Survival Analysis of a Nonautonomous Logistic Model with Stochastic Perturbation

Abstract

Taking white noise into account, a stochastic nonautonomous logistic model is proposed and investigated. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, stochastic permanence, and global asymptotic stability are established. Moreover, the threshold between weak persistence and extinction is obtained. Finally, we introduce some numerical simulink graphics to illustrate our main results.

1. Introduction

The classical nonautonomous logistic equation can be expressed as follows: 𝑑𝑥(𝑡)𝑑𝑡=𝑥(𝑡)[𝑎(𝑡)−𝑏(𝑡)𝑥(𝑡)].(1.1) for 𝑡≥0 with initial value 𝑥(0)=𝑥₀>0,𝑥(𝑡) is the population size at time 𝑡,𝑎(𝑡) denotes the rate of growth, and 𝑎(𝑡)/𝑏(𝑡) stands for the carrying capacity at time 𝑡. We refer the reader to May [1] for a detailed model construction. Obviously, system (1.1) has an equilibrium 𝑥^∗=𝑎(𝑡)/𝑏(𝑡), see the following (A1). Then, system (1.1) becomes the following equation: 𝑑𝑥(𝑡)𝑑𝑡=𝑎(𝑡)𝑥(𝑡)1−𝑥𝑥(^∗𝑡).(1.2)

Owing to its theoretical and practical significance, the deterministic system (1.1) and its generalization form have been extensively studied and many important results on the global dynamics of solutions have been founded, for example, Freedman and Wu [2], Golpalsamy [3], Kuang [4], Lisena [5], and the references therein. In particular, the books by Golpalsamy [3] and Kuang [4] are good references in this area.

In the real world, population dynamics is inevitably affected by environmental noise which is an important component in an ecosystem (see e.g., [6–9]). The deterministic models assume that parameters in the systems are all deterministic irrespective environmental fluctuations. Hence, they have some limitations in mathematical modeling of ecological systems, besides they are quite difficult to fitting data perfectly and to predict the future dynamics of the system accurately [8]. May [10] pointed out the fact that due to environmental noise, the birth rate, carrying capacity, competition coefficient, and other parameters involved in the system exhibit random fluctuation to a greater or lesser extent.

Recall that the parameters 𝑎(𝑡) represent the intrinsic growth rate. In practice, we usually estimate it by an average value plus an error term. In general, by the well-known central limit theorem, the error term follows a normal distribution and is sometimes dependent on how much the the current population sizes differ from the equilibrium state (see, e.g., [11–13]). In other words, we can replace the rate 𝑎(𝑡) by an average value plus a random fluctuation term: 𝑎(𝑡)⟶𝑎(𝑡)+𝛼(𝑡)𝑥(𝑡)−𝑥^∗𝐵̇(𝑡),(1.3) where 𝛼(𝑡) are continuous positive bounded function on 𝑅₊, and 𝛼²(𝑡) represents the intensity of the white noise at time 𝑡; 𝐵̇(𝑡) are the white noise, namely, 𝐵(𝑡) is a Brownian motion defined on a complete probability space (Ω,ℱ,𝒫) with a filtration {ℱ_𝑡}_𝑡∈𝑅+ satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ₀ contains all 𝒫-null sets). Then, by model (1.2), we obtain an Itŏ stochastic differential equation: 𝑑𝑥(𝑡)=𝑎(𝑡)𝑥(𝑡)1−𝑥(𝑡)𝑥^∗𝑑𝑡+𝛼(𝑡)𝑥(𝑡)𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑡).(1.4) Owing to the model (1.4) describes a population dynamics, it is necessary to investigate the survival of the logistic population which involves extinction, persistence, and global asymptotical stability (see, e.g., [14–16]). As far as we know, there are few results of this aspect for model (1.4). Furthermore, up to the authors' knowledge, all the publications have not obtained the persistence-extinction threshold for model (1.4). The aims of this work are to deal with the above problems one by one, which generalize the work of Wang and Liu (see, e.g., [17, 18]) where they mainly investigated survival analysis of population model with parameters perturbation.

