AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 241702 10.1155/2012/241702 241702 Research Article Stochastic Delay Logistic Model under Regime Switching Wu Zheng Huang Hao Wang Lianglong Braverman Elena School of Mathematical Science Anhui University Hefei, Anhui 230039 China ahut.edu.cn 2012 9 7 2012 2012 02 04 2012 14 06 2012 2012 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.

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 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 (1.1)dx(t)dt=x(t)[a-bx(t)+cx(t-τ)] 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 , Kolmanovskiĭ, and Myshkis , Kuang  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  (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 . 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 (1.2)a+σw˙(t), where w˙(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) (1.3)dx(t)=x(t)[(a-bx(t)+cx(t-τ))dt+σdw(t)]. We refer to  for more details.

To our knowledge, much attention to environmental noise is paid on white noise ( 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, ). 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),t0 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: (1.4)dx(t)=x(t)[(a(r(t))-b(r(t))x(t)+c(r(t))x(t-τ))dt+σ(r(t))dw(t)]. The mechanism of ecosystem described by (1.4) can be explained as follows. Assume that initially, the Markov chain r(0)=ιS, then the ecosystem (1.4) obeys the SDE (1.5)dx(t)=x(t)[(a(ι)-b(ι)x(t)+c(ι)x(t-τ))dt+σ(ι)dw(t)], until the Markov chain r(t) jumps to another state, say, ς. Then the ecosystem satisfies the SDE (1.6)dx(t)=x(t)[(a(ς)-b(ς)x(t)+c(ς)x(t-τ))dt+σ(ς)dw(t)], 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 extinction of a logistic model under regime switching was considered in , a new single-species model disturbed by both white noise and colored noise in a polluted environment was developed and analyzed in , a general stochastic logistic system under regime switching was proposed and was treated in .

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,,{t}t0,P) be a complete probability space with a filtration {t}t0 satisfying the usual conditions (i.e., it is right continuous and 0 contains all P-null sets ). Let w(t),t0, 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 (2.1)P{r(t+δ)=vr(t)=u}={γuvδ+o(δ),ifuv,1+γuvδ+o(δ),ifu=v, where δ>0, γuv is the transition rate from u to v and γuv0 if uv, while γuu=-vuγuv. 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: (2.2)πΓ=0, subject to (2.3)i=1nπi=1,πi>0,iS. We refer to [9, 24] for the fundamental theory of stochastic differential equations.

For convenience and simplicity in the following discussion, define (2.4)f^=miniSf(i),fˇ=maxiSf(i),f¯=maxiS|f(i)|, where {f(i)}iS 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 (2.5)maxiS(-b(i)+14θ(i)c2(i)+θˇ)0. Then, for any given initial data {x(t):-τt0}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):-τt0}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 (2.6)1k0<min-τt0x(t)max-τt0x(t)<k0. For each integer kk0, define the stopping time (2.7)τk=inf{t[0,τe):x(t)(1k,k)}, where throughout this paper we set inf= (as usual denotes the empty set). Clearly, τk is increasing as k. Set τ=limkτk, where ττe a.s. If we can show that τ= a.s., then τe= a.s. and x(t)R+ a.s. for all t0. In other words, we need to show τ= a.s. Define a C2-function V:R+R+ by (2.8)V(x)=x-1-logx, which is not negative on x>0. Let kk0 and T>0 be arbitrary. For 0tτkT, it is not difficult to show by the generalized Itô formula that (2.9)dV(x(t))=LV(x(t),x(t-τ),r(t))dt+σ(r(t))(x(t)-1)dw(t), where LV:R+×R+×SR is defined by (2.10)LV(x,y,i)=-a(i)+12σ2(i)+(a(i)+b(i))x-c(i)y-b(i)x2+c(i)xy. Using condition (2.5), we compute (2.11)-b(i)x2+c(i)xy-b(i)x2+14θ(i)c2(i)x2+θ(i)y2-θˇx2+θˇy2. Moreover, there is clearly a constant K1>0 such that (2.12)-a(i)+12σ2(i)+(a(i)+b(i))x-c(i)yK1(1+x+y). Substituting these into (2.10) yields (2.13)LV(x,y,i)K1(1+x+y)-θˇx2+θˇy2.

