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

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 𝑑𝑥(𝑡)(𝑡(𝑡)(𝑑𝑡=𝑥)(𝑎−𝑏𝑥+𝑐𝑥𝑡−𝜏)) (1) 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 𝑑𝑥(𝑡)𝑥𝑑𝑡=diag₁(𝑡),…,𝑥_𝑛(𝑡)(𝑏+𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏)), (2) where 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛, 𝑏=(𝑏₁,…,𝑏_𝑛)^𝑇∈𝑅^𝑛₊, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×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 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 𝑏_𝑖 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 𝑏_𝑖 by 𝑏_𝑖+𝜎_𝑖𝑤̇(𝑡), where 𝑤̇(𝑡) is a white noise (i.e., 𝑤(𝑡) is a Brownian motion) and 𝜎_𝑖≥0 represents the intensity of noise. As a result, (2) becomes a stochastic differential equation (SDE, in short) 𝑑𝑥(𝑡)=diag𝑥₁(𝑡),…,𝑥_𝑛(𝑡)×[(𝑏+𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏))𝑑𝑡+𝜎𝑑𝑤(𝑡)], (3) where 𝜎=(𝜎₁,…,𝜎_𝑛)^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: 𝑑𝑥(𝑡)=diag𝑥₁(𝑡),…,𝑥_𝑛(𝑡)×𝑏(𝑟(𝑡(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡.[())+𝐴))+𝐵))𝑡−𝜏))𝑑𝑡+𝜎))𝑑𝑤)] (4) 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 (𝑡)𝑥𝑑𝑥=diag₁(𝑡),…,𝑥_𝑛(𝑡)×[(𝑏(𝜄)+𝐴(𝜄)𝑥(𝑡)+𝐵(𝜄)𝑥(𝑡−𝜏))𝑑𝑡+𝜎(𝜄)𝑑𝑤(𝑡)], (5) until the Markov chain 𝑟(𝑡) jumps to another state, say 𝜍. Therefore, the ecosystem (4) satisfies the SDE (𝑡)𝑥𝑑𝑥=diag₁(𝑡),…,𝑥_𝑛(𝑡)×[(𝑏(𝜍)+𝐴(𝜍)𝑥(𝑡)+𝐵(𝜍)𝑥(𝑡−𝜏))𝑑𝑡+𝜎(𝜍)𝑑𝑤(𝑡)], (6) 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,{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 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 𝑅^𝑛, that is, 𝑅^𝑛₊={𝑥∈𝑅^𝑛∶𝑥_𝑖>0 for all 1≤𝑖≤𝑛} and denote by 𝑅^𝑛₊ the nonnegative cone in 𝑅^𝑛, that is, 𝑅^𝑛₊={𝑥∈𝑅^𝑛∶𝑥_𝑖≥0 for all 1≤𝑖≤𝑛}. If 𝐴 is a vector or matrix, its transpose is denoted be 𝐴^𝑇. If 𝐴 is a matrix, its trace norm is denoted by |𝐴|=trace(𝐴^𝑇𝐴), 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 𝑥^𝑇𝐴𝑥 for the state 𝑥∈𝑅^𝑛₊ only. Therefore, for a symmetric 𝑛×𝑛 matrix 𝐴, we naturally introduce the following definition: 𝜆⁺_max(𝐴)=sup_{𝑥∈𝑅𝑛+,|𝑥|=1}𝑥^𝑇𝐴𝑥. (7) For more properties of 𝜆⁺_max(𝐴), 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=(𝛾_𝑢𝑣) given by 𝑃{𝑟()(𝑡)}=𝛾𝑡+𝛿=𝑣∣𝑟=𝑢^𝑢𝑣𝛿+𝑜(𝛿),if𝑢≠𝑣,1+𝛾_𝑢𝑣𝛿+𝑜(𝛿),if𝑢=𝑣, (8) where 𝛿>0, 𝛾_𝑢𝑣 is the transition rate from 𝑢 to 𝑣 and 𝛾_𝑢𝑣≥0 if 𝑢≠𝑣, while 𝛾_𝑢𝑢=−∑_𝑣≠𝑢𝛾_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. This is equivalent to the condition that for, any 𝑢,𝑣∈𝑆, one can find finite numbers 𝑖₁,𝑖₂,…,𝑖_𝑘∈𝑆 such that 𝛾_𝑢𝑖1𝛾_𝑖1𝑖2,…,𝛾_𝑖𝑘𝑣>0. Under this condition, the Markov chain has a unique stationary (probability) distribution 𝜋=(𝜋₁,𝜋₂,…,𝜋_𝑁)∈𝑅^1×𝑁 which can be determined by solving the following linear equation: 𝜋Γ=0 (9) subject to ^𝑁_𝑖=1𝜋_𝑖=1,𝜋_𝑖>0,∀𝑖∈𝑆. (10) 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 𝑓_𝑖(𝑘), (1≤𝑖≤𝑛,𝑘∈𝑆), let 𝑓̌=max_{1≤𝑖≤𝑛,𝑘∈𝑆}𝑓_𝑖(𝑘),𝑓(̌𝑘)=max_{1≤𝑖≤𝑛}𝑓_𝑖(𝑘),𝑓=min_{1≤𝑖≤𝑛,𝑘∈𝑆}𝑓_𝑖(𝑘),𝑓(𝑘)=min_{1≤𝑖≤𝑛}𝑓_𝑖(𝑘). (11) To proceed, we first state a result, whose proof can be found in [28].