Throughout the paper, we always have some assumption and notations.(A1) It holds that 𝑎(𝑡) and 𝑏(𝑡) are continuous bounded function on 𝑅 with 𝑏(𝑡)>0. Moreover, 𝑎(𝑡)/𝑏(𝑡) is a constant.(A2)It holds that 𝑓^𝑢=sup_𝑡∈𝑅𝑓(𝑡),𝑓^𝑙=inf_𝑡∈𝑅𝑓(𝑡),⟨𝑥(𝑡)⟩=1𝑡^𝑡0𝑓𝑥(𝑠)𝑑𝑠,_∗=liminf_𝑡→+∞𝑓(𝑡),𝑓^∗=limsup_𝑡→+∞𝑓(𝑡),𝑅₊=(0,+∞),𝑅₊=[0,+∞),𝑎=limsup_𝑡→+∞1𝑡^𝑡0𝑎(𝑠)𝑑𝑠.(1.5)

The following definitions are commonly used and we list them here.

Definition 1.1. (1) The population 𝑥(𝑡) is said to be extinctive if lim_𝑡→+∞𝑥(𝑡)=0 a.s.
(2) The population 𝑥(𝑡) is said to be nonpersistent in the mean (see e.g., Huaping and Zhien [14]) if limsup_𝑡→+∞⟨𝑥(𝑡)⟩=0 a.s.
(3) The population 𝑥(𝑡) is said to be weakly persistent (see e.g., Hallam and Ma [15]) if limsup_𝑡→+∞𝑥(𝑡)>0 a.s.
(4) The population 𝑥(𝑡) is said to be permanence (see e.g., Jiang et al. [19]) if for arbitrary 𝜀>0, there are constants 𝛽>0,𝑀>0 such that liminf_𝑡→+∞𝑃{𝑥(𝑡)≥𝛽}≥1−𝜀 and liminf_𝑡→+∞𝑃{𝑥(𝑡)≤𝑀}≥1−𝜀.

It follows from the above definitions that stochastic permanence implies stochastic weak persistence, extinction means stochastic nonpersistence in the mean. But generally, the reverses are not true.

The rest of the paper is arranged as follows. In Section 2, sufficient criteria for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence of the population are established. In Section 3, we study global asymptotic stability of positive equilibrium. In Section 4, we work out some figures to illustrate the various theorems obtained in Section 3 and Section 4. The last section gives the conclusions and future directions of the research.

2. Persistence and Extinction

As 𝑥(𝑡) in system (1.2) denotes the population size, it should be nonnegative. So, for further study, we must firstly give some conditions under which system (1.2) has a global positive solution. Similar to Mao et al. [20], we have the following Lemma.

Lemma 2.1. For model (1.4), with any given initial value 𝑥(0)=𝑥₀>0, there is a unique solution 𝑥(𝑡) on 𝑡≥0 and the solution will remain in 𝑅₊ with probability 1.