Noticing that u2(u-1-logu)+2onu>0, we obtain that (2.14)LV(x,y,i)K2(1+V(x)+V(y))-θˇx2+θˇy2, where K2 is a positive constant. Substituting these into (2.9) yields (2.15)dV(x(t))[K2(1+V(x(t))+V(x(t-τ)))-θˇx2(t)+θˇx2(t-τ)]dt+σ(r(t))(x(t)-1)dw(t).

Now, for any t[0,T], we can integrate both sides of (2.15) from 0 to τkt and then take the expectations to get (2.16)EV(x(τkt))V(x(0))+E0τkt[K2(1+V(x(s))+V(x(s-τ)))-θˇx2(s)+θˇx2(s-τ)]ds. Compute (2.17)E0τktV(x(s-τ))ds=E-ττkt-τV(x(s))ds-τ0V(x(s))ds+E0τktV(x(s))ds, and, similarly (2.18)E0τktx2(s-τ)ds-τ0x2(s)ds+E0τktx2(s)ds. Substituting these into (2.16) gives (2.19)EV(x(τkt))K3+2K2E0τktV(x(s))dsK3+2K2E0tV(x(τks))ds=K3+2K20tEV(x(τks))ds, where K3=V(x(0))+K2T+K2-τ0V(x(s))ds+θˇ-τ0x2(s)ds.

By the Gronwall inequality, we obtain that (2.20)EV(x(τkT))K3e2TK2. Note that for every ω{τkT},x(τk,ω) equals either k or 1/k, thus (2.21)V(x(τk,ω))[(k-1-logk)(1k-1+logk)]. It then follows from (2.20) that (2.22)K3e2TK2E[1{τkT}(ω)V(x(τkT,ω))]=E[1{τkT}(ω)V(x(τk,ω))]P{τkT}[(k-1-logk)(1k-1+logk)], where 1{τkT} is the indicator function of {τkT}. Letting k gives limkP{τkT}=0 and hence P{τT}=0. Since T>0 is arbitrary, we must have P{τ<}=0, so P{τ=}=1 as required.

Corollary 2.2.

Assume that there is a positive number θ such that (2.23)maxiS(-b(i)+14θc2(i)+θ)0. 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 (2.24)-b^+c¯0. Then for any given initial data {x(t):-τt0}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.

The proof of this theorem is the same as that of the theorem above. Let (2.25)V(x)=x-1-logxonx>0, then we have (2.9) and (2.10). By (2.24), we get (2.26)LV(x,y,i)-a(i)+12σ2(i)+(a(i)+b(i))x-c(i)y-b(i)x2+c(i)xyK(1+x+y)+(-b(i)+c¯)x2-12c¯x2+12c¯y2K(1+x+y)-12c¯x2+12c¯y2, 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)c(i)xy12θ(i)c2(i)x2+θ(i)2y2,-b(i)x2+c(i)xy-b(i)x2+12θ(i)c2(i)x2+θ(i)2y2=(-b(i)+12θ(i)c2(i)+θˇ2)x2-θˇ2x2+θˇ2y2. Therefore, if we assume that (2.28)maxiS(-b(i)+12θ(i)c2(i)+θˇ2)0, then (2.29)-b(i)x2+c(i)xy-θˇ2x2+θˇ2y2, hence (2.30)LV(x,y,i)K1(1+x+y)-θˇ2x2+θˇ2y2, 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 θ(i)(i=1,2,,n) such that (2.31)maxiS(-b(i)+12θ(i)c2(i)+θˇ2)0. Then for any given initial data {x(t):-τt0}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 (2.32)maxiS(-b(i)+12θc2(i)+θ2)0. 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.