Assumption 1. Assume that there exist positive numbers 𝑐₁,…,𝑐_𝑛 such that max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+max_𝑘∈𝑆𝐶𝐵(𝑘)≤0, (12) where 𝐶=diag(𝑐₁,…,𝑐_𝑛).

Assumption 2. Assume that there exist positive numbers 𝑐₁,…,𝑐_𝑛 such that max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+max_𝑘∈𝑆𝐶𝐵(𝑘)<0, (13) where 𝐶=diag(𝑐₁,…,𝑐_𝑛).

Assumption 3. Assume that there exist positive numbers 𝑐₁,…,𝑐_𝑛 such that |𝑐|⁻¹max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+𝑐̂⁻¹max_𝑘∈𝑆𝐶𝐵(𝑘)≤0, (14) where 𝐶=diag(𝑐₁,…,𝑐_𝑛) and 𝑐̂=min_{1≤𝑖≤𝑛}𝑐_𝑖.

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 limsup_𝑡→∞𝐸|𝑥(𝑡)|^𝑝≤𝐾₁𝑝,limsup_𝑡→∞1𝑡^𝑡₀𝐸|𝑥(𝑠)|^𝑝+1𝑑𝑠≤𝐾₂𝑝. (15)(16)
(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 limsup_𝑡→+∞𝑃{|𝑥(𝑡)|>𝐻}<𝜀. (17)
(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 limsup_𝑡→∞1𝑡log|𝑥(𝑡)|≤^𝑛_𝑖=1𝜋_𝑘𝛽(𝑘)𝑎.𝑠., (18) where 𝛽(𝑘)=𝑏̌(𝑘)−(1/2)𝜎²(𝑘). Particularly, if  ∑^𝑁_𝑘=1𝜋_𝑘𝛽(𝑘)<0, then limsup_𝑡→∞1𝑡log|𝑥(𝑡)|<0𝑎.𝑠. (19)
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 liminf_𝑡→+∞𝑃{|𝑥(𝑡)|≤𝐻}≥1−𝜀,liminf_𝑡→+∞𝑃{|𝑥(𝑡)|≥𝛿}≥1−𝜀, (20) 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 𝛼(𝑘)=𝑏̂(𝑘)−12𝜎̌²(𝑘)(𝑘)=𝑏̌(𝑘)−1,𝛽2𝜎²(𝑘), (21) and we impose the following assumptions.

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

Assumption 5. ∑^𝑁_𝑘=1𝜋_𝑘𝛼(𝑘)>0.

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 𝑍^𝑁×𝑁=𝐴=𝑎_{𝑖𝑗𝑁×𝑁}∶𝑎_𝑖𝑗≤0,𝑖≠𝑗. (22)

Lemma 3 (see [29]). If 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁∈𝑍^𝑁×𝑁 has all of its row sums positive, that is, ^𝑁_𝑗=1𝑎_𝑖𝑗>0,∀1≤𝑖≤𝑁, (23) then 𝐴>0.

Lemma 4 (see [29]). If 𝐴∈𝑍^𝑁×𝑁, then the following statements are equivalent:(1)𝐴is a nonsingular 𝑀-matrix.(2)All of the principal minors of 𝐴 are positive; that is, 𝑎₁₁⋯𝑎_1𝑘𝑎⋮⋮⋮_1𝑘⋯𝑎_𝑘𝑘>0𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑘=1,2,…,𝑁. (24)(3)𝐴 is semipositive; that is, there exists 𝑥≫0 in 𝑅^𝑁 such that 𝐴𝑥≫0.

Lemma 5 (see [23]). Assumptions 4 and 5 imply that there exists a constant 𝜃>0 such that the matrix 𝐴(𝜃)=diag𝜉₁(𝜃),𝜉₂(𝜃),…,𝜉_𝑁(𝜃)−Γ (25) is a nonsingular 𝑀-matrix, where 𝜉_𝑘(𝜃)(𝑘)−1=𝜃𝛼2𝜃²𝜎̌²(𝑘),∀𝑘∈𝑆. (26)

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 limsup_𝑡→∞𝐸1|𝑥(𝑡)|^𝜃≤𝐻, (27) where 𝐻 is a fixed positive constant (defined by (42) in the proof).

