Stochastic Delay Population Dynamics under Regime Switching: Global Solutions and Extinction

Abstract

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. 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, and 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. We here only mention Ahmad and Rao [1], Bereketoglu and Győri [2], 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 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 term 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 of the attention paid 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 ([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 obtain 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].

Equation (4) describes the dynamics of populations. This paper is concerned with the positive global solutions, ultimate boundedness and extinction. The stochastic permanence and asymptotic estimations of solutions will be investigated in the next note [28].

This paper is organized as follows. In the next section, some sufficient conditions for global positive solutions for any initial positive value are given by using generalized Itô formula, Gronwall inequality, and 𝑉-function techniques. In Section 3, the stochastically ultimate boundedness of solutions is obtained by virtue of Young's inequality. Section 4 is devoted to the extinction of solutions. Finally, an example and its numerical simulation are given to illustrate our main results.

2. 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 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>0forall1≤𝑖≤𝑛}, and denote by 𝑅^𝑛₊ the nonnegative cone in 𝑅^𝑛, that is 𝑅^𝑛₊={𝑥∈𝑅^𝑛∶𝑥_𝑖≥0forall1≤𝑖≤𝑛}. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. If 𝐴 is a matrix, its trace norm is denoted by |𝐴|=trace(𝐴^𝑇𝐴), and its operator norm is denoted by ‖𝐴‖=sup{|𝐴𝑥|∶|𝑥|=1}. Moreover, let 𝜏>0 and denote by 𝐶([−𝜏,0];𝑅^𝑛₊) the family of continuous functions from [−𝜏,0] to 𝑅^𝑛₊.

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), refer to 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) We refer to [12, 29] for the fundamental theory of stochastic differential equations.

For convenience and simplicity in the following discussion, for any constant sequence 𝑓_𝑖(𝑘),(1≤𝑖≤𝑛,𝑘∈𝑆) let 𝑓^̌‌=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)

As 𝑥(𝑡) in system (4) denotes populations size at time 𝑡, it should be nonnegative. Thus for further study, we must give some condition under which (4) has a unique global positive solution.