Proof. Since the coefficients of (1.4) are locally Lipschitz continuous, for any given initial value 𝑥₀∈𝑅⁺, there is a unique maximal local solution 𝑥(𝑡) on 𝑡∈[−𝜏^𝑀,𝜏_𝑒), where 𝜏_𝑒 is the explosion time (cf. Mao [21, page 95]). To show this solution is global, we need to show that 𝜏_𝑒=+∞ a.s. Let 𝑘₀>0 be sufficiently large for 1𝑘₀<𝑥₀<𝑘₀.(2.1) For each time integer 𝑘≥𝑘₀, define the stopping time: 𝜏_𝑘=inf𝑡∈0,𝜏_𝑒∶𝑥(𝑡)≤1𝑘or𝑥(𝑡)≥𝑘,(2.2) 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, whence 𝜏_+∞≤𝜏_𝑒 a.s. for all 𝑡≥0. In other words, to complete the proof, all we need to show is that 𝜏_+∞=+∞ a.s. To show this statement, let us define a 𝐶² function 𝑉∶𝑅₊→𝑅₊ by 𝑉(𝑥)=√𝑥−1−0.5ln𝑥.(2.3) The nonnegativity of this function can be seen from √𝑢−1−0.5ln𝑢≥0on𝑢>0.(2.4) Let 𝑘≥𝑘₀ and 𝑇>0 be arbitrary. For 0≤𝑡≤𝜏_𝑘∧𝑇, we can apply the Itô formula to 𝑉(𝑥(𝑡)) to obtain that 𝑑𝑉(𝑡)=0.5𝑥^−0.5(𝑡)−𝑥⁻¹(𝑡)𝑥(𝑡)𝑎(𝑡)−𝑎(𝑡)𝑥^∗+0.5−0.25𝑥_−1.5𝑥(𝑡)𝑑𝑡+𝛼(𝑡)𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑡)=−0.125𝛼₂(𝑡)+0.5𝑥⁻²(𝑡)𝛼²(𝑡)𝑥²(𝑡)𝑥(𝑡)−𝑥^∗²𝑑𝑡(𝑡)𝑥^2.5(𝑡)+0.25𝛼²(𝑡)𝑥²(𝑡)+0.25𝛼²(𝑡)𝑥^∗−0.5𝑎(𝑡)𝑥^∗𝑥^1.5−0.125𝛼₂(𝑡)(𝑡)𝑥^∗−0.5𝑎(𝑡)𝑥^0.5(𝑡)+0.5𝑎(𝑡)𝑥^∗−0.5𝛼²𝑥^∗+0.25𝛼₂𝑥(𝑡)=𝐹(𝑥(𝑡))𝑑(𝑡𝑡+)0𝑥.^∗5𝛼²(−𝑡)0𝑥.5𝑎_0.5(𝑡)𝑑𝑡+0.5𝛼(𝑡)𝑥^0.5(𝑡)−1𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑡)(𝑡)−1𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑡),(2.5) where 𝐹(𝑥(𝑡))=−0.125𝛼²(𝑡)𝑥^2.5(𝑡)+0.25𝛼²(𝑡)𝑥²(𝑡)+0.25𝛼²(𝑡)𝑥^∗−0.5𝑎(𝑡)𝑥^∗𝑥^1.5+0.5𝑎(𝑡)(𝑡)𝑥^∗−0.5𝛼²𝑥^∗𝑥(𝑡)−0.125𝛼²+0.25𝛼₂(𝑡)𝑥^∗−0.5𝑎(𝑡)𝑥^0.5(𝑡)(𝑡)𝑥^∗²−0.5𝑎(𝑡).(2.6) It is easy to see that 𝐹(𝑥(𝑡)) is bounded, say by 𝐾, in 𝑅₊. We, therefore, obtain that 𝑑𝑉𝑥(𝑡)≤𝐾𝑑𝑡+0.5𝛼(𝑡)𝑥^0.5(𝑡)−1𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑡).(2.7) Integrating both sides from 0 to 𝜏_𝑘∧𝑇, and then taking expectations, yields 𝐸𝑉𝑥𝜏_𝑘∧𝑇≤𝑉(𝑥(0))+𝐾𝑇.(2.8) Note that, for every 𝜔∈{𝜏_𝑘≤𝑇}, 𝑥(𝜏_𝑘,𝜔) equals either 𝑘 or 1/𝑘, and hence 𝑉(𝑥(𝜏_𝑘,𝜔)) is no less than either √𝑘−1−0.5log(𝑘),(2.9) or 1𝑘−1−0.5log𝑘1=1𝑘−1+0.5log(𝑘).(2.10) Consequently, 𝑉𝑥𝜏_𝑘,𝜔≥√𝑘−1−0.5log(𝑘)∧1𝑘−1+0.5log(𝑘).(2.11) It then follows from (2.15) that 𝑉(𝑥(0))+𝐾𝑇≥≥𝑃𝐸1𝜏_{{𝜏𝑘≤𝑇}𝑘}𝑉𝑥𝜏_𝑘,𝜔≤𝑇√𝑘−1−0.5log(𝑘)∧1𝑘−1+0.5log(𝑘),(2.12) where 1_{{𝜏𝑘≤𝑇}} is the indicator function of {𝜏_𝑘}. Letting 𝑘→+∞ gives 𝑃𝜏_+∞≤𝑇=0.(2.13) Since 𝑇>0 is arbitrary, we must have 𝑃𝜏_+∞<+∞=0,(2.14) so 𝑃{𝜏_+∞=+∞}=1 as required.