Proof. Define 𝑉(𝑥)=∑^𝑛_𝑖=1𝑥_𝑖 on 𝑥∈𝑅^𝑛₊. Then 𝑑𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡𝑏(𝑟(𝑡(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡.)[())+𝐴))+𝐵))𝑡−𝜏))𝑑𝑡+𝜎))𝑑𝑤)] (28)
Define also 𝑈(𝑥)=1𝑉(𝑥)on𝑥∈𝑅^𝑛₊. (29) Let 𝑦(𝑡)=𝑥(𝑡−𝜏). Applying the generalized Ito formula, we derive from (28) that 𝑑𝑈=−𝑈²𝑑𝑉+𝑈³(𝑑𝑉)²=−𝑈²𝑥^𝑇𝑏(𝑟(𝑡(𝑟(𝑡(𝑟(𝑡𝑦(𝑟(𝑡(𝑡))+𝐴))𝑥+𝐵))𝑑𝑡+𝜎))𝑑𝑤)+𝑈³𝑥^𝑇𝜎(𝑟(𝑡))²=𝑑𝑡−𝑈²𝑥^𝑇𝑏(𝑟(𝑡))+𝐴(𝑟(𝑡))𝑥+𝐵(𝑟(𝑡))𝑦+𝑈³𝑥^𝑇𝜎(𝑟(𝑡))²𝑑𝑡−𝑈²𝑥^𝑇𝜎(𝑟(𝑡))𝑑𝑤(𝑡) (30) 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,…,𝑞_𝑁)^𝑇≫0 such that 𝜆=𝜆₁,𝜆₂,…,𝜆_𝑁^𝑇∶=𝐴(𝜃)𝑞≫0, (31) namely, 𝑞_𝑘(𝑘)−1𝜃𝛼2𝜃²𝜎̌²(𝑘)−^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙>0∀1≤𝑘≤𝑁. (32)
Define the function 𝑉∶𝑅₊×𝑆→𝑅₊ by 𝑉(𝑈,𝑘)=𝑞_𝑘(1+𝑈)^𝜃. It follows from the generalized Ito formula that 𝐸𝑉(𝑈(𝑡),𝑟(𝑡))=𝑉(𝑈(0),𝑟(0))+𝐸^𝑡₀𝐿𝑉(𝑈(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))𝑑𝑠, (33) where 𝐿𝑉𝑈,𝑥,𝑦,𝑘=𝑞_𝑘𝜃(1+𝑈)^𝜃−1−𝑈²𝑥^𝑇𝑏(𝑘)+𝐴(𝑘)𝑥+𝐵(𝑘)𝑦+𝑈³𝑥^𝑇𝜎(𝑘)²+𝑞_𝑘𝜃(𝜃−1)2×(1+𝑈)^𝜃−2𝑈⁴𝑥^𝑇𝜎(𝑘)²+^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙(1+𝑈)^𝜃=𝑞_𝑘𝜃(1+𝑈)^𝜃−2×−(1+𝑈)𝑈²𝑥^𝑇𝑏(𝑘)(𝑘)(𝑘)𝑦+()𝑈+𝐴𝑥+𝐵1+𝑈³𝑥^𝑇𝜎(𝑘)²+12(𝜃−1)𝑈⁴𝑥^𝑇𝜎(𝑘)²+^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙(1+𝑈)^𝜃. (34)
It is easy to see that, for all 𝑥∈𝑅^𝑛₊, −𝑥^𝑇𝐴(𝑘)𝑥𝑉²−𝑥≤𝐾,_𝑇𝑏(𝑘)𝑉+𝑥^𝑇𝜎(𝑘)²−𝑥^𝑇𝐴(𝑘)𝑥𝑉²≤𝐾, (35) where 𝐾 is a positive constant, while 𝑥^𝑇𝑏(𝑘)𝑉−12()𝑥𝜃+1^𝑇𝜎(𝑘)²𝑉²≥𝑏̂(𝑘)−12(𝜃+1)𝜎̌²(𝑘)=𝛽(𝑘)−12𝜃𝜎̌²(𝑘). (36)
Consequently, −(1+𝑈)𝑈²𝑥^𝑇𝑏(𝑘)(𝑘)(𝑘)𝑦+()𝑈+𝐴𝑥+𝐵1+𝑈³𝑥^𝑇𝜎(𝑘)²+12(𝜃−1)𝑈⁴𝑥^𝑇𝜎(𝑘)²=−𝑈²𝑥^𝑇𝑏(𝑘)−𝑈³𝑥^𝑇𝑏(𝑘)−𝑈²𝑥^𝑇𝐴(𝑘)𝑥−𝑈³𝑥^𝑇𝐴(𝑘)𝑥−𝑈²𝑥^𝑇𝐵(𝑘)𝑦−𝑈³𝑥^𝑇𝐵(𝑘)𝑦+𝑈³𝑥^𝑇𝜎(𝑘)²+12(𝜃+1)𝑈⁴𝑥^𝑇𝜎(𝑘)²𝑥≤−^𝑇𝑏(𝑘)𝑉−12()𝑥𝜃+1^𝑇𝜎(𝑘)²𝑉²𝑈²+−𝑥^𝑇𝑏(𝑘)𝑉+𝑥^𝑇𝜎(𝑘)²−𝑥^𝑇𝐴(𝑘)𝑥𝑉²𝑈−𝑥^𝑇𝐴(𝑘)𝑥𝑉²𝛼(𝑘)−1≤−2𝜃𝜎̌²(𝑘)𝑈²+𝐾(1+𝑈). (37)
