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

Authors: Zheng Wu; Hao Huang; Lianglong Wang
Journal: Abstract and Applied Analysis
 (2013)

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## 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 ofV-function technique, M-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.


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## Body

## 1. Introduction

The delay differential equation(1)dx(t)dt=x(t)(a-bx(t)+cx(t-τ))
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
(2)dx(t)dt=diag(x1(t),…,xn(t))(b+Ax(t)+Bx(t-τ)),
where x=(x1,…,xn)T∈Rn, b=(b1,…,bn)T∈R+n, A=(aij)n×n∈Rn×n, B=(bij)n×n∈Rn×n. 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 statex(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 bi+σiw˙(t), where w˙(t) 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)
(3)dx(t)=diag(x1(t),…,xn(t))=×[(b+Ax(t)+Bx(t-τ))dt+σdw(t)],
where σ=(σ1,…,σn)T. 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:
(4)dx(t)=diag(x1(t),…,xn(t))×[(b(r(t))+A(r(t))x(t)+B(r(t))x(t-τ))dt+σ(r(t))dw(t)].
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
(5)dx(t)=diag(x1(t),…,xn(t))×[(b(ι)+A(ι)x(t)+B(ι)x(t-τ))dt+σ(ι)dw(t)],
until the Markov chain r(t) jumps to another state, say ς. Therefore, the ecosystem (4) satisfies the SDE
(6)dx(t)=diag(x1(t),…,xn(t))×[(b(ς)+A(ς)x(t)+B(ς)x(t-τ))dt+σ(ς)dw(t)],
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(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+n the positive cone in Rn, that is, R+n={x∈Rn:xi>0 for all 1≤i≤n} and denote by R-+n the nonnegative cone in Rn, that is, R-+n={x∈Rn:xi≥0 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 |A|=trace(ATA), 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 formxTAx for the state x∈R+n only. Therefore, for a symmetric n×n matrix A, we naturally introduce the following definition:
(7)λmax+(A)=supx∈R+n,|x|=1xTAx.
For more properties of λmax+(A), please see the appendix in [7].Letr(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
(8)P{r(t+δ)=v∣r(t)=u}={γuvδ+o(δ),ifu≠v,1+γuvδ+o(δ),ifu=v,
where δ>0, γuv is the transition rate from u to v and γuv≥0 if u≠v, while γuu=-∑v≠u‍γuv. 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. This is equivalent to the condition that for, any u,v∈S, one can find finite numbers i1,i2,…,ik∈S such that γui1γi1i2,…,γikv>0. 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:
(9)πΓ=0
subject to
(10)∑i=1N‍πi=1,πi>0,∀i∈S.
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 sequencefi(k), (1≤i≤n,k∈S), let
(11)fˇ=max1≤i≤n,k∈Sfi(k),fˇ(k)=max1≤i≤nfi(k),f^=min1≤i≤n,k∈Sfi(k),f^(k)=min1≤i≤nfi(k).
To proceed, we first state a result, whose proof can be found in [28].Assumption 1.
Assume that there exist positive numbersc1,…,cn such that
(12)maxk∈S{λmax+[12(C-A(k)+A(k)C-)]}+maxk∈S∥C-B(k)∥≤0,
where C-=diag(c1,…,cn).Assumption 2.
Assume that there exist positive numbersc1,…,cn such that
(13)maxk∈S{λmax+[12(C-A(k)+A(k)C-)]}+maxk∈S∥C-B(k)∥<0,
where C-=diag(c1,…,cn).Assumption 3.
Assume that there exist positive numbersc1,…,cn such that
(14)|c|-1maxk∈S{λmax+[12(C-A(k)+A(k)C-)]}+c^-1maxk∈S∥C-B(k)∥≤0,
where C-=diag(c1,…,cn) and c^=min1≤i≤nci.Theorem 1.
(
1
) Under Assumption 1, for any given initial data {x(t):-τ≤t≤0}∈C([-τ,0];R+n), there is a unique solution x(t) to (4) on t≥-τ and the solution will remain in R+n with probability 1, namely, x(t)∈R+n for all t≥-τ almost surely.