Theorem 1. Assume that there are positive numbers 𝑐₁,…,𝑐_𝑛 and 𝜃 such that max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶+14𝜃𝐶𝐵(𝑘)𝐵^𝑇(𝑘)𝐶+𝜃𝐼≤0, (12) where 𝐶=diag(𝑐₁,…,𝑐_𝑛). Then 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 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^𝑛₊), there is a unique maximal local solution 𝑥(𝑡) on 𝑡∈[−𝜏,𝜏_𝑒), where 𝜏_𝑒 is the explosion time. To show that the solution is global, we need to show that 𝜏_𝑒=∞ a.s.
Let 𝑘₀>0 be sufficiently lager such that 1𝑘₀≤min_{−𝜏≤𝑡≤0}|𝑥(𝑡)|≤max_{−𝜏≤𝑡≤0}|𝑥(𝑡)|≤𝑘₀. (13) For each integer 𝑘≥𝑘₀, define the stopping time 𝜏_𝑘=inf𝑡∈0,𝜏_𝑒∶𝑥_𝑖(𝑡)∉1𝑘,,𝑘forsome𝑖=1,2,…,𝑛 (14) where throughout this paper we set inf∅=∞ (as usual ∅ denotes the empty set). Clearly, 𝜏_𝑘 is increasing as 𝑘→∞. Set 𝜏_∞=lim_𝑘→∞𝜏_𝑘, where 𝜏_∞≤𝜏_𝑒 a.s. If 𝜏_∞=∞ a.s., then 𝜏_𝑒=∞ a.s. and 𝑥(𝑡)∈𝑅^𝑛₊ a.s. for all 𝑡≥0. In other words, to complete the proof, one should show that 𝜏_∞=∞ a.s. Define 𝑉∶𝑅^𝑛₊→𝑅₊ by 𝑉(𝑥)=^𝑛_𝑖=1𝑐_𝑖𝑥_𝑖−1−log𝑥_𝑖. (15) The nonnegativity of this function can be seen from 𝑢−1−log𝑢≥0on𝑢>0. Let 𝑘≥𝑘₀ and 𝑇>0 be arbitrary. For 0≤𝑡≤𝜏_𝑘∧𝑇, it is easy to see by the generalized Itô formula that 𝐸𝑉𝑥𝜏_𝑘∧𝑡=𝑉(𝑥(0))+𝐸^𝜏𝑘₀^∧𝑡𝐿𝑉(𝑥(𝑠),𝑥(𝑠−𝜏),𝑟(𝑠))𝑑𝑠, (16) where 𝐿𝑉∶𝑅^𝑛₊×𝑅^𝑛₊×𝑆→𝑅 is defined by 𝐿𝑉𝑥,𝑦,𝑘=𝑥^𝑇𝐶𝑏(𝑘)+𝑥^𝑇𝐶𝐴(𝑘)𝑥+𝑥^𝑇𝐶𝐵(𝑘)𝑦−𝑐^𝑇𝑏(𝑘)(𝑘)(𝑘)𝑦+1+𝐴𝑥+𝐵2𝜎^𝑇(𝑘)𝐶𝜎(𝑘), (17) and 𝑐=(𝑐₁,…,𝑐_𝑛)^𝑇. Using condition (12) we compute 𝑥^𝑇𝐶𝐴(𝑘)𝑥+𝑥^𝑇(𝑘)𝑦≤1𝐶𝐵2𝑥^𝑇𝐶𝐴(𝑘)+𝐴(𝑘)𝐶𝑥+1𝑥4𝜃^𝑇𝐶𝐵(𝑘)𝐵^𝑇(𝑘)𝐶𝑥+𝜃𝑦²=𝑥^𝑇12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+14𝜃𝐶𝐵(𝑘)𝐵^𝑇(𝑘)𝐶+𝜃𝐼𝑥−𝜃|𝑥|²+𝜃𝑦²≤−𝜃|𝑥|²+𝜃𝑦². (18)
Moreover, there is a constant 𝐾₁>0 such that max_𝑘∈𝑆𝑥^𝑇𝐶𝑏(𝑘)+𝑐^𝑇𝐴(𝑘)𝑥+𝑐^𝑇𝐵(𝑘)𝑦−𝑐^𝑇𝑏(𝑘)+12𝜎^𝑇(𝑘)𝐶𝜎(𝑘)≤𝐾₁1+|𝑥|+𝑦. (19)
Substituting these inequalities into (17) yields 𝐿𝑉𝑥,𝑦,𝑖≤𝐾₁1+|𝑥|+𝑦−𝜃|𝑥|²+𝜃𝑦². (20) Noticing that 𝑢≤2(𝑢−1−log𝑢)+2on𝑢>0, we compute |𝑥|≤^𝑛_𝑖=1𝑥_𝑖≤^𝑛_𝑖=12𝑥_𝑖−1−log𝑥_𝑖2+2≤2𝑛+𝑐̂^𝑛_𝑖=1𝑐_𝑖𝑥_𝑖−1−log𝑥_𝑖2=2𝑛+𝑐̂𝑉(𝑥). (21)
It follows from (20) and (21) that 𝐿𝑉𝑥,𝑦,𝑘≤𝐾₂1+𝑉(𝑥)+𝑉𝑦−𝜃|𝑥|²+𝜃𝑦², (22) where 𝐾₂ is a positive constant. Substituting this inequality into (16) yields 𝐸𝑉𝑥𝜏_𝑘(𝑥(0∧𝑡≤𝑉))+𝐾₂𝐸^𝜏𝑘₀^∧𝑡[1+𝑉(𝑥(𝑠))+𝑉(𝑥(𝑠−𝜏))]𝑑𝑠+𝐸^𝜏𝑘₀^∧𝑡−𝜃𝑥²(𝑠)+𝜃𝑥²(𝑠−𝜏)𝑑𝑠. (23) Compute 𝐸^𝜏𝑘₀^∧𝑡𝑉(𝑥(𝑠−𝜏))𝑑𝑠=𝐸^{𝜏𝑘∧(𝑡−𝜏)}_−𝜏𝑉(𝑥(𝑠≤))𝑑𝑠_0−𝜏𝑉(𝑥(𝑠))𝑑𝑠+𝐸^𝜏𝑘₀^∧𝑡𝑉(𝑥(𝑠))𝑑𝑠 (24) and, similarly 𝐸^𝜏𝑘₀^∧𝑡|𝑥(𝑠−𝜏)|²𝑑𝑠≤⁰_−𝜏|𝑥(𝑠)|²𝑑𝑠+𝐸^𝜏𝑘₀^∧𝑡|𝑥(𝑠)|²𝑑𝑠. (25) Substituting these inequalities into (23) gives 𝐸𝑉𝑥𝜏_𝑘∧𝑡≤𝐾₃+2𝐾₂𝐸^𝜏𝑘₀^∧𝑡𝑉(𝑥(𝑠))𝑑𝑠≤𝐾₃+2𝐾₂𝐸^𝑡₀𝑉𝑥𝜏_𝑘∧𝑠𝑑𝑠≤𝐾₃+2𝐾₂^𝑡₀𝐸𝑉𝑥𝜏_𝑘∧𝑠𝑑𝑠, (26) where 𝐾₃=𝑉(𝑥(0))+𝐾₂𝑇+𝐾₂∫⁰_−𝜏𝑉(𝑥(𝑠))𝑑𝑠+𝜃∫⁰_−𝜏|𝑥(𝑠)|²𝑑𝑠.
By the Gronwall inequality, we obtain that 𝐸𝑉𝑥𝜏_𝑘∧𝑇≤𝐾₃𝑒^2𝑇𝐾2. (27) Noting that for every 𝜔∈{𝜏_𝑘≤𝑇}, 𝑉𝑥𝜏_𝑘,𝜔≥𝑐̂𝑘−1−log𝑘∧1/𝑘−1+log𝑘, (28) one has by (27) that 𝐾₃𝑒^2𝑇𝐾2≥𝐸𝑉𝑥𝜏_𝑘1∧𝑇≥𝐸_{{𝜏𝑘≤𝑇}}(𝜔)𝑉𝑥𝜏_𝑘𝟏∧𝑇,𝜔=𝐸_{{𝜏𝑘≤𝑇}}(𝜔)𝑉𝑥𝜏_𝑘≥̂𝜏,𝜔𝑐𝑃_𝑘×∧,≤𝑇𝑘−1−log𝑘1/𝑘−1+log𝑘 (29) where 𝟏_{{𝜏𝑘≤𝑇}} is the indicator function of {𝜏_𝑘≤𝑇}. Letting 𝑘→∞ gives lim_𝑘→∞𝑃{𝜏_𝑘≤𝑇}=0 and hence 𝑃{𝜏_∞≤𝑇}=0. Since 𝑇>0 is arbitrary, we must have 𝑃{𝜏_∞<∞}=0, so 𝑃{𝜏_∞=∞}=1 as required.