Substituting (37) into (34) yields 𝐿𝑉𝑈,𝑥,𝑦,𝑘≤𝑞_𝑘𝜃(1+𝑈)^𝜃−2−𝛼(𝑘)−12𝜃𝜎̌²(𝑘)𝑈²(++𝐾1+𝑈)_𝑁𝑙=1𝛾_𝑘𝑙𝑞_𝑙(1+𝑈)^𝜃=(1+𝑈)^𝜃−2×−𝑞_𝑘𝜃𝛼(𝑘)−12𝜃𝜎̌²(𝑘)−^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙𝑈²+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙𝑈+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙. (38)
Now, choose a constant 𝜅>0 sufficiently small such that it satisfies 𝜆−𝜅𝑞≫0, that is, 𝑞_𝑘(𝑘)−1𝜃𝛼2𝜃²𝜎̌²(𝑘)−^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙−𝜅𝑞_𝑘>0∀1≤𝑘≤𝑁. (39)
Then, by the generalized Itô formula again, 𝐸𝑒^𝜅𝑡𝑉(𝑈(𝑡)(𝑡=,𝑟))𝑉(𝑈(0),𝑟(0))+𝐸^𝑡₀𝜅𝑒^𝜅𝑡𝑉(𝑈(𝑠),𝑟(𝑠))+𝑒^𝜅𝑡𝐿𝑉(𝑈(𝑠),𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))𝑑𝑠. (40)
It is computed that 𝜅𝑒^𝜅𝑡𝑉(𝑈,𝑖)+𝑒^𝜅𝑡𝐿𝑉𝑈,𝑥,𝑦,𝑖≤𝑒^𝜅𝑡(1+𝑈)^𝜃−2×𝜅𝑞_𝑘(1+𝑈)²−𝑞_𝑘𝜃𝛼(𝑘)−12𝜃𝜎̌²(𝑘)−^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙𝑈²+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙𝑈+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙≤𝑒^𝜅𝑡(1+𝑈)^𝜃−2×−𝑞_𝑘𝜃𝛼(𝑘)−12𝜃𝜎̌²(𝑘)−^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙−𝜅𝑞_𝑘𝑈²+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙+2𝜅𝑞_𝑘𝑈+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙+𝜅𝑞_𝑘≤𝑛^−𝜃𝑞𝜅𝐻𝑒^𝜅𝑡, (41) where 1𝐻=𝑛𝑞𝜅^𝜃max_𝑖∈𝑆sup_𝑈∈𝑅+(1+𝑈)^𝜃−2×−𝑞_𝑘𝜃𝛼(𝑘)−12𝜃𝜎̌²(𝑘)−^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙−𝜅𝑞_𝑘𝑈²+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙+2𝜅𝑞_𝑘𝑈+𝑞_𝑘𝜃𝐾+2^𝑁_𝑙=1𝛾_𝑘𝑙𝑞_𝑙+𝜅𝑞_𝑘∨1. (42)
This implies limsup_𝑡→∞𝐸𝑈^𝜃(𝑥(𝑡))≤limsup_𝑡→∞𝐸(1+𝑈(𝑥(𝑡)))^𝜃≤𝑛^−𝜃𝐻. (43)
For 𝑥(𝑡)∈𝑅^𝑛₊, note that (∑^𝑛_𝑖=1𝑥_𝑖(𝑡))^𝜃≤(𝑛max_{1≤𝑖≤𝑛}𝑥_𝑖(𝑡))^𝜃≤𝑛^𝜃|𝑥(𝑡)|^𝜃. Consequently, limsup_𝑡→∞𝐸1|𝑥(𝑡)|^𝜃≤𝐻. (44)
The required assertion (27) is obtained.