(
2
) Under Assumption 2, for any given initial data {x(t):-τ≤t≤0}∈C([-τ,0];R+n) 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
(15)limsupt→∞E|x(t)|p≤K1(p),(16)limsupt→∞1t∫0t‍E|x(s)|p+1ds≤K2(p).
(
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
(17)limsupt→+∞P{|x(t)|>H}<ε.
(
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
(18)limsupt→∞1tlog|x(t)|≤∑i=1n‍πkβ(k)a.s.,
where β(k)=bˇ(k)-(1/2)σ^2(k). Particularly, if  ∑k=1N‍πkβ(k)<0, then
(19)limsupt→∞1tlog|x(t)|<0a.s.
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
(20)liminft→+∞P{|x(t)|≤H}≥1-ε,liminft→+∞P{|x(t)|≥δ}≥1-ε,
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(21)α(k)=b^(k)-12σˇ2(k),β(k)=bˇ(k)-12σ^2(k),
and we impose the following assumptions.Assumption 4.
For someu∈S, γiu>0 (for all i≠u).Assumption 5.
∑
k
=
1
N
‍
π
k
α
(
k
)
>
0.Assumption 6.
For eachk∈S, α(k)>0.LetG 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
(22)ZN×N={A=(aij)N×N:aij≤0,i≠j}.Lemma 3 (see [29]).
IfA=(aij)N×N∈ZN×N has all of its row sums positive, that is,
(23)∑j=1N‍aij>0,∀1≤i≤N,
then A>0.Lemma 4 (see [29]).
IfA∈ZN×N, then the following statements are equivalent:(1)
Ais a nonsingular M-matrix.(2)
All of the principal minors ofA are positive; that is,
(24)|a11⋯a1k⋮⋮⋮a1k⋯akk|>0foreveryk=1,2,…,N.(3)
A is semipositive; that is, there exists x≫0 in RN such that Ax≫0.Lemma 5 (see [23]).
Assumptions4 and 5 imply that there exists a constant θ>0 such that the matrix
(25)A(θ)=diag(ξ1(θ),ξ2(θ),…,ξN(θ))-Γ
is a nonsingular M-matrix, where
(26)ξk(θ)=θα(k)-12θ2σˇ2(k),∀k∈S.Lemma 6 (see [23]).
Assumption6 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 constantθ>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
(27)limsupt→∞E(1|x(t)|θ)≤H,
where H is a fixed positive constant (defined by (42) in the proof).Proof.
DefineV(x)=∑i=1n‍xi on x∈R+n. Then
(28)dV(x(t))=xT(t)[(b(r(t))+A(r(t))x(t)+B(r(t))x(t-τ))dt+σ(r(t))dw(t)].
Define also(29)U(x)=1V(x)onx∈R+n.
Let y(t)=x(t-τ). Applying the generalized Ito^ formula, we derive from (28) that
(30)dU=-U2dV+U3(dV)2dU=-U2xT{[b(r(t))+A(r(t))x+B(r(t))y]dt=-U2xTd+σ(r(t))dw(t)[b(r(t))+A(r(t))x+B(r(t))y]}+U3|xTσ(r(t))|2dtdU={|xTσ(r(t))|2-U2xT[b(r(t))+A(r(t))x+B(r(t))y]dU+U3|xTσ(r(t))|2}dt-U2xTσ(r(t))dw(t)
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 q→=(q1,q2,…,qN)T≫0 such that
(31)λ→=(λ1,λ2,…,λN)T:=A(θ)q→≫0,
namely,
(32)qk(θα(k)-12θ2σˇ2(k))-∑l=1N‍γklql>0∀1≤k≤N.
Define the functionV-:R+×S→R+ by V-(U,k)=qk(1+U)θ. It follows from the generalized Ito^ formula that
(33)EV-(U(t),r(t))=V-(U(0),r(0))=EV-(U(t),r(t))+E∫0t‍LV-(U(s),x(s-τ),r(s))ds,
where
(34)LV-(U,x,y,k)=qkθ(1+U)θ-1{|xTσ(k)|2-U2xT[b(k)+A(k)x+B(k)y]+U3|xTσ(k)|2}+qkθ(θ-1)2×(1+U)θ-2U4|xTσ(k)|2+∑l=1N‍γklql(1+U)θ=qkθ(1+U)θ-2×{12-(1+U)U2xT[b(k)+A(k)x+B(k)y]+(1+U)U3|xTσ(k)|2+12(θ-1)U4|xTσ(k)|2}+∑l=1N‍γklql(1+U)θ.