For each iS,b(i)>0,and-b^+c¯0.

For each iS,b(i)>0,and-b^+c¯<0.

For each iS,b(i)>0,c(i)0and-b^+cˇ<0.

For some uS,γiu>0(iu).

i=1nπi[a(i)-(1/2)σ2(i)]>0.

i=1nπi[a(i)-(1/2)σ2(i)]<0.

For each iS,a(i)-(1/2)σ2(i)>0.

For each iS,a(i)-(1/2)σ2(i)<0.

Definition 3.1.

Equation (1.4) is said to be stochastically permanent if for any ɛ(0,1), there exist positive constants H=H(ɛ),δ=δ(ɛ) such that (3.1)liminft+P{x(t)H}1-ɛ,liminft+P{x(t)δ}1-ɛ, where x(t) 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 ɛ(0,1), there exists a positive constant H=H(ɛ), such that the solutions of (1.4) with any positive initial value have the property that (3.2)limsupt+P{x(t)>H}<ɛ.

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 data {x(t):-τt0}C([-τ,0];R+), the solution x(t) of (1.4) has the properties that (3.3)limsuptE(xp(t))K1(p),(3.4)limsupt1t0tE(xp+1(s))K2(p), where both K1(p) and K2(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 (3.5)-λ=(1+c¯p+1)-1(-b^+c¯),(3.6)γ=τ-1log(1+λ). Define the function V:R+×R+R+ by (3.7)V(x,t)=eγtxp. By the generalized Itô formula, we have (3.8)dV(x(t),t)=LV(x(t),x(t-τ),t,r(t))dt+peγtσ(r(t))xpdw(t), where LV:R+×R+×R+×SR is defined by (3.9)LV(x,y,t,i)=eγt(γ+pa(i)+12p(p-1)σ2(i))xp-peγtb(i)xp+1+peγtc(i)xpy. By (3.5) and Young's inequality, we obtain that (3.10)LV(x,y,t,i)eγt(γ+pa(i)+12p(p-1)σ2(i))xp-peγtb(i)xp+1+peγtc¯(pp+1xp+1+1p+1yp+1)eγt[(γ+paˇ+12p(p-1)σ2(i))xp-λpxp+1]+eγtpc¯p+1[-(1+λ)xp+1+yp+1]H1eγt+eγtpc¯p+1(-eγτxp+1+yp+1), where H1=maxiS{supxR+[(γ+paˇ+(1/2)p(p-1)σ2(i))xp-λpxp+1]}. Moreover, (3.11)0teγsxp+1(s-τ)dseγτ-τ0xp+1(s)ds+eγτ0teγsxp+1(s)ds. By (3.10) and (3.11), one has (3.12)eγtE(xp(t))xp(0)+H1γ(eγt-1)+pc¯p+1eγτ-τ0xp+1(s)ds, which yields (3.13)limsuptE(xp(t))K1(p), where (3.14)K1(p)=maxiS{supxR+γ-1[(γ+paˇ+12p(p-1)σ2(i))xp-λpxp+1]}.

By the generalized Itô formula, Young's inequality and (3.5) again, it follows (3.15)0E(xp(t))x0p+E0tpxp(s)[12(p-1)σ2(r(s))+a(r(s))]-pb(r(s))xp+1(s)+pc¯xp(s)x(s-τ)dsx0p+E0t{1+c¯p+112p(p-1)σ2(r(s))xp(s)+pa(r(s))xp(s)x0p+E0t+p[-(1+c¯p+1)λ]xp+1(s)+pc¯p+1(-xp+1(s)+xp+1(s-τ))}dsH2+E0t{12p(p-1)σ2(r(s))xp(s)+pa(r(s))xp+p[-(1+c¯p+1)λ]xp+1(s)}ds, where H2=xp(0)+(pc¯/(p+1))-τ0xp+1(s)ds. This implies (3.16)pλE0txp+1(s)dsH2+E0t12p(p-1)σ2(r(s))xp(s)+pa(r(s))xp(s)-λpc¯p+1xp+1(s)ds.