Assumption 7. Assume that there exist positive numbers 𝑐₁,…,𝑐_𝑛 such that max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+max_𝑘∈𝑆𝐶𝐵(𝑘)<0, (45) where 𝐶=diag(𝑐₁,…,𝑐_𝑛). 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 limsup_𝑡→∞log(𝑥(𝑡))log𝑡≤1𝑎.𝑠. (46)

Proof. By Theorem 1 (1), the solution 𝑥(𝑡) will remain in 𝑅^𝑛₊ for all 𝑡≥−𝜏 with probability 1. Denote 𝑉(𝑥)=∑^𝑛_𝑖=1𝑥_𝑖, on 𝑥∈𝑅^𝑛₊. It is known that 𝑑𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡𝑏(𝑟(𝑡(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡.)[())+𝐴))+𝐵))𝑡−𝜏))𝑑𝑡+𝜎))𝑑𝑤)] (47) We can also derive from this that 𝐸sup_{𝑡≤𝑢≤𝑡+1}𝑉(𝑥(𝑢(𝑥(𝑡))≤𝐸𝑉))+max_𝑘∈𝑆|𝑏(𝑘)|_𝑡^𝑡+1𝐸|𝑥(𝑠)|𝑑𝑠+max_𝑘∈𝑆|𝐴(𝑘)|_𝑡^𝑡+1𝐸|𝑥(𝑠)|²𝑑𝑠+max_𝑘∈𝑆|𝐵(𝑘×)|_𝑡𝑡+1𝐸(|𝑥(𝑠)||𝑥(𝑠−𝜏)|)𝑑𝑠+𝐸sup_{𝑡≤𝑢≤𝑡+1}^𝑢_𝑡𝑥^𝑇(𝑠)𝜎(𝑟(𝑠))𝑑𝑤(𝑠). (48)
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 𝐸sup_{𝑡≤𝑢≤𝑡+1}^𝑢_𝑡𝑥^𝑇(𝑠)𝜎(𝑟(𝑠))𝑑𝑤(𝑠)≤3max_𝑘∈𝑆|𝑏(𝑘)|𝐸_𝑡^𝑡+1|𝑥(𝑠)|²𝑑𝑠^1/2≤𝐸9𝜎̌_𝑡^𝑡+1𝑥²(𝑠)𝑑𝑠^1/2≤𝐸sup_{𝑡≤𝑢≤𝑡+1}𝑥(𝑢)⋅9𝜎̌_𝑡^𝑡+1𝑥(𝑠)𝑑𝑠^1/21≤𝐸2sup_{𝑡≤𝑢≤𝑡+1}𝑥(𝑢)²+9𝜎̌_𝑡^𝑡+1(𝑥(𝑠)𝑑𝑠)^21/21≤𝐸2sup_{𝑡≤𝑢≤𝑡+1}𝑥(𝑢)+9𝜎̌²_𝑡^𝑡+1𝑥(𝑠)≤1𝑑𝑠2𝐸sup_{𝑡≤𝑢≤𝑡+1}𝑥(𝑢)+9𝜎̌²_𝑡^𝑡+1𝐸(𝑥(𝑠))𝑑𝑠. (49)
Combining the inequality above with _𝑡^𝑡+1𝐸𝑥(𝑠𝑥(≤1(|)||𝑠−𝜏)|)𝑑𝑠2_𝑡^𝑡+1𝐸|𝑥(𝑠)|²1𝑑𝑠+2_𝑡^𝑡+1𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠, (50) we get that 𝐸sup_{𝑡≤𝑢≤𝑡+1}𝑉(𝑥(𝑢(𝑡)))≤𝐸𝑉𝑥+max_𝑘∈𝑆|𝑏(𝑘)|_𝑡^𝑡+1𝐸|𝑥(𝑠+)|𝑑𝑠max_𝑘∈𝑆|𝐴(𝑘+1)|2max_𝑘∈𝑆|𝐵(𝑘)|_𝑡^𝑡+1𝐸|𝑥(𝑠)|²+1𝑑𝑠2max_𝑘∈𝑆|𝐵(𝑘)|_𝑡^𝑡+1𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠+3max_𝑘∈𝑆|𝑏(𝑘)|𝐸_𝑡^𝑡+1|𝑥(𝑠)|²𝑑𝑠^1/2. (51)
Recalling the following inequality |𝑥|≤∑^𝑛_𝑖=1𝑥_𝑖≤𝑉(𝑥) for any  𝑥∈𝑅^𝑛₊, we obtain limsup_𝑡→∞𝐸sup_{𝑡≤𝑢≤𝑡+1}𝑥(𝑢)≤max_𝑘∈𝑆|𝑏(𝑘)|𝐾₁(1)+max_𝑘∈𝑆|𝐴(𝑘)|+max_𝑘∈𝑆|𝐵(𝑘)|𝐾₁(2)+3max_𝑘∈𝑆|𝑏(𝑘)|(𝐾(2))^1/2. (52)
It is following from (52) that there is a positive constant 𝑀 such that 𝐸sup_{𝑘≤𝑡≤𝑘+1}|𝑥(𝑡)|≤𝑀,𝑘=1,2,…. (53)
Let 𝜀>0 be arbitrary. Then, by Chebyshev’s inequality, we have 𝑃sup_{𝑘≤𝑡≤𝑘+1}|𝑥(𝑢)|>𝑘^1+𝜀≤𝑀𝑘^1+𝜀,𝑘=1,2,…. (54)
Applying the well-known Borel-Cantelli lemma [24], we obtain that for almost all 𝜔∈Ωsup_{𝑘≤𝑡≤𝑘+1}|𝑥(𝑢)|≤𝑘^1+𝜀 (55) 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, log(|𝑥(𝑡)|)≤()log𝑡1+𝜀log𝑘log𝑘=1+𝜀. (56)
Therefore, limsup_𝑡→∞(log(|𝑥(𝑡)|)/log𝑡)≤1+𝜀a.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 liminf_𝑡→∞log(|𝑥(𝑡)|)1log𝑡≥−𝜃𝑎.𝑠. (57)