It is easy to see that, for allx∈R+n,
(35)-xTA(k)xV2≤K,-xTb(k)V+|xTσ(k)|2-xTA(k)xV2≤K,
where K is a positive constant, while
(36)xTb(k)V-12(θ+1)|xTσ(k)|2V2≥b^(k)-12(θ+1)σˇ2(k)=β^(k)-12θσˇ2(k).
Consequently,(37)-(1+U)U2xT[b(k)+A(k)x+B(k)y]+(1+U)U3|xTσ(k)|2+12(θ-1)U4|xTσ(k)|2=-U2xTb(k)-U3xTb(k)-U2xTA(k)x-U3xTA(k)x-U2xTB(k)y-U3xTB(k)y+U3|xTσ(k)|2+12(θ+1)U4|xTσ(k)|2≤-(xTb(k)V-12(θ+1)|xTσ(k)|2V2)U2+(-xTb(k)V+|xTσ(k)|2-xTA(k)xV2)U-xTA(k)xV2≤-(α(k)-12θσˇ2(k))U2+K(1+U).
Substituting (37) into (34) yields
(38)LV-(U,x,y,k)≤qkθ(1+U)θ-2{-(α(k)-12θσˇ2(k))U2+K(1+U)}≤+∑l=1N‍γklql(1+U)θ=(1+U)θ-2×{-[qkθ(α(k)-12θσˇ2(k))-∑l=1N‍γklql]U2ddd+(qkθK+2∑l=1N‍γklql)Uddd+(qkθK+2∑l=1N‍γklql)}.
Now, choose a constantκ>0 sufficiently small such that it satisfies λ→-κq→≫0, that is,
(39)qk(θα(k)-12θ2σˇ2(k))-∑l=1N‍γklql-κqk>0∀1≤k≤N.
Then, by the generalized Itô formula again,(40)E[eκtV-(U(t),r(t))]=V-(U(0),r(0))+E∫0t‍[κeκtV-(U(s),r(s))+eκtLV-(U(s),x(s),x(s-τ),r(s))]ds.
It is computed that(41)κeκtV-(U,i)+eκtLV-(U,x,y,i)≤eκt(1+U)θ-2×{κqk(1+U)2(qkθK+2∑l=1N‍γklql)-[qkθ(α(k)-12θσˇ2(k))-∑l=1N‍γklql]U2+(qkθK+2∑l=1N‍γklql)U+(qkθK+2∑l=1N‍γklql)}≤eκt(1+U)θ-2×{-[qkθ(α(k)-12θσˇ2(k))-∑l=1N‍γklql-κqk]U2+(qkθK+2∑l=1N‍γklql+2κqk)U+(qkθK+2∑l=1N‍γklql+κqk)}≤n-θq^κHeκt,
where
(42)H=1q^κnθmaxi∈S{supU∈R+(1+U)θ-2=H1q^κnθmaxi∈Sd×{∑l=1N‍-[∑l=1N‍γklql-κqkqkθ(α(k)-12θσˇ2(k))=H1q^κnθmaxi∈Sd-∑l=1N‍γklql-κqk]U2=H1q^κnθmaxi∈Sd+(qkθK+2∑l=1N‍γklql+2κqk)U=H1q^κnθmaxi∈Sd+(qkθK+2∑l=1N‍γklql+κqk)}∨1}.
This implies(43)limsupt→∞E[Uθ(x(t))]≤limsupt→∞E[(1+U(x(t)))θ]≤n-θH.
Forx(t)∈R+n, note that (∑i=1n‍xi(t))θ≤(nmax1≤i≤nxi(t))θ≤nθ|x(t)|θ. Consequently,
(44)limsupt→∞E(1|x(t)|θ)≤H.