The inequality above implies (3.17)limsupt1tE0txp+1(s)H3pλ, where H3=maxiS{supxR+[(1/2)p(p-1)σ2(i)xp+pa(i)xp-(λpc¯/(p+1))xp+1]} and the desired assertion (3.4) follows by setting K2(p)=H3/pλ.

Remark 3.4.

From (3.3) of Theorem 3.3, there is a T>0 such that (3.18)E(xp(t))2K1(p)tT. Since E(xp(t)) is continuous, there is a K¯1(p,x0) such that (3.19)E(xp(t))K¯1(p,x0)fort[0,T]. Taking L(p,x0)=max(2K1(p),K¯1(p,x0)), we have (3.20)E(xp(t))L(p,x0)t[0,). This means that the pth 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 pth (p>1) moment of solutions of (1.4) is bounded.

Theorem 3.6.

Solutions of (1.4) are stochastically ultimately bounded under (A1).

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 (3.21)β(i)=a(i)-12σ2(i). Under (A3), it has i=1nπiβ(i)>0. Moreover, let G be a vector or matrix. By G0 we mean all elements of G are positive. We also adopt here the traditional notation by letting (3.22)Zn×n={A=(aij)n×n:aij0,ij}. We will also need some useful results.

Lemma 3.7 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

If AZn×n, then the following statements are equivalent.

A is a nonsingular M-matrix (see  for definition of M-matrix).

All of the principal minors of A are positive; that is, (3.23)|a11a1kak1akk|>0 forevery k=1,2,,n.

A is semipositive, that is, there exists x0 in Rn such that Ax0.

Lemma 3.8 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

(i) Assumptions (A2) and (A3) imply that there exists a constant θ>0 such that the matrix (3.24)A(θ)=diag(ξ1(θ),ξ2(θ),,ξn(θ))-Γ is a nonsingular M-matrix, where ξi(θ)=θβ(i)-(1/2)θ2σ2(i),iS.

(ii) Assumption (A4) implies that there exists a constant θ>0 such that the matrix A(θ) is a nonsingular M-matrix.

Lemma 3.9.

If there exists a constant θ>0 such that A(θ) is a nonsingular M-matrix and c(i)0(i=1,2,,n), then the global positive solution x(t) of (1.4) has the property that (3.25)limsuptE(|x(t)|-θ)H, where H is a fixed positive constant (defined by (3.35) in the proof).

Proof.

Let U(t)=x-1(t)  on  t0. Applying the generalized Itô formula, we have (3.26)dU(t)=U(t)(-a(r(t))+σ2(r(t))+b(r(t))x(t)-c(r(t))x(t-τ))dt-σU(t)dw(t).

By Lemma 3.7, for the given θ, there is a vector q=(q1,q2,,qn)T0 such that (3.27)λ=(λ1,λ2,,λn)T=A(θ)q0, namely, (3.28)qi(θβ(i)-12θ2σ2(i))-j=1nγijqj>01in.

Define the function V:R+×SR+ by V(U,i)=qi(1+U)θ. Applying the generalized Itô formula again, we have (3.29)EV(U(t),r(t))=V(U(0),r(0))+E0tLV(U(s),x(s-τ),r(s))ds, where LV:R+×R+×SR is defined by (3.30)LV(U,y,i)=(1+U)θ-2{-U2[qi(θβ(i)-12θ2σ2(i))-j=1nγijqj](1+U)θ-2+U[qiθ(b(i)-a(i)+σ2(i))+2j=1nγijqj](1+U)θ-2+[qiθb(i)+j=1nγijqj-qiθc(i)(1+U)Uy]}.