Proof. Let 𝑈∶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 𝑑(1+𝑈(𝑡))^𝜃≤𝜃(1+𝑈(𝑡))^𝜃−2×−𝑈²(𝑡)𝛼(𝑟(𝑡−1))2𝜃𝜎̌²(𝑟(𝑡))+𝐾₁𝑈(𝑡)+𝐾₁+𝑏̌((𝑡𝑑𝑡−𝜃1+𝑈))^𝜃−1𝑈²(𝑡)𝜎(𝑟(𝑡))𝑑𝑤(𝑡). (58)
By (43), there exists a positive constant 𝑀 such that 𝐸(1+𝑈(𝑡))^𝜃≤𝑀on𝑡≥0. (59)
Let 𝛿>0 be sufficiently small such that 𝜃|𝛼|+12𝜃𝜎̌²+𝐾₁𝛿+3max_𝑘∈𝑆|𝜎(𝑘)|𝛿^1/2<12. (60)
Then (58) implies that 𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑡))^𝜃≤𝐸(1+𝑈((𝑘−1)𝛿))^𝜃+𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}^𝑡_{()𝛿𝑘−1}𝜃(1+𝑈(𝑠))^𝜃−2×−𝑈²(𝑠)𝛼(𝑟(𝑠−1))2𝜃𝜎̌²(𝑟(𝑠))+𝐾₁(𝑈(𝑠)+1)𝑑𝑠+𝐸sup_{(𝑘−1)𝛿≤𝑡≤𝑘𝛿}^𝑡_{(𝑘−1)𝛿}𝜃(1+𝑈(𝑠))^𝜃−1𝑈²×(𝑠)𝑥^𝑇(𝑠)𝜎(𝑟(𝑠))𝑑𝑤(𝑠). (61) It is computed that 𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}^𝑡_{()𝛿𝑘−1}𝜃(1+𝑈(𝑠))^𝜃−2×−𝑈²(𝑠𝛼(𝑟(𝑠−1)))2𝜃𝜎̌²(𝑟(𝑠))+𝐾₁(𝑈(𝑠)+1)𝑑𝑠≤𝐸^𝑘𝛿_{(𝑘−1)𝛿}𝜃(1+𝑈(𝑠))^𝜃−2−𝑈²(𝑠)1𝛼−2𝜃𝜎̌²+𝐾₁(𝑈(𝑠)+1)𝑑𝑠≤𝜃𝐸_()𝛿^𝑘𝛿_𝑘−1|𝛼|+12𝜃𝜎̌²+𝐾₁(1+𝑈(𝑠))^𝜃|𝛼|+1𝑑𝑠≤𝜃2𝜃𝜎̌²+𝐾₁𝐸^𝑡_{()𝛿𝑘−1}sup_{()𝑘−1𝛿≤𝑠≤𝑘𝛿}(1+𝑈(𝑠))^𝜃|𝛼|+1𝑑𝑠≤𝜃2𝜃𝜎̌²+𝐾₁𝛿𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑡))^𝜃. (62)
On the other hand, by the BDG’s inequality, we derive that 𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿()𝛿}^𝑘𝛿_𝑘−1𝜃(1+𝑈(𝑠))^𝜃−1𝑈²×(𝑠)𝑥^𝑇(𝑠)𝜎(𝑟(𝑠))𝑑𝑤(𝑠)≤3𝐸_()𝛿^𝑘𝛿_𝑘−1𝜃(1+𝑈(𝑠))^𝜃−1𝑈²(𝑠)²𝑥^𝑇𝜎(𝑟(𝑠))^21/2≤3𝜃𝐸_()𝛿^𝑘𝛿_𝑘−1(1+𝑈(𝑠))^2(𝜃−1)𝑈²(𝑠)|𝑥(𝑠)|²|𝜎(𝑟(𝑠))|²|𝑥(𝑠)|^21/2≤3𝜃max_𝑘∈𝑆|𝜎(𝑘)|𝐸_()𝛿^𝑘𝛿_𝑘−1(1+𝑈(𝑠))^2𝜃𝑑𝑠^1/2≤3𝜃max_𝑘∈𝑆|𝜎(𝑘)|𝛿^1/2𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑠))^2𝜃1/2≤3𝜃max_𝑘∈𝑆|𝜎(𝑘)|𝛿^1/2𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑠))^𝜃. (63)
Substituting this and (62) into (61) gives 𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑡))^𝜃≤𝐸(1+𝑈((𝑘−1)𝛿))^𝜃1+𝜃𝛼+2𝜃𝜎̌²+𝐾₁𝛿+3max_𝑘∈𝑆|𝜎(𝑘)|𝛿^1/2×𝐸sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑠))^𝜃. (64) Making use of (59) and (60), we obtain that sup_{(𝑘−1)𝛿≤𝑡≤𝑘𝛿}𝐸(1+𝑈(𝑡))^𝜃≤2𝑀on𝑡≥0. (65)
Let 𝜀>0 be arbitrary. Then, by Chebyshev inequality, we have 𝑃𝜔∶sup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑡))^𝜃>(𝑘𝛿)^1+𝜀≤2𝑀(𝑘𝛿)^1+𝜀,𝑘=1,2,…. (66) Applying the Borel-Cantelli lemma, we obtain that for almost all 𝜔∈Ωsup_{()𝑘−1𝛿≤𝑡≤𝑘𝛿}(1+𝑈(𝑡))^𝜃≤(𝑘𝛿)^1+𝜀 (67) 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, log(1+𝑈(𝑡))^𝜃≤()()log𝑡1+𝜀log𝑘𝛿log((𝑘−1)𝛿)≤1+𝜀. (68) Therefore limsup_𝑡→∞(log(1+𝑈(𝑡))^𝜃/log𝑡)≤1+𝜀a.s. Let 𝜀→0, we obtain the desired assertion limsup_𝑡→∞log(1+𝑈(𝑡))^𝜃log𝑡≤1a.s. (69) Recalling the definition of 𝑈(𝑡), this yields limsup_𝑡→∞(log|𝑥(𝑡)|^−𝜃/log𝑡)≤1a.s., which further implies liminf_𝑡→∞log(|𝑥(𝑡)|)1log𝑡≥−𝜃a.s. (70) This is our required assertion (57).