From Lemma 2.1, we know that solutions of system (1.4) will remain in the positive cone 𝑅₊. This nice positive property provides us with a great opportunity to construct different types of Lyapunov functions to discuss how the solutions vary in 𝑅₊ in more details. Now we will study the persistence and extinction of system (1.4).

Theorem 2.2. If 𝑎<0, then the population 𝑥(𝑡) by (1.4) goes to extinction.

Proof. Applying Itŏ’s formula to (1.4) leads to 𝑑ln𝑥(𝑡)=𝑎(𝑡)−𝑎(𝑡)𝑥^∗𝑥(𝑡)−𝛼²(𝑡)𝑥(𝑡)−𝑥^∗²2𝑑𝑡+𝛼(𝑡)𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑡)⋅(2.15) Integrating both sides of (2.15) from 0 to 𝑡, we have ln𝑥(𝑡)−ln𝑥₀=^𝑡0𝑎(𝑠)−𝑏(𝑠)𝑥(𝑠)−𝛼²(𝑠)𝑥(𝑠)−𝑥^∗²2𝑑𝑠+𝑀(𝑡),(2.16) where 𝑀(𝑡)=∫^𝑡0𝛼(𝑠)(𝑥(𝑠)−𝑥^∗)𝑑𝐵(𝑠). The quadratic variation of 𝑀(𝑠) is ⟨𝑀(𝑡),𝑀(𝑡)⟩=∫^𝑡0𝛼²(𝑠)(𝑥(𝑠)−𝑥^∗)²𝑑𝑠. By virtue of the exponential martingale inequality (see, e.g., [22] on page 36), for any positive constants 𝑇₀,𝛼, and 𝛽, we have 𝑃sup_{0≤𝑡≤𝑇0}𝑀(𝑡)−𝛼2⟨𝑀(𝑡),𝑀(𝑡)⟩>𝛽≤𝑒^−𝛼𝛽.(2.17) Choose 𝑇₀=𝑘,𝛼=1,𝛽=2ln𝑘, then it follows that 𝑃sup_{0≤𝑡≤𝑘}𝑀(𝑡)−12⟨𝑀(𝑡),𝑀(𝑡)⟩>2ln𝑘≤𝑘1².(2.18) Making use of Borel-Cantelli lemma (see, e.g., [22] on page 10) yields that for almost all 𝜔∈Ω, there is a random integer 𝑘₀=𝑘₀(𝜔) such that for 𝑘≥𝑘₀, sup_{0≤𝑡≤𝑘}𝑀(𝑡)−12⟨𝑀(𝑡),𝑀(𝑡)⟩≤2ln𝑘.(2.19) This is to say 𝑀(𝑡)≤2ln𝑘+12⟨𝑀(𝑡),𝑀(𝑡)⟩=2ln𝑘+12^𝑡0𝛼(𝑠)𝑥(𝑠)−𝑥^∗²𝑑𝑠,(2.20) for all 0≤𝑡≤𝑘,𝑘≥𝑘₀ a.s. Substituting this inequality into (2.16), we can obtain that ln𝑥(𝑡)−ln𝑥₀≤^𝑡0𝑎(𝑠)−𝑎𝑥(^∗𝑠)𝑥(𝑠)𝑑𝑠+2ln𝑘,(2.21) for all 0≤𝑡≤𝑘,𝑘≥𝑘₀ a.s. In other words, we have shown that, for 0<𝑘−1≤𝑡≤𝑘,𝑘≥𝑘₀, 𝑡⁻¹ln𝑥(𝑡)−ln𝑥₀≤𝑡⁻¹^𝑡0𝑎(𝑠)−𝑎𝑥(^∗𝑠)𝑥(𝑠)𝑑𝑠+2𝑡⁻¹ln𝑘.(2.22) Taking superior limit on both sides yields limsup_𝑡→+∞𝑥(𝑡)≤𝑎. That is to say, if 𝑎<0, one can see that lim_𝑡→+∞𝑥(𝑡)=0.