Assumption 1. Assume that there exist positive numbers 𝑐₁,…,𝑐_𝑛 such that max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶+max_𝑘∈𝑆𝐶𝐵(𝑘)≤0, (30) where 𝐶=diag(𝑐₁,…,𝑐_𝑛).

The following theorem is easy to verify in applications, which will be used in the sections below.

Theorem 2. 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 a.s.

Proof. Define 𝑉∶𝑅^𝑛₊→𝑅₊ by 𝑉(𝑥)=∑^𝑛_𝑖=1𝑐_𝑖(𝑥_𝑖−1−log𝑥_𝑖). The non-negativity of this function can be seen from 𝑢−1−log𝑢≥0on𝑢>0, and then we have (16) and (17).
If 𝐵(𝑘)≠0,𝑘∈𝑆, then ‖𝐶𝐵(𝑘)‖≠0. Consequently 𝑥^𝑇𝐶𝐴(𝑘)𝑥+𝑥^𝑇(𝑘)𝑦≤1𝐶𝐵2𝑥^𝑇𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶𝑥+12(𝑘𝑥𝐶𝐵)^𝑇𝐶𝐵(𝑘)𝐵^𝑇(𝑘)+1𝐶𝑥2𝐶𝐵(𝑘)𝑦²=12𝑥^𝑇𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶𝑥+12𝐶𝐵(𝑘)|𝑥|²+12𝐶𝐵(𝑘)𝑦². (31) Otherwise ‖𝐶𝐵(𝑘)‖=0 for 𝐵(𝑘)=0,𝑘∈𝑆. In this case, we also have that 𝑥^𝑇𝐶𝐴(𝑘)𝑥+𝑥^𝑇(𝑘)𝑦≤1𝐶𝐵2𝑥^𝑇𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶1𝑥+2𝐶𝐵(𝑘)|𝑥|²+12𝐶𝐵(𝑘)𝑦². (32) Thus, 𝑥^𝑇𝐶𝐴(𝑘)𝑥+𝑥^𝑇(𝑘)𝑦≤1𝐶𝐵2𝑥^𝑇𝐶𝐴(𝑘)+𝐴(𝑘)𝐶1𝑥+2𝐶𝐵(𝑘)|𝑥|²+12𝐶𝐵(𝑘)𝑦². (33) Denote 𝜂=max_𝑘∈𝑆‖𝐶𝐵(𝑘)‖. By (33) and Assumption 1, one has 𝑥^𝑇𝐶𝐴(𝑘)𝑥+𝑥^𝑇(𝑘)𝑦≤1𝐶𝐵2𝑥^𝑇𝐶𝐴(𝑘)+𝐴(𝑘)𝐶1𝑥+2𝜂|𝑥|²+12𝜂𝑦²≤max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶|𝑥|²+12𝜂|𝑥|²+12𝜂𝑦²1≤−2𝜂|𝑥|²+12𝜂𝑦². (34) The rest of the proof is similar to that of Theorem 1 and omitted.

3. Ultimate Boundness

Theorem 2 shows that solutions of the SDE (4) will remain in the positive cone 𝑅^𝑛₊. This nice property provides us with a great opportunity to discuss how solutions vary in 𝑅^𝑛₊ in detail. In this section, we give the definitions of stochastically ultimate boundedness of the SDE (4) and some sufficient conditions under which solutions of SDE (4) are stochastically ultimate bounded.

Definition 3. The solutions of (4) are called stochastically ultimately bounded, if for any 𝜀∈(0,1), there exists a positive constant 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. (35)

Assumption 2. Assume that there exist positive numbers 𝑐₁,…,𝑐_𝑛 such that −𝜆=max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶+max_𝑘∈𝑆𝐶𝐵(𝑘)<0, (36) where 𝐶=diag(𝑐₁,…,𝑐_𝑛).

Theorem 4. 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 constants 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. (37)(38)