The required assertion (27) is obtained.Assumption 7.
Assume that there exist positive numbersc1,…,cn such that
(45)maxk∈S{λmax+[12(C-A(k)+A(k)C-)]}+maxk∈S∥C-B(k)∥<0,
where C-=diag(c1,…,cn). Moreover for each k∈S, B(k)≥0.Theorem 8.
Under Assumptions4, 5, and 7, (4) is stochastically permanent.The proof is a simple application of the Chebyshev’s inequality, Lemmas5 and 7, and Theorem 1(3). Similarly, we have the following result.Theorem 9.
Under Assumptions6 and 7, (4) is stochastically permanent.


## 4. Asymptotic Properties

Lemma 10.
Under Assumption2, 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
(46)limsupt→∞log(x(t))logt≤1a.s.Proof.
By Theorem1 (1), the solution x(t) will remain in R+n for all t≥-τ with probability 1. Denote V(x)=∑i=1n‍xi, on x∈R+n. It is known that
(47)dV(x(t))=xT(t)[(b(r(t))+A(r(t))x(t)+B(r(t))x(t-τ))dt+σ(r(t))dw(t)].
We can also derive from this that
(48)E(supt≤u≤t+1V(x(u)))≤EV(x(t))+maxk∈S|b(k)|∫tt+1‍E|x(s)|ds+maxk∈S|A(k)|∫tt+1‍E|x(s)|2ds+maxk∈S|B(k)|×∫tt+1‍E(|x(s)||x(s-τ)|)ds+E(supt≤u≤t+1∫tu‍xT(s)σ(r(s))dw(s)).
From (15), we know that limsupt→∞E|x(t)|≤K1(1) and limsupt→∞E|x(t)|2≤K1(2). By the well-known BDG’s inequality [29] and the Hölder’s inequality, we derive that
(49)E(supt≤u≤t+1∫tu‍xT(s)σ(r(s))dw(s))≤3maxk∈S|b(k)|E(∫tt+1|x(s)|2ds‍)1/2≤E(9σˇ∫tt+1‍x2(s)ds)1/2≤E(supt≤u≤t+1x(u)·9σˇ∫tt+1‍x(s)ds)1/2≤E[(12supt≤u≤t+1x(u))2+9σˇ∫tt+1‍(x(s)ds)2]1/2≤E(12supt≤u≤t+1x(u)+9σˇ2∫tt+1‍x(s)ds)≤12E(supt≤u≤t+1x(u))+9σˇ2∫tt+1‍E(x(s))ds.
Combining the inequality above with(50)∫tt+1‍E(|x(s)||x(s-τ)|)ds≤12∫tt+1‍E|x(s)|2ds+12∫tt+1‍E|x(s-τ)|2ds,
we get that
(51)E(supt≤u≤t+1V(x(u)))≤EVx(t)+maxk∈S|b(k)|∫tt+1‍E|x(s)|ds+(maxk∈S|A(k)|+12maxk∈S|B(k)|)∫tt+1‍E|x(s)|2ds+12maxk∈S|B(k)|∫tt+1‍E|x(s-τ)|2ds+3maxk∈S|b(k)|E(∫tt+1‍|x(s)|2ds)1/2.
Recalling the following inequality|x|≤∑i=1n‍xi≤V(x) for any  x∈R+n, we obtain
(52)limsupt→∞E(supt≤u≤t+1x(u))≤maxk∈S|b(k)|K1(1)+(maxk∈S|A(k)|+maxk∈S|B(k)|)K1(2)+3maxk∈S|b(k)|(K(2))1/2.
It is following from (52) that there is a positive constant M such that
(53)E(supk≤t≤k+1|x(t)|)≤M,k=1,2,….
Letε>0 be arbitrary. Then, by Chebyshev’s inequality, we have
(54)P(supk≤t≤k+1|x(u)|>k1+ε)≤Mk1+ε,k=1,2,….
Applying the well-known Borel-Cantelli lemma [24], we obtain that for almost all ω∈Ω(55)supk≤t≤k+1|x(u)|≤k1+ε
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,
(56)log(|x(t)|)logt≤(1+ε)logklogk=1+ε.