Assumption 8. Assume that there exist positive numbers 𝑐₁,…,𝑐_𝑛 such that −𝜆=|𝑐|⁻¹max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+𝑐̂⁻¹max_𝑘∈𝑆𝐶𝐵(𝑘)<0, (71) where 𝐶=diag(𝑐₁,…,𝑐_𝑛) and 𝑐̂=min_{1≤𝑖≤𝑛}𝑐_𝑖. 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 limsup_𝑡→∞1𝑡^𝑡₀|𝑥(𝑠1)|𝑑𝑠≤𝜆^𝑁_𝑖=𝑘𝜋_𝑘𝛽(𝑘)𝑎.𝑠.,liminf_𝑡→∞1𝑡^𝑡₀|𝑥(𝑠2𝑐̂)|𝑑𝑠≥𝜆^𝑛_𝑘=1𝜋_𝑘𝛼(𝑘)𝑎.𝑠., (72)
where −𝜆=min_𝑘∈𝑆[𝜆_min(𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶)]<0.

Proof. By Theorem 1(1), the solution 𝑥(𝑡) will remain in 𝑅₊ for all 𝑡≥−𝜏 with probability 1. Define 𝑉(𝑥)=𝑐^𝑇𝑥=∑^𝑛_𝑖=1𝑐_𝑖𝑥_𝑖, for 𝑥∈𝑅^𝑛₊. By generalized Ito formula, one has 𝑑𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝐶𝑏(𝑟(𝑡(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡.[())+𝐴))+𝐵))𝑡−𝜏))𝑑𝑡+𝜎))𝑑𝑤)] (73) From Lemmas 5, 10, and 11, it follows that lim_𝑡→∞log(𝑉(𝑥(𝑡)))𝑡=0a.s. (74) By (73), it has 𝑑log𝑉(𝑥(𝑡))=𝑉⁻¹(𝑥(𝑡))𝑥^𝑇(𝑡)𝐶𝑏(𝑟(𝑡(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡[())+𝐴))+𝐵))𝑡−𝜏))𝑑𝑡+𝜎))𝑑𝑤)]−0.5𝑉⁻²(𝑥(𝑡))𝑥^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))²𝑑𝑡. (75) Meanwhile, 𝛼(𝑟(𝑡))≤𝑉⁻¹(𝑥(𝑡))𝐶𝑏(𝑟(𝑡))−0.5𝑉⁻²(𝑥(𝑡))𝑥^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))²(𝑟(𝑡,𝑐̂≤𝛽))−0.5⁻¹𝜆|𝑥(𝑡)|≤𝑉⁻¹(𝑥(𝑡))𝑥^𝑇(𝑡)𝐶(𝐴(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥(≤|𝑐|))+𝐵))𝑡−𝜏))₋₁max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶+𝑐̂⁻¹max_𝑘∈𝑆𝐶𝐵(𝑘)|𝑥(𝑡)|+0.5𝑐̂⁻¹max_𝑘∈𝑆(𝑘×(−|𝑥(𝑡+|𝑥(.𝐶𝐵))|𝑡−𝜏)|) (76) Substituting (76) into (75) yields (𝑥(𝑡(𝑟(𝑡|𝑥(𝑡𝑐̂𝑑log𝑉))≤𝛽))𝑑𝑡−𝜆)|𝑑𝑡+0.5⁻¹max_𝑘∈𝑆𝐶𝐵(𝑘)(−|𝑥(𝑡)|+|𝑥(𝑡−𝜏)|)+𝑉⁻¹(𝑥(𝑡))𝑥^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))𝑑𝑤(𝑡). (77) Hence, log𝑉(𝑥(𝑡))+𝜆^𝑡₀|𝑥(𝑠(𝑥(0+)|𝑑𝑠≤log𝑉))^𝑡₀𝛽(𝑟(𝑠))𝑑𝑠+⁰_−𝜏|𝑥(𝑠+)|𝑑𝑠_𝑡0𝑉⁻¹(𝑥(𝑠))𝑥^𝑇(𝑠)𝐶𝜎(𝑟(𝑠))𝑑𝑤(𝑠). (78) Applying the strong law of large numbers for martingales, we have lim_𝑡→∞1𝑡^𝑡₀𝑉⁻¹(𝑥(𝑠))𝑥^𝑇(𝑠)𝐶𝜎(𝑟(𝑠))𝑑𝑤(𝑠)=0a.s. (79) Dividing both sides of (78) by 𝑡 and letting 𝑡→∞, we obtain that 𝜆limsup_𝑡→+∞1𝑡^𝑡₀|𝑥(𝑠)|𝑑𝑠≤^𝑁_𝑘=1𝜋_𝑘𝛽(𝑘)a.s., (80) which implies the required assertion (72).