Proof. By Theorem 2, the solution 𝑥(𝑡) will remain in 𝑅^𝑛₊ for all 𝑡≥−𝜏 with probability 1. If max_𝑘∈𝑆‖𝐶𝐵(𝑘)‖>0, we let 𝜂=(𝑝+1)⁻¹max_𝑘∈𝑆‖𝐶𝐵(𝑘)‖ and 𝛾=𝜏⁻¹log((𝜆+2𝜂)/2𝜂)>0. Define 𝑉(𝑥,𝑡)=𝑒^𝛾𝑡(∑^𝑛_𝑖=1𝑐_𝑖𝑥_𝑖)^𝑝=𝑒^𝛾𝑡(𝑐^𝑇𝑥)^𝑝. It has by the generalized Itô formula that 𝑑𝑉(𝑥(𝑡),𝑡)=𝐿𝑉(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡,𝑟(𝑡))𝑑𝑡+𝑝𝑒^𝛾𝑡𝑐^𝑇𝑥(𝑡)^𝑝−1𝑥^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))𝑑𝑤(𝑡), (39) where 𝐿𝑉∶𝑅^𝑛₊×𝑅^𝑛₊×𝑅₊×𝑆→𝑅 is defined by 𝐿𝑉𝑥,𝑦,𝑡,𝑘=𝑒^𝛾𝑡𝛾𝑐^𝑇𝑥^𝑝+𝑝𝑐^𝑇𝑥^𝑝−1𝑥^𝑇𝐶𝑏(𝑘)(𝑘)(𝑘)𝑦+1+𝐴𝑥+𝐵2𝑝𝑝−1𝑐^𝑇𝑥^𝑝−2𝑥^𝑇𝐶𝜎(𝑘)². (40) Meanwhile, by Assumption 2 and Young’s inequality, one gets 𝐿𝑉𝑥,𝑦,𝑡,𝑘≤𝑒^𝛾𝑡𝛾|𝑐|^𝑝|𝑥|^𝑝+𝑝|𝑐|^𝑝|𝑏(𝑘)||𝑥|^𝑝+12𝑝𝑝−1|𝑐|^𝑝|𝜎(𝑘)|²|𝑥|^𝑝+𝑝𝑐^𝑇𝑥^𝑝−1𝑥^𝑇𝐶𝐴(𝑘)𝑥+𝐵(𝑘)𝑦≤𝑒^𝛾𝑡𝐾𝑝|𝑥|^𝑝+12𝑝𝑐^𝑇𝑥^𝑝−1𝑥×𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶𝑥^𝑇+𝑝𝑐^𝑇𝑥^𝑝−1𝐶𝐵(𝑘)|𝑥|𝑦≤𝑒^𝛾𝑡𝐾𝑝|𝑥|^𝑝+𝑒^𝛾𝑡𝑝|𝑐|^𝑝−1×max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶|𝑥|^𝑝+1+𝑒^𝛾𝑡𝑝|𝑐|^𝑝−1𝐶𝐵(𝑘)|𝑥|^𝑝𝑦≤𝑒^𝛾𝑡𝐾𝑝|𝑥|^𝑝+𝑒^𝛾𝑡𝑝|𝑐|^𝑝−1×max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶|𝑥|^𝑝+1+𝑒^𝛾𝑡𝑝|𝑐|^𝑝−1(𝑘𝑝𝐶𝐵)|𝑥|𝑝+1^𝑝+1+1𝑦𝑝+1^𝑝+1≤𝑒^𝛾𝑡𝐾𝑝|𝑥|^𝑝+𝑝|𝑐|^𝑝−1−𝜆+𝜂|𝑥|^𝑝+1+𝜂𝑦^𝑝+1≤𝑒^𝛾𝑡𝐾𝑝|𝑥|^𝑝−12𝑝𝜆|𝑐|^𝑝−1|𝑥|^𝑝+1+𝑝|𝑐|^𝑝−1×−12𝜆+𝜂|𝑥|^𝑝+1+𝜂𝑦^𝑝+1≤𝐻𝑝𝑒^𝛾𝑡+𝑝𝜂|𝑐|^𝑝−1𝑒^𝛾𝑡−𝑒^𝛾𝜏|𝑥|^𝑝+1+𝑦^𝑝+1, (41) where 𝐾𝑝=max_𝑘∈𝑆𝛾|𝑐|^𝑝+𝑝|𝑐|^𝑝|𝑏(𝑘+1)|2𝑝𝑝−1|𝑐|^𝑝|𝜎(𝑘)|²,𝐻𝑝=sup_𝑥∈𝑅+𝐾𝑝|𝑥|^𝑝−12𝑝𝜆|𝑐|^𝑝−1|𝑥|^𝑝+1∨1. (42) On the other hand, ^𝑡₀𝑒^𝛾𝑠|𝑥(𝑠−𝜏)|^𝑝+1𝑑𝑠=𝑒^𝛾𝜏𝑡₀𝑒^{𝛾(𝑠−𝜏)}|𝑥(𝑠−𝜏)|^𝑝+1𝑑𝑠=𝑒^{𝛾𝜏𝑡−𝜏}_−𝜏𝑒^𝛾𝑠|𝑥(𝑠)|^𝑝+1𝑑𝑠≤𝑒^𝛾𝜏0_−𝜏|𝑥(𝑠)|^𝑝+1𝑑𝑠+𝑒^𝛾𝜏𝑡₀𝑒^𝛾𝑠|𝑥(𝑠)|^𝑝+1𝑑𝑠, (43) by (41) and (43), we obtain that 