Therefore,limsupt→∞(log(|x(t)|)/logt)≤1+εa.s. Letting ε→0, we obtain the desired assertion (46).Lemma 11.
If there exists a constantθ>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
(57)liminft→∞log(|x(t)|)logt≥-1θa.s.Proof.
LetU:R+n→R+n be the same as defined by (29); for convenience, we write U(x(t))=U(t). Applying the generalized Ito^ formula, for the fixed constant θ>0, we derive from (37) that
(58)d(1+U(t))θ≤θ(1+U(t))θ-2d(1+U(t))θ×[-U2(t)(α(r(t))-12θσˇ2(r(t)))d(1+U(t))θ+K1U(t)+K1+bˇ12]dtd(1+U(t))θ-θ(1+U(t))θ-1U2(t)σ(r(t))dw(t).
By (43), there exists a positive constant M such that
(59)E(1+U(t))θ≤Mont≥0.
Letδ>0 be sufficiently small such that
(60)θ[(|α^|+12θσˇ2+K1)δ+3maxk∈S|σ(k)|δ1/2]<12.
Then (58) implies that
(61)E[sup(k-1)δ≤t≤kδ(1+U(t))θ]≤E[(1+U((k-1)δ))θ]+E{sup(k-1)δ≤t≤kδ|∫(k-1)δt‍θ(1+U(s))θ-2×[-U2(s)(12α^(r(s))-12θσˇ2(r(s)))+K1(U(s)+1)-U2(s)(12α^(r(s))]ds∫(k-1)δt‍θ(1+U(s))θ-2|sup(k-1)δ≤t≤kδ}+E{sup(k-1)δ≤t≤kδ|∫(k-1)δt‍θ(1+U(s))θ-1U2×(s)xT(s)σ(r(s))dw(s)∫(k-1)δt‍|sup(k-1)δ≤t≤kδ}.
It is computed that
(62)E{sup(k-1)δ≤t≤kδ|K1(U(s)+1)12∫(k-1)δt‍θ(1+U(s))θ-2×[-U2(s)(α^(r(s))-12θσˇ2(r(s)))+K1(U(s)+1)12∫(k-1)δt‍θ(1+U(s))θ-2]dssup(k-1)δ≤t≤kδ|}≤E{∫(k-1)δkδ‍|θ(1+U(s))θ-2[-U2(s)(α^-12θσˇ2)+K1(U(s)+1)-U2(s)(α^-12θσˇ2)]θ(1+U(s))θ-2|ds∫(k-1)δkδ‍}≤θE{∫(k-1)δkδ‍(|α^|+12θσˇ2+K1)(1+U(s))θds}≤θ(|α^|+12θσˇ2+K1)E[∫(k-1)δt‍sup(k-1)δ≤s≤kδ(1+U(s))θds]≤θ(|α^|+12θσˇ2+K1)δE[sup(k-1)δ≤t≤kδ(1+U(t))θ].
On the other hand, by the BDG’s inequality, we derive that(63)E{sup(k-1)δ≤t≤kδ|∫(k-1)δkδ‍θ(1+U(s))θ-1U2×(s)xT(s)σ(r(s))dw(s)∫(k-1)δkδ‍|sup(k-1)δ≤t≤kδ}≤3E{∫(k-1)δkδ‍[θ(1+U(s))θ-1U2(s)]2|xTσ(r(s))|2}1/2≤3θE{∫(k-1)δkδ‍(1+U(s))2(θ-1)U2(s)|x(s)|2|σ(r(s))|2|x(s)|2}1/2≤3θmaxk∈S|σ(k)|E{∫(k-1)δkδ‍(1+U(s))2θds}1/2≤3θmaxk∈S|σ(k)|δ1/2E{sup(k-1)δ≤t≤kδ(1+U(s))2θ}1/2≤3θmaxk∈S|σ(k)|δ1/2E{sup(k-1)δ≤t≤kδ(1+U(s))θ}.