Theorem 2.3. If 𝑎=0, then the population 𝑥(𝑡) by (1.4) is nonpersistent in the mean.

Proof. For all 𝜀>0, ∃𝑇₁ such that 𝑡⁻¹∫^𝑡0𝑎(𝑠)𝑑𝑠≤𝑎+𝜀/2=𝜀/2 for 𝑡>𝑇₁, substituting this inequality into (2.22) gives 𝑡⁻¹ln𝑥(𝑡)−ln𝑥₀≤𝑡⁻¹^𝑡0𝑎(𝑠)−𝑎(𝑠𝑥)𝑥^∗(𝑠)𝑑𝑠+2𝑡⁻¹≤𝜀ln𝑘2−𝑡⁻¹^𝑡0𝑎(𝑠)𝑥(𝑠)𝑥^∗𝑑𝑠+2𝑡⁻¹ln𝑘,(2.23) for all 𝑇₁<𝑡≤𝑘,𝑘≥𝑘₀ a.s. Note that for sufficiently large 𝑡 satisfying 𝑇₁<𝑇<𝑘−1≤𝑡≤𝑘 and 𝑘≥𝑘₀, we have ln𝑘/𝑡≤𝜀/4. In other words, we have already shown that ln𝑥(𝑡)−ln𝑥₀<𝜀𝑡−^𝑡0𝑎(𝑠)𝑥(𝑠)𝑥^∗𝑑𝑠;𝑡>𝑇.(2.24) Define ℎ(𝑡)=∫^𝑡0𝑥(𝑠)𝑑𝑠 and 𝑁=inf_𝑠∈𝑅{𝑎(𝑠)/𝑥^∗}, then we have ln𝑑ℎ𝑑𝑡<𝜀𝑡−𝑁ℎ(𝑡)+ln𝑥⁰;𝑡>T.(2.25) Consequently, 𝑒^𝑁𝑑ℎ𝑑𝑡<𝑥⁰𝑒^𝜀𝑡;𝑡>𝑇.(2.26) Integrating this inequality from 𝑇 to 𝑡 results in 𝑁⁻¹𝑒^{𝑁ℎ(𝑡)}−𝑒^{𝑁ℎ(𝑇)}<𝑥₀𝜀⁻¹𝑒^𝜀𝑡−𝑒^𝜀𝑇.(2.27) Rewriting this inequality, one then obtains 𝑒^{𝑁ℎ(𝑡)}<𝑒^{𝑁ℎ(𝑇)}+𝑥₀𝑁𝜀⁻¹𝑒^𝜀𝑡−𝑥₀𝑁𝜀⁻¹𝑒^𝜀𝑇.(2.28) Taking the logarithm of both sides leads to ℎ(𝑡)<𝑁⁻¹ln𝑥₀𝑁𝜀⁻¹𝑒^𝜀𝑡+𝑒^{𝑁ℎ(𝑇)}−𝑥₀𝑁𝜀⁻¹𝑒^𝜀𝑇.(2.29) In other words, we have shown that 𝑡⁻¹^𝑡0𝑥(𝑠)𝑑𝑠≤𝑡⁻¹𝑁⁻¹ln𝑥₀𝑁𝜀⁻¹𝑒^𝜀𝑡+𝑒^{𝑁ℎ(𝑇)}−𝑥₀𝑁𝜀⁻¹𝑒^𝜀𝑇^∗.(2.30) An application of the L'Hospital's rule, one can derive ⟨𝑥⟩^∗≤𝑁⁻¹𝑡⁻¹ln𝑥₀𝑁𝜀⁻¹𝑒^𝜀𝑡^∗=𝜀𝑁.(2.31) Since 𝜀 is arbitrary, we have ⟨𝑥⟩^∗≤0, which is the required assertion.