𝑒^𝛾𝑡𝐸[𝑉(𝑥(𝑡(𝑥(0))]≤𝑉))+^𝑡₀𝐻𝑝𝑒^𝛾𝑠𝑑𝑠+𝑝|𝑐|^𝑝−1𝜂×^𝑡₀𝑒^𝛾𝑠−𝑒^𝛾𝜏|𝑥(𝑠)|^𝑝+1+|𝑥(𝑠−𝜏)|^𝑝+1(𝑥(0+𝐻𝑝𝑑𝑠≤𝑉))𝛾𝑒^𝛾𝑡−1+𝑝|𝑐|^𝑝−1𝜂𝑒^𝛾𝜏0_−𝜏|𝑥(𝑠)|^𝑝+1𝑑𝑠, (44) which yields limsup_𝑡→∞(𝑥(𝑡≤𝐻𝑝𝐸𝑉))𝛾. (45) Since |𝑥(𝑡)|≤∑^𝑛_𝑖=1𝑥_𝑖(𝑡)≤𝑉(𝑥(𝑡))/𝑐̂, it has limsup_𝑡→∞𝐸|𝑥(𝑡)|^𝑝≤𝐻(𝑝)/𝑐𝛾̂ and the desired assertion (37) follows by setting 𝐾₁(𝑝)=𝐻(𝑝)/𝑐𝛾̂. It is easy to verify this result, if max_𝑘∈𝑆‖𝐶𝐵(𝑘)‖=0. We omit its proof here.
Define 𝑉(𝑥)=(𝑐^𝑇𝑥)^𝑝. By the generalized Itô formula, it follows 𝑑𝑉(𝑥(𝑡))=𝐿𝑉(𝑥(𝑡)()(𝑡𝑐,𝑥𝑡−𝜏,𝑟))𝑑𝑡+𝑝^𝑇𝑥(𝑡)^𝑝−1𝑥^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))𝑑𝑤(𝑡), (46) where 𝐿𝑉∶𝑅^𝑛₊×𝑅^𝑛₊×𝑆→𝑅 is defined by 𝐿𝑉𝑥,𝑦,𝑘=𝑝𝑐^𝑇𝑥^𝑝−1𝑥^𝑇𝐶𝑏(𝑘)(𝑘)(𝑘)𝑦+1+𝐴𝑥+𝐵2𝑝𝑝−1𝑐^𝑇𝑥^𝑝−2𝑥^𝑇𝐶𝜎(𝑘)². (47) By Assumption 2 and Young’s inequality again, 𝐿𝑉|𝑐|𝑥,𝑦,𝑘≤𝑝^𝑝|𝑏(𝑘)||𝑥|^𝑝+12𝑝𝑝−1|𝑐|^𝑝|𝜎(𝑘)|²|𝑥|^𝑝+𝑝𝑐^𝑇𝑥^𝑝−1𝑥^𝑇𝐶(𝐴(𝑘)(𝑘𝑦|𝑐|𝑥+𝐵))≤𝑝^𝑝|𝑏(𝑘)||𝑥|^𝑝+12𝑝𝑝−1|𝑐|^𝑝|𝜎(𝑘)|²|𝑥|^𝑝+𝑝|𝑐|^𝑝−1−𝜆+𝜂|𝑥|^𝑝+1+𝜂𝑦^𝑝+1. (48) It is easy to compute 0≤𝐸𝑉(𝑥(0))+𝐸^𝑡₀𝑝𝑏^̌‌|𝑐|^𝑝|𝑥(𝑠)|^𝑝+12𝑝𝑝−1|𝑐|𝜎^𝑝̌2|𝑥(𝑠)|^𝑝−𝑝|𝑐|^𝑝−1𝜆+𝜂|𝑥(𝑠)|^𝑝+1+𝑝|𝑐|^𝑝−1𝜂|𝑥(𝑠−𝜏)|^𝑝+1𝑑𝑠. (49) Moreover, ^𝑡₀|𝑥(𝑠−𝜏)|^𝑝+1𝑑𝑠≤⁰_−𝜏|𝑥(𝑠)|^𝑝+1𝑑𝑠+^𝑡₀|𝑥(𝑠)|^𝑝+1𝑑𝑠, (50) hence, we get 12𝜆𝑝|𝑐|^𝑝−1𝐸^𝑡₀|𝑥(𝑠)|^𝑝+1𝑑𝑠≤𝐸𝑉(𝑥(0))+𝐸^𝑡₀𝑝|𝑐|𝑏^𝑝̌|𝑥(𝑠)|^𝑝+12𝑝𝑝−1|𝑐|𝜎^𝑝̌2|𝑥(𝑠)|^𝑝𝜆−𝑝2+𝜂|𝑐|^𝑝−1|𝑥(𝑠)|^𝑝+1+𝑝𝜂|𝑐|^𝑝−1|𝑥(𝑠−𝜏)|^𝑝+1𝑑𝑠≤𝐸𝑉(𝑥(0))+𝑝𝜂|𝑐|^𝑝−10_−𝜏|𝑥(𝑠)|^𝑝+1𝑑𝑠+𝐸^𝑡₀𝑝|𝑐|𝑏^𝑝̌|𝑥(𝑠)|^𝑝+12𝑝𝑝−1|𝑐|𝜎^𝑝̌2|𝑥(𝑠)|^𝑝−12𝑝𝜆|𝑐|^𝑝−1|𝑥(𝑠)|^𝑝+1𝑑𝑠≤𝐸𝑉(𝑥(0))+𝑝𝜂|𝑐|^𝑝−1×⁰_−𝜏|𝑥(𝑠)|^𝑝+1𝑑𝑠+𝐻𝑝𝑡, (51) where 𝐻(𝑝)=sup_𝑥∈𝑅+(𝑝|𝑐|𝑏^𝑝̌|𝑥|^𝑝+(1/2)𝑝|𝑝−1||𝑐|𝜎^𝑝̌2|𝑥(𝑠)|^𝑝−(1/2)𝜆𝑝|𝑐|^𝑝−1|𝑥|^𝑝+1). This implies immediately that limsup_𝑡→∞1𝑡𝐸^𝑡₀|𝑥(𝑠)|^𝑝+1𝑝𝑑𝑠≤2𝐻𝑝𝜆|𝑐|^𝑝−1 (52) and the desired assertion (38) follows by setting 𝐾₂(𝑝)=2𝐻(𝑝)/𝑝𝜆|𝑐|^𝑝−1.