Now, choose a constant κ>0 sufficiently small such that (3.31)λ-κq0, that is, (3.32)qi(θβ(i)-12θ2σ2(i))-j=1nγijqj-κqi>01in. Then, by the generalized Itô formula again, (3.33)E[eκtV(U(t),r(t))]=V(U(0),r(0))+E0t[κeκtV(U(s),r(s))+eκtLV(U(s),x(s-τ),r(s))]ds. It is computed that (3.34)κeκtV(U,i)+eκtLV(U,y,i)eκt(1+U)θ-2{-U2[qi(θβ(i)-12θ2σ2(i))-j=1nγijqj-κqi]eκt(1+U)θ-2+U[qiθ(b(i)-a(i)+σ2(i))+2j=1nγijqj+2κqi]eκt(1+U)θ-2+qiθb(i)+j=1nγijqj+κqi}q^κHeκt, where (3.35)H=1q^κmaxiS{supUR+(1+U)θ-2{-U2[qi(θβ(i)-12θ2σ2(i))-j=1nγijqj-κqi]eκt(1+U)θ-2(1+U)θ-2+U[qiθ(b(i)-a(i)+σ2(i))+2j=1nγijqj+2κqi]eκt(1+U)θ-2(1+U)θ-2+qiθb(i)+j=1nγijqj+κqi},1}, which implies (3.36)q^E[eκt(1+U(t))θ]qˇ(1+x-1(0))θ+q^Heκt. Then (3.37)limsuptE(Uθ(t))limsuptE[(1+U(t))θ]H. Recalling the definition of U(t), 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 a>0,  b>0 and 0c<b, then the solution x(t) of (1.1) is persistent, namely, (3.38)liminftx(t)>0. Furthermore, we consider its associated stochastic delay equation (1.4), that is, (3.39)dx(t)=x(t)[(a(r(t))-b(r(t))x(t)+c(r(t))x(t-τ))dt+σ(r(t))dw(t)], where a(i)>0,b(i)>0,c(i)0,  for  iS,  and  cˇ<b^. 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 (3.40)i=1nπi[a(i)-12σ2(i)]>0ora(i)-12σ2(i)>0,foreachiS.

Corollary 3.13.

Assume that for some iS,b(i)>0,-b(i)+|c(i)|0, and a(i)-(1/2)σ2(i)>0. Then the subsystem (3.41)dx(t)=x(t)[(a(i)-b(i)x(t)+c(i)x(t-τ))dt+σ(i)dw(t)] 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 {x(t):-τt0}C([-τ,0];R+), the solution x(t) of (1.4) has the property that (4.1)limsuptlogx(t)ti=1nπi[a(i)-12σ2(i)]a.s.

Proof.

By Theorem 2.3, the solution x(t) will remain in R+ for all t-τ with probability 1. We have by the generalized Itô formula and (A1) that (4.2)dlogx(t)(a(r(t))-12σ2(r(t))-b^x(t)+b^x(t-τ))dt+σ(r(t))dw(t), where (A1) is used in the last step. Then, (4.3)log(V(x(t)))log(V(x(0)))+b^-τ0x(s)ds+0t(a(r(s))-12σ2(r(s)))ds+M(t), where M(t)=0tσ(r(t))dw(t). The quadratic variation of M(t) is given by (4.4)M,Mt=0tσ2(r(s))dsσ¯2t. Therefore, applying the strong law of large numbers for martingales , we obtain (4.5)limtM(t)t=0a.s.

It finally follows from (4.3) by dividing t on the both sides and then letting t that (4.6)limsuptlogx(t)tlimsupt1t0t[a(r(s))-12σ2(r(s))]ds=i=1nπi[a(i)-12σ2(i)]a.s., 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 {x(t):-τt0}C([-τ,0];R+), the solution x(t) of (1.4) has the property that (4.7)limsuptlogx(t)t<0a.s. 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 {x(t):-τt0}C([-τ,0];R+), the solution x(t) of (1.4) has the property that (4.8)limsuptlogx(t)t-φ2a.s., where  φ=miniS(σ2(i)-2a(i))>0. 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 (4.9)i=1nπi[a(i)-12σ2(i)]<0ora(i)-12σ2(i)<0,foreachiS, 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.

