ , r t dt σ r t x t − 1 dw t , 2.9 Abstract and Applied Analysis 5 where LV : R × R × S → R is defined byand Applied Analysis 5 where LV : R × R × S → R is defined by LV ( x, y, i ) −a i 1 2 σ2 i a i b i x − c i y − b i x2 c i xy. 2.10 Using condition 2.5 , we compute −b i x2 c i xy ≤ −b i x2 1 4θ i c2 i x2 θ i y2 ≤ −θ̌x2 θ̌y2. 2.11 Moreover, there is clearly a constant K1 > 0 such that −a i 1 2 σ2 i a i b i x − c i y ≤ K1 ( 1 x y ) . 2.12 Substituting these into 2.10 yields LV ( x, y, i ) ≤ K1 ( 1 x y ) − θ̌x2 θ̌y2. 2.13 Noticing that u ≤ 2 u − 1 − logu 2 on u > 0, we obtain that LV ( x, y, i ) ≤ K2 ( 1 V x V ( y )) − θ̌x2 θ̌y2, 2.14 where K2 is a positive constant. Substituting these into 2.9 yields dV x t ≤ [ K2 1 V x t V x t − τ − θ̌x2 t θ̌x2 t − τ ] dt σ r t x t − 1 dw t . 2.15 Now, for any t ∈ 0, T , we can integrate both sides of 2.15 from 0 to τk ∧ t and then take the expectations to get EV x τk ∧ t ≤ V x 0 E ∫ τk∧t 0 [ K2 1 V x s V x s − τ − θ̌x2 s θ̌x2 s − τ ] ds.


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 , Kolmanovskiȋ, and Myshkis 2 , Kuang 3 among many others. In 1.1 , the state x t 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 a > 0, b > 0, and c < b 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 a 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 a by a σẇ t , 1.2 whereẇ t is a white noise i.e., w t is a Brownian motion and σ is a positive number representing the intensity of noise. As a result, 1.1 becomes a stochastic differential equation SDE, in short dx t x t a − bx t cx t − τ dt σdw t .

1.3
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 r t , t ≥ 0 with a finite state space S {1, 2, . . . , n}. 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 r 0 ι ∈ S, then the ecosystem 1.4 obeys the SDE until the Markov chain r t jumps to another state, say, ς. Then the ecosystem satisfies the SDE for a random amount of time until r t 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 3 extinction of a logistic model under regime switching was considered in 18 , a new singlespecies 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.

Global Positive Solution
Throughout this paper, unless otherwise specified, let Ω, F, {F t } t≥0 , P be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions i.e., it is right continuous and F 0 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 R the interval 0, ∞ and denote by R the interval 0, ∞ . Moreover, let τ > 0 and denote by C −τ, 0 ; R the family of continuous functions from −τ, 0 to R .
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 γ uv given by where δ > 0, γ uv is the transition rate from u to v and γ uv ≥ 0 if u / v, while γ uu − v / u γ uv . 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 R . 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. Under this condition, the Markov chain has a unique stationary probability distribution π π 1 , π 2 , . . . , π n ∈ R 1×n which can be determined by solving the following linear equation: We refer to 9, 24 for the fundamental theory of stochastic differential equations.

Abstract and Applied Analysis
For convenience and simplicity in the following discussion, define where {f i } i∈S is a constant vector. As x t in model 1.4 denotes population size at time t, it should be nonnegative. Thus, for further study, we must give some condition under which 1.4 has a unique global positive solution.
Then, for any given initial data {x t : −τ ≤ t ≤ 0} ∈ C −τ, 0 ; R , there is a unique solution x t to 1.4 on t ≥ −τ and the solution will remain in R with probability 1, namely, x t ∈ R for all t ≥ −τ a.s.
Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given where τ e is the explosion time. To show that this solution is global, we need to prove τ e ∞ a.s. Let k 0 > 0 be sufficiently large for For each integer k ≥ k 0 , define the stopping time where throughout this paper we set inf ∅ ∞ as usual ∅ denotes the empty set . Clearly, τ k is increasing as s. If we can show that τ ∞ ∞ a.s., then τ e ∞ a.s. and x t ∈ R a.s. for all t ≥ 0. In other words, we need to Abstract and Applied Analysis Using condition 2.5 , we compute Moreover, there is clearly a constant K 1 > 0 such that Substituting these into 2.10 yields where K 2 is a positive constant. Substituting these into 2.9 yields