Remark 5. From (37) of Theorem 4, there is a 𝑇>0 such that 𝐸|𝑥(𝑡)|^𝑝≤2𝐾₁𝑝,∀𝑡≥𝑇. (53) Since 𝐸|𝑥(𝑡)|^𝑝 is continuous, there is a 𝐾₁(𝑝,𝑥₀) such that 𝐸|𝑥(𝑡)|^𝑝≤𝐾₁𝑝,𝑥₀for𝑡∈[0,𝑇]. (54) Let 𝐿(𝑝,𝑥₀)=max(2𝐾₁(𝑝),𝐾₁(𝑝,𝑥₀)), we have 𝐸|𝑥(𝑡)|^𝑝≤𝐿𝑝,𝑥₀,∀𝑡∈[0,∞), (55) which implies that the 𝑝th moment of any positive solution of (4) is bounded.

Remark 6. Conclusion (38) of Theorem 4 shows that the average in time of the 𝑝th (𝑝>1) moment of solutions of (4) will be bounded.

Theorem 7. Solutions of (4) are stochastically ultimately bounded under Assumption 2.

Proof. This can be easily verified by Chebyshev’s inequality and Theorem 4.

4. Extinction

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, (56) where 𝐶=diag(𝑐₁,…,𝑐_𝑛) and 𝑐̂=min_{1≤𝑖≤𝑛}𝑐_𝑖.

Theorem 8. Under Assumption 3, for any given initial data {𝑥(𝑡)∶−𝜏≤𝑡≤0}∈𝐶([−𝜏,0];𝑅^𝑛₊), the solution 𝑥(𝑡) of (4) has the properties that limsup_𝑡→∞1𝑡log|𝑥(𝑡)|≤^𝑛_𝑖=1𝜋_𝑘𝛽(𝑘)a.s., (57) where 𝛽(𝑘)=𝑏^̌‌(𝑘)−(1/2)𝜎²(𝑘). Particularly, if∑^𝑁_𝑘=1𝜋_𝑘𝛽(𝑘)<0, then limsup_𝑡→∞1𝑡log|𝑥(𝑡)|<0a.s. (58) That is, the population will become extinct exponentially with probability 1.