Remark 4.5.

Let A1′′ and A2 hold, i=1nπi[a(i)-(1/2)σ2(i)]0. Then, SDE (1.4) is either stochastically permanent or extinctive. That is, it is stochastically permanent if and only if i=1nπi[a(i)-(1/2)σ2(i)]>0, while it is extinctive if and only if i=1nπi[a(i)-(1/2)σ2(i)]<0.

Corollary 4.6.

Assume that for some iS, (4.10)-b(i)+|c(i)|0,a(i)-12σ2(i)<0. Then for any given initial data {x(t):-τt0}C([-τ,0];R+), the solution x(t) of subsystem (4.11)dx(t)=x(t)[(a(i)-b(i)x(t)+c(i)x(t-τ))dt+σ(i)dw(t)] tend to zero a.s.

5. Asymptotic Properties Lemma 5.1.

Assume that (A1′) holds. Then for any given initial data {x(t):-τt0}C([-τ,0];R+), the solution x(t) of (1.4) has the property (5.1)limsuptlog(x(t))logt1a.s.

Proof.

By Theorem 2.3, the solution x(t) will remain in R+ for all t-τ with probability 1. It is known that (5.2)dx(t)(aˇx(t)+c¯x(t)x(t-τ))dt+σ(r(t))dw(t),E(suptut+1x(u))E(x(t))+aˇtt+1E(x(s))ds+c¯tt+1E(x(s)x(s-τ))ds+E(suptut+1tuσ(r(s))x(s)dw(s)). From (3.3) of Theorem 3.3, it has (5.3)limsuptE(x(t))K1(1),limsuptE(x2(t))K1(2).

By the well-known BDG's inequality  and the Hölder's inequality, we obtain (5.4)E(suptut+1tuσ(r(s))x(s)dB(s))3E[tt+1(σ(r(s))x(s))2ds]1/2E(suptut+1x(u)9σˇtt+1x(s)ds)1/212E(suptut+1x(u))+9σˇ2tt+1E(x(s))ds. Note that (5.5)tt+1E[x(s)x(s-τ)]ds12tt+1E(x2(s))ds+12tt+1E(x2(s-τ))ds. Therefore, (5.6)E(suptut+1x(u))2E(x(t))+2aˇtt+1E(x(s))ds+c¯tt+1E(x2(s))ds+c¯tt+1E(x2(s-τ))ds+18σˇ2tt+1E(x(s))ds. This, together with (5.3), yields (5.7)limsuptE(suptut+1x(u))2(1+aˇ+18σˇ2)K1(1)+2c¯K1(2).

From (5.7), there exists a positive constant M such that (5.8)E(supktk+1x(t))M,k=1,2,. Let ɛ>0 be arbitrary. Then, by Chebyshev's inequality, (5.9)P(supktk+1x(u)>k1+ɛ)Mk1+ɛ,k=1,2,. Applying the well-known Borel-Cantelli lemma , we obtain that for almost all ωΩ(5.10)supktk+1x(u)k1+ɛ, for all but finitely many k. Hence, there exists a k0(ω), for almost all ωΩ, for which (5.10) holds whenever kk0. Consequently, for almost all ωΩ, if kk0 and ktk+1, then (5.11)log(x(t))logt(1+ɛ)logklogk=1+ɛ. Therefore, (5.12)limsuptlog(x(t))logt1+ɛa.s. Letting ɛ0, we obtain the desired assertion (5.1).

Lemma 5.2.

If there exists a constant θ>0 such that A(θ) is a nonsingular M-matrix and c(i)0(i=1,2,,n), then the global positive solution x(t) of SDE (1.4) has the property that (5.13)liminftlog(x(t))logt-1θa.s.

Proof.