Theorem 2.4. If 𝑎−liminf_𝑡→+∞⟨(𝑥^∗𝛼(𝑡))²/2⟩>0, then the population 𝑥(𝑡) by (1.4) is weakly persistent.

Proof. To begin with, let us show that limsup_𝑡→+∞𝑡⁻¹ln𝑥(𝑡)≤0a.s.(2.32) Applying Itŏ’s formula to (1.2) results in 𝑑𝑒^𝑡ln𝑥(𝑡)==𝑒𝑒^𝑡_𝑡ln𝑥(𝑡)𝑑𝑡+𝑒^𝑡𝑑ln𝑥(𝑡)ln𝑥(𝑡)+𝑎(𝑡)−𝑎𝑥(^∗𝑡)𝑥(𝑡)−𝛼²(𝑡)𝑥(𝑡)−𝑥^∗²2𝑑𝑡+𝑒^𝑡𝛼(𝑡)𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑡).(2.33) Thus, we have shown that 𝑒^𝑡ln𝑥(𝑡)−ln𝑥₀=^𝑡0𝑒^𝑠ln𝑥(𝑠)+𝑎(𝑠)−𝑎𝑥(^∗𝑠)𝑥(𝑠)−𝛼²(𝑠)𝑥(𝑠)−𝑥^∗²2𝑑s+𝑁(𝑡),(2.34) where 𝑁(𝑡)=^𝑡0𝑒^𝑠𝛼(𝑠)𝑥(𝑡)−𝑥^∗𝑑𝐵(𝑠).(2.35) Note that 𝑁(𝑡) is a local martingale with the quadratic form: ⟨𝑁(𝑡),𝑁(𝑡)⟩=^𝑡0𝑒^2𝑠𝛼²(𝑠)𝑥(𝑡)−𝑥^∗²𝑑𝑠.(2.36) It then follows from the exponential martingale inequality (2.17) by choosing 𝑇₀=𝜇𝑘,𝛼=𝑒^−𝜇𝑘,𝛽=𝜌𝑒^𝜇𝑘ln𝑘 that 𝑃sup_{0≤𝑡≤𝜇𝑘}𝑁(𝑡)−0.5𝑒^−𝜇𝑘⟨𝑁(𝑡),𝑁(𝑡)⟩>𝜌𝑒^𝜇𝑘ln𝑘≤𝑘^−𝜌,(2.37) where 𝜌>1 and 𝜇>1. In view of Borel-Cantelli lemma, for almost all 𝜔∈Ω, there exists a 𝑘₀(𝜔) such that, for every 𝑘≥𝑘₀(𝜔), 𝑁(𝑡)≤0.5𝑒^−𝜇𝑘⟨𝑁(𝑡),𝑁(𝑡)⟩+𝜌𝑒^𝜇𝑘ln𝑘,0≤𝑡≤𝜇𝑘.(2.38) Substituting the above inequalities into (2.34) yields 𝑒^𝑡ln𝑥(𝑡)−ln𝑥₀≤^𝑡0𝑒^𝑠ln𝑥(𝑠)+𝑎(𝑠)−𝑎𝑥(^∗𝑠)𝑥(𝑠)−𝛼²(𝑠)𝑥(𝑠)−𝑥^∗²+𝑒2𝑑𝑠_−𝜇𝑘2^𝑡0=𝑒^2𝑠𝛼²(𝑠)𝑥(𝑠)−𝑥^∗²𝑑𝑠+𝜌𝑒^𝜇𝑘ln𝑘_𝑡0𝑒^𝑠ln𝑥(𝑠)+𝑎(𝑠)−𝑎𝑥(^∗𝑠)𝑥(𝑠)−𝛼²(𝑠)𝑥(𝑠)−𝑥^∗²1−𝑒^{𝑠−𝜇𝑘}2𝑑𝑠+𝜌𝑒^𝜇𝑘ln𝑘.(2.39) It is easy to see that there exists a constant 𝐶 independent of 𝑘 such that ln𝑥(𝑠)+𝑎(𝑠)−𝑎(𝑠)𝑥^∗𝑥(𝑠)−𝛼²(𝑠)𝑥(𝑠)−𝑥^∗²1−𝑒^{𝑠−𝜇𝑘}2≤𝐶(2.40) for any 0≤𝑠≤𝜇𝑘 and 𝑥>0. In other words, we have 𝑒^𝑡ln𝑥(𝑡)−ln𝑥₀≤𝐶𝑒^𝑡−1+𝜌𝑒^𝜇𝑘ln𝑘(2.41) for any 0≤𝑡≤𝜇𝑘.
This is to say ln𝑥(𝑡)≤𝑒^−𝑡ln𝑥₀+𝐶1−𝑒^−𝑡+𝜌𝑒^{−𝑡+𝜇𝑘}ln𝑘.(2.42) Consequently, if 𝜇(𝑘−1)<𝑡≤𝜇𝑘 and 𝑘≥𝑘₀(𝜔), one can observe that 𝑡⁻¹ln𝑥(𝑡)≤𝑡⁻¹𝑒^−𝑡ln𝑥₀+𝑡⁻¹𝐶1−𝑒^−𝑡+𝑡⁻¹𝜌𝑒^{−𝜇(𝑘−1)+𝜇𝑘}ln𝑘,(2.43) which becomes the desired assertion (2.32) by letting 𝑘→+∞.
Now suppose that 𝑎>0, we will prove limsup_𝑡→+∞𝑥(𝑡)>0 a.s. If this assertion is not true, let 𝐹={limsup_𝑡→+∞𝑥(𝑡)=0} and suppose 𝑃(𝐹)>0. In view of (2.16), ln𝑥(𝑡)𝑡=ln𝑥⁰𝑡+𝑡⁻¹^𝑡0𝑎(𝑠)−𝑎𝑥(^∗𝑠)𝑥(𝑠)−𝛼²(𝑠)𝑥(𝑠)−𝑥^∗²2𝑑𝑠,+𝑀𝑡(𝑡).(2.44) On the other hand, for all 𝜔∈𝐹, we have lim_𝑡→+∞𝑥(𝑡,𝜔)=0. Then, the law of large numbers for local martingales (see, e.g., [22, page 12]) indicates that lim_𝑡→+∞(𝑀(𝑡)/𝑡)=0. Substituting this equality into (2.44) results in limsup_𝑡→+∞𝑡⁻¹ln𝑥(𝑡,𝜔)=𝑎−liminf_𝑡→+∞𝑥^∗𝛼(𝑡)²2>0.(2.45) Then, 𝑃(limsup_𝑡→+∞[𝑡⁻¹ln𝑥(𝑡)]>0)>0, which contradicts (2.32).