Proof. By Theorem 2, the solution 𝑥(𝑡) will remain in 𝑅^𝑛₊ for all 𝑡≥−𝜏 with probability 1. Define 𝑉(𝑥)=𝑐^𝑇𝑥=^𝑛_𝑖=1𝑐_𝑖𝑥_𝑖on𝑥∈𝑅^𝑛₊, (59) where 𝑐=(𝑐₁,…,𝑐_𝑛)^𝑇. Then 𝑑𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝐶𝑏(𝑟(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡.[(+𝐴))+𝐵))𝑡−𝜏)))𝑑𝑡+𝜎))𝑑𝑤)] (60) By the generalized Itô formula, (𝑥(𝑡=1𝑑log𝑉))𝑉(𝑥(𝑡(𝑥(𝑡−1))𝑑𝑉))2𝑉²(𝑥(𝑡((𝑥(𝑡))𝑑𝑉)))²=1𝑉(𝑥(𝑡𝑥))^𝑇(𝑡)𝐶×𝑏(𝑟(𝑡)(𝑟(𝑡𝑥(𝑡)(𝑟(𝑡𝑥((𝑟(𝑡(𝑡−1[(+𝐴))+𝐵))𝑡−𝜏)))𝑑𝑡+𝜎))𝑑𝑤)]2𝑉²(𝑥(𝑡𝑥))^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))²𝑑𝑡. (61) It is computed 𝑥^𝑇(𝑡)𝐶𝐴(𝑟(𝑡))𝑥(𝑡)𝑉(𝑥(𝑡+𝑥))^𝑇(𝑡)𝐶𝐵(𝑟(𝑡))𝑥(𝑡−𝜏)𝑉(𝑥(𝑡≤𝑥))_𝑇(𝑡)𝐶𝐴(𝑟(𝑡))+𝐴^𝑇(𝑟(𝑡))𝐶𝑥(𝑡)(𝑥(𝑡+2𝑉))𝐶𝐵(𝑟(𝑡))|𝑥(𝑡−𝜏)|𝑐̂≤|𝑐|⁻¹max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴(𝑘)𝐶+𝑐̂⁻¹max_𝑘∈𝑆(𝑘𝑥(𝑡+𝑐̂𝐶𝐵)|)|₋₁max_𝑘∈𝑆(𝑘−|𝑥(𝑡+|𝑥(,𝑥𝐶𝐵)()|𝑡−𝜏)|)_𝑇(𝑡)𝐶𝑏(𝑟(𝑡))𝑉(𝑥(𝑡−𝑥))^𝑇(𝑡)𝐶𝜎(𝑟(𝑡))²2𝑉²(𝑡)≤𝑏^̌‌(𝑟(𝑡−1))2𝜎²(𝑟(𝑡))=𝛽(𝑟(𝑡)). (62)(63) Substituting these two inequalities into (61) yields (𝑥(𝑡(𝑥(0log𝑉))≤log𝑉))+^𝑡₀𝛽(𝑟(𝑠))𝑑𝑠+𝑐̂⁻¹max_𝑘∈𝑆(𝑘×𝐶𝐵)_𝑡0[−|𝑥(𝑠+|𝑥((𝑡)(𝑥(0+𝑐̂)|𝑠−𝜏)|]𝑑𝑠+𝑀≤log𝑉))⁻¹max_𝑘∈𝑆𝐶𝐵(𝑘)⁰_−𝜏𝑥(𝑠)+𝑑𝑠_𝑡0𝛽(𝑟(𝑠))𝑑𝑠+𝑀(𝑡), (64) where 𝑀(𝑡) is a martingale defined by 𝑀(𝑡)=^𝑡₀𝑥^𝑇(𝑠)𝐶𝜎(𝑟(𝑠))𝑉(𝑥(𝑠(𝑡).))𝑑𝑤 (65) The quadratic variation of this martingale is ⟨𝑀,𝑀⟩_𝑡=^𝑡₀𝑥^𝑇(𝑠)𝐶𝜎(𝑟(𝑠))²𝑉²(𝑥(𝑠))𝑑𝑠≤𝜎^̌‌2𝑡, (66) hence limsup_𝑡→∞⟨𝑀,𝑀⟩_𝑡𝑡≤𝜎^̌‌2a.s. (67) Applying the strong law of large numbers for martingales [29], we therefore have lim_𝑡→∞𝑀(𝑡)𝑡=0a.s. (68) It finally follows from (64) by dividing 𝑡 on the both sides and then letting 𝑡→∞ that limsup_𝑡→∞log𝑉(𝑥(𝑡))𝑡≤limsup_𝑡→∞1𝑡^𝑡₀𝛽(𝑟(𝑠=))𝑑𝑠_𝑁𝑘=1𝜋_𝑘𝛽(𝑘)a.s., (69) which is the required assertion (57).

Similarly, we can prove the following conclusions.

Theorem 9. Assume that Assumption 3 holds. Assume moreover that the noise intensities 𝜎(𝑖) are sufficiently large in the sense that 𝜎_𝑖(𝑘)𝜎_𝑗(𝑘)−𝑏_𝑖(𝑘)−𝑏_𝑗(𝑘)>0,1≤𝑖,𝑗≤𝑛,foreach𝑘∈𝑆, (70) then for any given initial data {𝑥(𝑡)∶−𝜏≤𝑡≤0}∈𝐶([−𝜏,0];𝑅^𝑛₊), the solution 𝑥(𝑡) of (4) has the properties that limsup_𝑡→∞1𝑡|𝑥(𝑡1log)|≤−2^𝑁_𝑘=1𝜋_𝑘𝜑(𝑘)a.s., (71) where 𝜑(𝑘)=min_{1≤𝑖,𝑗≤𝑛}𝜎_𝑖(𝑘)𝜎_𝑗(𝑘)−𝑏_𝑖(𝑘)−𝑏_𝑗(𝑘)>0. (72) That is, the population will become extinct exponentially with probability 1.