Applying the generalized Itô formula, for the fixed constant θ>0, we derive from (3.26) that (5.14)d[(1+U(t))θ]θ(1+U(t))θ-2[-U2(t)(β^-12σˇ2)+U(t)(bˇ+σˇ2)+bˇ]dt-θσ(r(t))(1+U(t))θ-1U(t)dw(t), where U(t)=1/x(t) on t>0. By (3.37), there exists a positive constant M such that (5.15)E[(1+U(t))θ]Mont0.

Let δ>0 be sufficiently small for (5.16)θ{[β^+2bˇ+12(θ+2)σˇ2]δ+3aˇδ1/2}<12. Then (5.14) implies that (5.17)E[sup(k-1)δtkδ(1+U(t))θ]E[(1+U((k-1)δ))θ]+E{sup(k-1)δtkδ|(k-1)δtθ(1+U(s))θ-2[-U2(s)(β^-12θσ˘2)+U(s)(b˘+σ˘2)+b˘]ds|}+E{sup(k-1)δtkδ|(k-1)δtθσ(r(s))(1+U(s))θ-1U(s)dw(s)|}. By directly computing, we have (5.18)E{sup(k-1)δtkδ|(k-1)δtθ(1+U(s))θ-2[-U2(s)(β^-12θσˇ2)+U(s)(bˇ+σˇ2)+bˇ]ds|}θE{(k-1)δt[β^+2bˇ+12(θ+2)σˇ2](1+U(s))θds}θ[β^+2bˇ+12(θ+2)σˇ2]δE[sup(k-1)δtkδ(1+U(t))θ]. By the BDG's inequality, it follows (5.19)E{sup(k-1)δtkδ|(k-1)δkδθσ(r(s))(1+U(s))θ-1U(s)dw(s)|}3θσˇδ1/2E{sup(k-1)δtkδ(1+U(s))θ}. Substituting this and (5.18) into (5.17) gives (5.20)E[sup(k-1)δtkδ(1+U(t))θ]E[(1+U((k-1)δ))θ]+θ{[β^+2bˇ+12(θ+2)σˇ2]δ+3σˇδ1/2}E{sup(k-1)δtkδ(1+U(s))θ}. Making use of (5.15) and (5.16), we obtain (5.21)E[sup(k-1)δtkδ(1+U(t))θ]2Mont0.

Let ɛ>0 be arbitrary. Then, we have by Chebyshev's inequality that (5.22)P{ω:sup(k-1)δtkδ(1+U(t))θ>(kδ)1+ɛ}2M(kδ)1+ɛ,k=1,2,. Applying the Borel-Cantelli lemma, we obtain that for almost all ωΩ, (5.23)sup(k-1)δtkδ(1+U(t))θ(kδ)1+ɛ holds for all but finitely many k. Hence, there exists an integer k0(ω)>1/δ+2, for almost all ωΩ, for which (5.23) holds whenever kk0. Consequently, for almost all ωΩ, if kk0 and (k-1)δtkδ, (5.24)log(1+U(t))θlogt(1+ɛ)log(kδ)log((k-1)δ)1+ɛ. Therefore, (5.25)limsuptlog(1+U(t))θlogt1+ɛa.s. Let ɛ0, we obtain (5.26)limsuptlog(1+U(t))θlogt1a.s. Recalling the definition of U(t), this yields (5.27)limsuptlog(1/xθ(t))logt1a.s., which further implies (5.28)liminftlog(x(t))logt-1θa.s. This is our required assertion (5.13).

Theorem 5.3.

Assume that (A1′′), (A2), and (A3) hold. Then for any given initial data {x(t):-τt0}C([-τ,0];R+), the solution x(t) of (1.4) obeys (5.29)limsupt1t0tx(s)ds1b^-c¯i=1nπi(a(i)-12σ2(i))a.s.,(5.30)liminft1t0tx(s)ds1bˇi=1nπi(a(i)-12σ2(i))a.s.

Proof.