2.15
Now, for any t ∈ 0, T , we can integrate both sides of 2.15 from 0 to τ k ∧ t and then take the expectations to get Abstract and Applied Analysis and, similarly Substituting these into 2. 16 gives 2.20 Note that for every ω ∈ {τ k ≤ T }, x τ k , ω equals either k or 1/k, thus It then follows from 2.20 that Then the conclusions of Theorem 2.1 hold.
Abstract and Applied Analysis 7 The following theorem is easy to verify in applications, which will be used in the sections below.
Then for any given initial data 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 K 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

2.27
Therefore, if we assume that Then for any given initial data {x t : −τ ≤ t ≤ 0} ∈ C −τ, 0 ; R , there is a unique solution x t to 1.4 on t ≥ −τ and the solution will remain in R with probability 1, namely, x t ∈ R for all t ≥ −τ a.s.
Similarly, we can establish a corollary as follows.

Corollary 2.5.
Assume that there is a positive number θ such that Then the conclusions of Theorem 2.4 hold.

Asymptotic Bounded Properties
For convenience and simplicity in the following discussion, we list the following assumptions.
where x t is the solution of 1.4 with any positive initial value. 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 A1 hold and p is an arbitrary given positive constant. Then for any given initial
where both K 1 p and K 2 p are positive constants defined in the proof.
Proof. By Theorem 2.3, the solution x t will remain in R for all t ≥ −τ with probability 1. Let

3.16
The inequality above implies where H 3 max i∈S {sup x∈R 1/2 p p − 1 σ 2 i x p pa i x p − λpc/ p 1 x p 1 } and the desired assertion 3.4 follows by setting K 2 p H 3 /pλ.

Remark 3.4. From 3.3 of Theorem 3.3, there is a T > 0 such that
Since E x p t is continuous, there is a K 1 p, x 0 such that Taking L p, x 0 max 2K 1 p , K 1 p, x 0 , we havefor the fundamental theory of This means that the pth moment of any positive solution of 1.4 is bounded. 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 n i 1 π i β i > 0. Moreover, let G be a vector or matrix. By G 0 we mean all elements of G are positive. We also adopt here the traditional notation by letting Z n×n A a ij n×n : a ij ≤ 0, i / j .

3.22
We will also need some useful results.

Lemma 3.7 see 24 .
If A ∈ Z n×n , then the following statements are equivalent.
1 A is a nonsingular M-matrix (see [24] for definition of M-matrix).

3.23
3 A is semipositive, that is, there exists x 0 in R n such that Ax 0.

Lemma 3.8 see 18 . (i) Assumptions A2 and A3 imply that there exists a constant θ > 0 such that the matrix
where H is a fixed positive constant (defined by 3.35 in the proof).

Abstract and Applied Analysis 13
Proof. Let U t x −1 t on t ≥ 0. Applying the generalized Itô formula, we have dU t U t −a r t σ 2 r t b r t x t − c r t x t − τ dt − σU t dw t .

3.28
Define the function V : R × S → R by V U, i q i 1 U θ . Applying the generalized Itô formula again, we have

3.33
It is computed that

3.34
where Then lim sup Recalling the definition of U t , we obtain the required assertion. 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. x t > 0. 3.38 Furthermore, we consider its associated stochastic delay equation 1.4 , that is,

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.
then the population x t represented by 1.4 will become extinct exponentially with probability 1. However, the original delay equation 1.1 may be persistent without environmental noise.
Then for any given initial data tend to zero a.s.