On the other hand, it is observed from (75)-(76) that 𝑑log𝑉(𝑥(𝑡))≥𝛼(𝑟(𝑡))𝑑𝑡−0.5𝑐̂⁻¹𝜆|𝑥(𝑡)|𝑑𝑡+𝑉⁻¹(𝑥(𝑡))𝑥^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))𝑑𝑤(𝑡). (81) Hence, log𝑉(𝑥(𝑡))+0.5𝑐̂⁻¹𝜆^𝑡₀|𝑥(𝑠(𝑥(0+)|𝑑𝑠≥log𝑉))^𝑡₀𝛼(𝑟(𝑠+))𝑑𝑠_𝑡0𝑉⁻¹(𝑥(𝑠))𝑥^𝑇(𝑠)𝐶𝜎(𝑟(𝑠))𝑑𝑤(𝑠). (82) Consequently, one gets that 0.5𝑐̂⁻¹𝜆liminf_𝑡→+∞1𝑡^𝑡₀|𝑥(𝑠)|𝑑𝑠≥^𝑁_𝑘=1𝜋_𝑘𝛼(𝑘)a.s., (83) 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 limsup_𝑡→∞1𝑡^𝑡₀|𝑥(𝑠1)|𝑑𝑠≤𝜆^𝑁_𝑖=𝑘𝜋_𝑘𝛽(𝑘)𝑎.𝑠.,liminf_𝑡→∞1𝑡^𝑡₀|𝑥(𝑠2𝑐̂)|𝑑𝑠≥𝜆^𝑛_𝑘=1𝜋_𝑘𝛼(𝑘)𝑎.𝑠., (84) where −𝜆=min_𝑘∈𝑆[𝜆_min(𝐶𝐴(𝑘)+𝐴^𝑇(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 𝑑𝑥(𝑡)=diag𝑥₁(𝑡),𝑥₂(𝑡)×𝑏(𝑟(𝑡(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡,[())+𝐴))+𝐵))𝑡−𝜏))𝑑𝑡+𝜎))𝑑𝑤)] (85) 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, 𝐴(𝑟(𝑡=𝑎))¹¹(𝑟(𝑡))𝑎₁₂(𝑟(𝑡𝑎))₂₁(𝑟(𝑡))𝑎₂₂(𝑟(𝑡,𝐵(𝑟(𝑡=𝑏))))¹¹(𝑟(𝑡))𝑏₁₂(𝑟(𝑡𝑏))₂₁(𝑟(𝑡))𝑏₂₂(𝑟(𝑡,)) (86) and 𝑟(𝑡) is a right-continuous Markov chain taking values in 𝑆={1,2}, and 𝑟(𝑡) and 𝑤(𝑡) are independent. Here 𝑏₁(1)=5,𝑎₁₁(1)=−5,𝑎₁₂(1)=√𝑏10,₂(1)=8,𝑎₂₁(1)=√10,𝑎₂₂(1)𝑏=−5,₁(2)=4,𝑎₁₁(2)=−3,𝑎₁₂(2)=√𝑏2,₂(2)=5,𝑎₂₁(2)=√2,𝑎₂₂(2)𝑏=−3,₁₁(1)=0,𝑏₁₂(1)=12,𝜎₁(1)=√𝑏2,₂₁(1)=1,𝑏₂₂(1)=0,𝜎₂(1)𝑏=2,₁₁(2)=0,𝑏₁₂(2)=√22,𝜎₁(2)=√𝑏14,₂₁(2)=√22,𝑏₂₂(2)=0,𝜎₂(2)=4. (87) Let 𝐶=𝐼∈𝑅^2×2. It is easy to compute that |𝑐|=√2,𝑐̂=1,max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶≤−3+√−√2,𝜆=−5−10<0,max_𝑘∈𝑆(𝑘≤√𝐶𝐵)52. (88) Then −𝜆=|𝑐|⁻¹max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+𝑐̂⁻¹max_𝑘∈𝑆𝐶𝐵(𝑘)<0. (89) 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 Γ=−112−2. (90) By solving the linear equation 𝜋Γ=0, we obtain the unique stationary (probability) distribution 𝜋=(𝜋₁,𝜋₂)=(2/3,1/3). Then ²₁₌₁𝜋_𝑘𝛼(𝑘)=23>0,²₁₌₁𝜋_𝑘𝛽(𝑘)=4>0. (91) Therefore, by Theorems 8 and 12, (85) is stochastically permanent and the solutions have the following properties: 2𝜆≤liminf_𝑡→∞1𝑡^𝑡₀|𝑥(𝑠)|𝑑𝑠≤limsup_𝑡→∞1𝑡^𝑡₀|𝑥(𝑠4)|𝑑𝑠≤𝜆a.s. (92)

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).