By Theorem 2.3, the solution x(t) will remain in R+ for all t-τ with probability 1. From Lemmas 3.8, 5.1, and 5.2, it follows (5.31)limtlog(x(t))t=0a.s. By generalized Itô formula, one has (5.32)logx(t)=logx0+0t(a(r(s))-12σ2(r(s)))ds-0tb(r(s))x(s)ds+0tc(r(s))x(s-τ)ds+0tσ(r(s))dw(s). Dividing by t on both sides, then we have (5.33)logx(t)tlogx0t+1t0t(a(r(s))-12σ2(r(s)))ds+(-b^+c¯)1t0tx(s)ds+c¯t-τ0x(s)ds+1t0tσ(r(s))dw(s).

Let t, by the strong law of large numbers for martingales and (5.31), we therefore have (5.34)limsupt1t0tx(s)ds1b^-c¯i=1nπi(a(i)-12σ2(i))a.s., which is the required assertions (5.29). And we also have (5.35)logx(t)t1tlogx0+1t0t(a(r(s))-12σ2(r(s)))ds-bˇt0tx(s)ds+1t0tσ(r(s))dw(s). Let t, by the strong law of large numbers for martingales and (5.21), we therefore have (5.36)liminft1t0tx(s)ds1bˇi=1nπi(a(i)-12σ2(i))a.s., 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 {x(t):-τt0}C([-τ,0];R+), the solution x(t) of (1.4) obeys (5.37)limsupt1t0tx(s)ds1b^-c¯i=1nπi(a(i)-12σ2(i))a.s.,liminft1t0tx(s)ds1bˇi=1nπi(a(i)-12σ2(i))a.s.

Corollary 5.5.

If that for some iS, (5.38)b(i)>0,b(i)>|c(i)|,a(i)-12σ2(i)>0, then the solution with positive initial value to subsystem (5.39)dx(t)=x(t)[(a(i)-b(i)x(t)+c(i)x(t-τ))dt+σ(i)dw(t)] has the property that (5.40)a(i)-(1/2)σ2(i)b(i)liminft1t0tx(s)dslimsupt1t0tx(s)dsa(i)-(1/2)σ2(i)b(i)-|c(i)|a.s.

Remark 5.6.

If c¯=0, (1.4) will be written by (5.41)dx(t)=x(t)[(a(r(t))-b(r(t))x(t))dt+σ(r(t))dw(t)], which is investigated in . 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 τ0.

6. Examples Example 6.1.

Consider a 2-dimensional stochastic differential equation with Markovian switching of the form (6.1)dx(t)=x(t)[(a(r(t))-b(r(t))x(t)+c(r(t))x(t-τ))dt+σ(r(t))dw(t)]ont0, where r(t) is a right-continuous Markov chain taking values in S={1,2}, and r(t) and w(t) are independent. Here (6.2)a(1)=2,b(1)=3,c(1)=1,σ(1)=1,a(2)=1,b(2)=2,c(2)=32,σ(2)=2. It can be computed that (6.3)b^=2;cˇ=32;a(1)-12σ2(1)=32;a(2)-12σ2(2)=-1.

By Theorem 2.3, the solution x(t) of (6.1) will remain in R+ for all t-τ with probability 1.

Case 1.

Let the generator of the Markov chain r(t) be (6.4)Γ=(-112-2). By solving the linear equation πΓ=0, we obtain the unique stationary (probability) distribution (6.5)π=(π1,π2)=(23,13). Therefore, (6.6)i=12πi(a(i)-12σ2(i))=23>0. By Theorems 3.10 and 5.3, (6.1) is stochastically permanent and its solution x(t) with any positive initial value has the following properties: (6.7)13liminft1t0tx(s)dslimsupt1t0tx(s)ds43a.s.

Case 2.

Let the generator of the Markov chain r(t) be (6.8)Γ=(-221-1). By solving the linear equation πΓ=0, we obtain the unique stationary (probability) distribution (6.9)π=(π1,π2)=(13,23). So, (6.10)i=12πi(a(i)-12σ2(i))=-16<0. 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).

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