Substituting this and (62) into (61) gives
(64)E[sup(k-1)δ≤t≤kδ(1+U(t))θ]≤E[(1+U((k-1)δ))θ]+θ{[α^+12θσˇ2+K1]δ+3maxk∈S|σ(k)|δ1/2}×E{sup(k-1)δ≤t≤kδ(1+U(s))θ}.
Making use of (59) and (60), we obtain that
(65)sup(k-1)δ≤t≤kδE[(1+U(t))θ]≤2Mont≥0.
Letε>0 be arbitrary. Then, by Chebyshev inequality, we have
(66)P{ω:sup(k-1)δ≤t≤kδ(1+U(t))θ>(kδ)1+ε}≤2M(kδ)1+ε,k=1,2,….
Applying the Borel-Cantelli lemma, we obtain that for almost all ω∈Ω(67)sup(k-1)δ≤t≤kδ(1+U(t))θ≤(kδ)1+ε
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,
(68)log(1+U(t))θlogt≤(1+ε)log(kδ)log((k-1)δ)≤1+ε.
Therefore limsupt→∞(log(1+U(t))θ/logt)≤1+εa.s. Let ε→0, we obtain the desired assertion
(69)limsupt→∞log(1+U(t))θlogt≤1a.s.
Recalling the definition of U(t), this yields limsupt→∞(log|x(t)|-θ/logt)≤1a.s., which further implies
(70)liminft→∞log(|x(t)|)logt≥-1θa.s.
This is our required assertion (57).Assumption 8.
Assume that there exist positive numbersc1,…,cn such that
(71)-λ=|c|-1maxk∈S{λmax+[12(C-A(k)+A(k)C-)]}-λ=+c^-1maxk∈S∥C-B(k)∥<0,
where C-=diag(c1,…,cn) and c^=min1≤i≤nci. Moreover for each k∈S, B(k)≥0.Theorem 12.
Under Assumptions4, 5, and 8, for any given initial data {x(t):-τ≤t≤0}∈C([-τ,0];R+), the solution x(t) of (4) obeys
(72)limsupt→∞1t∫0t‍|x(s)|ds≤1λ∑i=kN‍πkβ(k)a.s.,liminft→∞1t∫0t‍|x(s)|ds≥2c^λ^∑k=1n‍πkα(k)a.s.,
where-λ^=mink∈S[λmin(C-A(k)+AT(k)C-)]<0.Proof.
By Theorem1(1), the solution x(t) will remain in R+ for all t≥-τ with probability 1. Define V(x)=cTx=∑i=1n‍cixi, for x∈R+n. By generalized Ito^ formula, one has
(73)dV(x(t))=xT(t)C-[(b(r(t))+A(r(t))x(t)+B(r(t))x(t-τ))dt+σ(r(t))dw(t)].
From Lemmas 5, 10, and 11, it follows that
(74)limt→∞log(V(x(t)))t=0a.s.
By (73), it has
(75)dlogV(x(t))=V-1(x(t))xT(t)C-[(b(r(t))+A(r(t))x(t)+B(r(t))x(t-τ))dt+σ(r(t))dw(t)]-0.5V-2(x(t))|xT(t)C-σ(r(t))|2dt.
Meanwhile,
(76)α(r(t))≤V-1(x(t))C-b(r(t))-0.5V-2(x(t))|xT(t)C-σ(r(t))|2≤β(r(t)),-0.5c^-1λ^|x(t)|≤V-1(x(t))xT(t)C-(A(r(t))x(t)+B(r(t))x(t-τ))≤|c|-1maxk∈S{λmax+[12(C-A(k)+AT(k)C-)]}+c^-1maxk∈S∥C-B(k)∥|x(t)|+0.5c^-1maxk∈S∥C-B(k)∥×(-|x(t)|+|x(t-τ)|).
Substituting (76) into (75) yields
(77)dlogV(x(t))≤β(r(t))dt-λ|x(t)|dtdlogV(x(t))+0.5c^-1maxk∈S∥C-B(k)∥(-|x(t)|+|x(t-τ)|)dlogV(x(t))+V-1(x(t))xT(t)C-σ(r(t))dw(t).