Remark 2.5. Theorems 2.2–2.4 have an obvious and interesting biological interpretation. It is easy to see that the extinction and persistence of population 𝑥(𝑡) modeled by (1.4) depend on 𝑎 and 𝑎−liminf_𝑡→+∞⟨(𝑥^∗𝛼(𝑡))²/2⟩. If 𝑎−liminf_𝑡→+∞⟨(𝑥^∗𝛼(𝑡))²/2⟩>0, the population 𝑥(𝑡) will be weakly persistent; If 𝑎<0, the population 𝑥(𝑡) will go to extinction. That is to say, if liminf_𝑡→+∞⟨(𝑥^∗𝛼(𝑡))²/2⟩=0, then 𝑎 is the threshold between weak persistence and extinction for the population 𝑥(𝑡).

Remark 2.6. From the condition 𝑎−liminf_𝑡→+∞⟨(𝑥^∗𝛼(𝑡))²/2⟩>0 in Theorems 2.4, we know that the white noise 𝛼(𝑡) is of disadvantage to the survival of the population.
On the other hand, it is well known that in the study of population systems, stochastic permanence, which means that the population will survive forever, is one of the most important and interesting topic owing to its theoretical and practical significance. So now let us show that population 𝑥(𝑡) modeled by (1.4) is stochastically permanent in some cases.