Proof. Let 𝑉∶𝑅^𝑛₊→𝑅₊ be the same as defined in the proof of Theorem 8, so we have (60), (61), and (62). It is also computed 𝑥^𝑇(𝑡)𝐶𝑏(𝑟(𝑡))𝑉(𝑥(𝑡−𝑥))^𝑇(𝑥(𝑡))𝐶𝜎(𝑟(𝑡))²2𝑉²(𝑥(𝑡=))2𝑥^𝑇(𝑡)𝐶𝑏(𝑟(𝑡))𝑐^𝑇𝑥(𝑡)2𝑉²(𝑥(𝑡−𝑥))_𝑇(𝑡)𝐶𝜎(𝑟(𝑡))𝜎^𝑇(𝑟(𝑡))𝐶𝑥(𝑡)2𝑉²(𝑥(𝑡=))2𝑥^𝑇(𝑡)𝐶𝑏(𝑟(𝑡))1𝐶𝑥(𝑡)2𝑉²(𝑥(𝑡−𝑥))_𝑇(𝑡)𝐶𝜎(𝑟(𝑡))𝜎^𝑇(𝑟(𝑡))𝐶𝑥(𝑡)2𝑉²(𝑥(𝑡=𝑥))_𝑇(𝑡)𝐶𝑏(𝑟(𝑡))1+1^𝑇𝑏^𝑇(𝑟(𝑡))𝐶𝑥(𝑡)2𝑉²(𝑥(𝑡−𝑥))_𝑇(𝑡)𝐶𝜎(𝑟(𝑡))𝜎^𝑇(𝑟(𝑡))𝐶𝑥(𝑡)2𝑉²(𝑥(𝑡𝑥))=−^𝑇(𝑡)𝐶𝑄(𝑟(𝑡))𝐶𝑥(𝑡)2𝑉²(𝑥(𝑡,)) (73) where 1=(1,…,1) and 𝑄(𝑘)=𝜎(𝑘)𝜎^𝑇(𝑘)−(𝑏(𝑘)1+1^𝑇𝑏^𝑇(𝑘)). Substituting (62) and (73) into (61) yields (𝑥(𝑡(𝑥(0−log𝑉))≤log𝑉))^𝑡₀𝑥^𝑇(𝑠)𝐶𝑄(𝑟(𝑠))𝐶𝑥(𝑠)2𝑉²(𝑥(𝑠+𝑐̂))𝑑𝑠₋₁max_𝑘∈𝑆𝐶𝐵(𝑘)^𝑡₀[−|𝑥(𝑠)|+|𝑥(𝑠−𝜏)|]𝑑𝑠+𝑀(𝑡). (74) Note that 𝜎_𝑖(𝑘)𝜎_𝑗(𝑘)−𝑏_𝑖(𝑘)−𝑏_𝑗(𝑘), the 𝑖𝑗th element of the matrix 𝑄(𝑘) is positive by (70). It is therefore easy to verify 𝑥^𝑇(𝑡)𝐶𝑄(𝑘)𝐶𝑥(𝑡)≥𝜑(𝑘)𝑉²(𝑥(𝑡)), (75) where 𝜑(⋅) has been defined in the statement of the theorem. Substituting this inequality into (74) yields (𝑥(𝑡(𝑥(0log𝑉))≤log𝑉))−^𝑡₀12𝜑(𝑘)𝑑𝑠+𝑐̂⁻¹max_𝑘∈𝑆(𝑘×𝐶𝐵)_𝑡0(−|𝑥(𝑠)|+|𝑥(𝑠−𝜏)|)𝑑𝑠+𝑀(𝑡). (76) The rest of the proof is similar to that of Theorem 8 and omitted.

5. Examples

In this section, an example and corresponding numerical simulations are given to illustrate our main results.

Example 10. 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)] (77) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡))^𝑇, 𝑏(𝑟(𝑡))=(𝑏₁(𝑟(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)) (78) and 𝑟(𝑡) is a right-contiuous 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)𝑏=3,₁₁(1)=0,𝑏₁₂(1)=12,𝜎₁(1)=√𝑏2,₂(1)=8,𝑎₂₁(1)=3,𝑎₂₂(1)𝑏=−5,₂₁(1)=1,𝑏₂₂(1)=0,𝜎₂(1)𝑏=2,₁(2)=4,𝑎₁₁(2)=−3,𝑎₁₂(2)𝑏=1,₁₁(2)=0,𝑏₁₂(2)=1,𝜎₁(2)=√𝑏14,₂(2)=5,𝑎₂₁(2)=1,𝑎₂₂(2)𝑏=−3,₂₁(2)=12,𝑏₂₂(2)=0,𝜎₂(2)=4. (79) Let 𝐶=𝐼∈𝑅^2×2, the identity matrix. It is easy to compute |𝑐|=√2,𝑐̂=1,𝛽(1)=7,𝛽(2)=−2,max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶≤−2,max_𝑘∈𝑆(𝑘≤√𝐶𝐵)52. (80) Then |𝑐|⁻¹max_𝑘∈𝑆𝜆⁺_max12𝐶𝐴(𝑘)+𝐴^𝑇(𝑘)𝐶+𝑐̂⁻¹max_𝑘∈𝑆𝐶𝐵(𝑘)<0. (81) By Theorems 2 and 7, the solutions of (77) will remain in 𝑅²₊ for all 𝑡≥−𝜏 with probability 1 and are stochastically ultimately bounded.
Let the generator of the Markov chain 𝑟(𝑡) be Γ=−441−1. (82) By solving the linear equation 𝜋Γ=0, we obtain the unique stationary (probability) distribution 𝜋=(𝜋₁,𝜋₂)=(1/5,4/5). Then ∑²_𝑘=1𝜋_𝑘𝛽(𝑘)=−1/5<0. Therefore, by Theorems 8, (77) is extinctive, shown in Figure 1.

(a)

(b)

Figure 1 

In Figure 1, for numerical solutions of (77), step size Δ𝑡=0.001, delay 𝜏=1. Initial datum of (𝑥₁(𝑡),𝑥₂(𝑡)) are random numbers in [1,200]×[1,600]. Initial datum are not shown in Figure 1.

6. Conclusion

This work is concerned with delay Lotka-Volterra model under regime switching diffusion in random environment. It should be pointed out that (77) is more difficult to handle than (3) in [23]. Fortunately, the difficulties caused by delay term are overcome by using Young's inequality. The model in [7] is similar to (4), while the coefficients in (4) are varied with regime switching. Similar results are technically obtained by making use of comparison principle.

Acknowledgments

The authors are grateful to the editor Prof. Zhiming Guo and anonymous referees for their helpful comments and suggestions which have improved the quality of this paper. The authors are indebted to Professor Tao Wu, Anhui University, for giving ideas on simulations of the example. 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, no. KJ20 12A19), 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 1308085MA01, no. 1308085QA15, and no. 1208085QA15), Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).