Hence,
(78)logV(x(t))+λ∫0t‍|x(s)|ds≤logV(x(0))+∫0t‍β(r(s))ds+∫-τ0‍|x(s)|ds+∫0t‍V-1(x(s))xT(s)C-σ(r(s))dw(s).
Applying the strong law of large numbers for martingales, we have
(79)limt→∞1t∫0t‍V-1(x(s))xT(s)C-σ(r(s))dw(s)=0a.s.
Dividing both sides of (78) by t and letting t→∞, we obtain that
(80)λlimsupt→+∞1t∫0t‍|x(s)|ds≤∑k=1N‍πkβ(k)a.s.,
which implies the required assertion (72).
On the other hand, it is observed from (75)-(76) that
(81)dlogV(x(t))≥α(r(t))dt-0.5c^-1λ^|x(t)|dt+V-1(x(t))xT(t)C-σ(r(t))dw(t).
Hence,
(82)logV(x(t))+0.5c^-1λ^∫0t‍|x(s)|ds≥logV(x(0))+∫0t‍α(r(s))ds+∫0t‍V-1(x(s))xT(s)C-σ(r(s))dw(s).
Consequently, one gets that
(83)0.5c^-1λ^liminft→+∞1t∫0t‍|x(s)|ds≥∑k=1N‍πkα(k)a.s.,
which implies the other required assertion (4.12).Similarly, using Lemmas6, 10, and 11, we can show the followingTheorem 13.
Under Assumptions5 and 8, for any given initial data {x(t):-τ≤t≤0}∈C([-τ,0];R+), the solution x(t) of (4) obeys
(84)limsupt→∞1t∫0t‍|x(s)|ds≤1λ∑i=kN‍πkβ(k)a.s.,liminft→∞1t∫0t‍|x(s)|ds≥2c^λ^∑k=1n‍πkα(k)a.s.,
where -λ^=mink∈S[λmin(C-A(k)+AT(k)C-)]<0.


## 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(85)dx(t)=diag(x1(t),x2(t))×[(b(r(t))+A(r(t))x(t)+B(r(t))x(t-τ))dt+σ(r(t))dw(t)],
where x(t)=(x1(t),x2(t))T, b(r(t))=(b1r(t)), b2(r(t)))T, σ(r(t))=(σ1(r(t)),σ2(r(t)))T,
(86)A(r(t))=(a11(r(t))a12(r(t))a21(r(t))a22(r(t))),B(r(t))=(b11(r(t))b12(r(t))b21(r(t))b22(r(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
(87)b1(1)=5,a11(1)=-5,a12(1)=10,b2(1)=8,a21(1)=10,a22(1)=-5,b1(2)=4,a11(2)=-3,a12(2)=2,b2(2)=5,a21(2)=2,a22(2)=-3,b11(1)=0,b12(1)=12,σ1(1)=2,b21(1)=1,b22(1)=0,σ2(1)=2,b11(2)=0,b12(2)=22,σ1(2)=14,b21(2)=22,b22(2)=0,σ2(2)=4.
Let C-=I∈R2×2. It is easy to compute that
(88)|c|=2,c^=1,maxk∈S{λmax+[12(C-A(k)+A(k)C-)]}≤-3+2,-λ^=-5-10<0,maxk∈S∥C-B(k)∥≤52.
Then
(89)-λ=|c|-1maxk∈S{λmax+[12(C-A(k)+A(k)C-)]}+c^-1maxk∈S∥C-B(k)∥<0.
Moreover, α(1)=3, α(2)=-4, β(1)=7, and β(2)=-2.
By Theorem1(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
(90)Γ=(-112-2).
By solving the linear equation πΓ=0, we obtain the unique stationary (probability) distribution π=(π1,π2)=(2/3,1/3). Then
(91)∑1=12‍πkα(k)=23>0,∑1=12‍πkβ(k)=4>0.
Therefore, by Theorems 8 and 12, (85) is stochastically permanent and the solutions have the following properties:
(92)2λ^≤liminft→∞1t∫0t‍|x(s)|ds≤limsupt→∞1t∫0t‍|x(s)|ds≤4λa.s.



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*Source: 129072-2013-05-28.xml*