Theorem 2.7. If  liminf_𝑡→+∞{𝑎(𝑡)−(𝛼^𝑢)²(𝑥^∗)²/2}>0, then the population 𝑥(𝑡) by (1.4) will be stochastic permanent.

Proof. First, we prove that for arbitrary 𝜀>0, there is a constant 𝑀>0 such that liminf_𝑡→+∞𝑃{𝑥(𝑡)≤𝑀}≥1−𝜀. Define 𝑉(𝑥)=𝑥^𝑝 for 𝑥∈𝑅₊, where 0<𝑝<1. Then, it follows from Itô formula that 𝑑𝑉(𝑥)=𝑝𝑥^𝑝−1𝑑𝑥+𝑝(𝑝−1)2𝑥^𝑝−2(𝑑𝑥)²=𝑝𝑥^𝑝−1𝑎(𝑡)𝑥1−𝑥+𝑝(𝑝−1)𝑥^∗𝑑𝑡+𝛼(𝑡)𝑥𝑥−𝑥^∗𝑑𝐵(𝑡)2𝑥^𝑝−2𝛼²(𝑡)𝑥²𝑥−𝑥^∗²=𝑝𝑥^𝑝−1𝑎(𝑡)𝑥1−𝑥𝑥^∗𝑑𝑡+𝑝(𝑝2−1)𝛼²(𝑡)𝑥𝑥−𝑥^∗²+𝛼(𝑡)𝑥_𝑝(𝑡)𝑥−𝑥^∗𝑑𝐵(𝑡).(2.46) Let 𝑘₀>0 be so large that 𝑥₀ lying within the interval [1/𝑘₀,𝑘₀]. For each integer 𝑘≥𝑘₀, define the stopping time 𝜏_𝑘=inf{𝑡≥0∶𝑥(𝑡)∉(1/𝑘,𝑘)}. Clearly, 𝜏_𝑘→+∞ almost surely as 𝑘→+∞. Applying Itô formula again to 𝑒^𝑡𝑉(𝑥) gives 𝑑𝑒^𝑡𝑉(𝑥)==𝑒𝑒^𝑡_𝑡𝑉(𝑥)𝑑𝑡+𝑒^𝑡𝑑𝑉(𝑥)𝑥^𝑝1+𝑝(𝑎(𝑡)−𝑏(𝑡)𝑥)−𝑝(1−𝑝)𝛼²(𝑡)+𝑒_𝑡2𝑥−𝑥^∗²𝑑𝑡=𝑒_𝑡𝛼(𝑡)𝑥^𝑝(𝑡)𝑥−𝑥^∗𝑑𝐵(𝑡)𝑥^𝑝1+𝑝𝑎(𝑡)−𝑝(1−𝑝)𝛼²(𝑡)2𝑥^∗²−𝑝𝑏(𝑡)𝑥+𝑝(1−𝑝)𝛼²(𝑡)𝑥𝑥^∗−𝑝(1−𝑝)𝛼²(𝑡)2𝑥²𝑑𝑡+𝑒^𝑡≤𝑒_𝑡𝛼(𝑡)𝑥^𝑝(𝑡)𝑥−𝑥^∗𝑑𝐵(𝑡)𝐾₁𝑑𝑡+𝑒^𝑡𝛼(𝑡)𝑥^𝑝(𝑡)𝑥−𝑥^∗𝑑𝐵(𝑡),(2.47) where 𝐾₁ is a positive constant. Integrating this inequality and then taking expectations on both sides, one can see that 𝐸𝑒^{𝑡∧𝜏𝑘}𝑥^𝑝𝑡∧𝜏_𝑘−𝑥^𝑝0≤^{𝑡∧𝜏𝑘}₀𝑒^𝑠𝐾₁𝑑𝑠≤𝐾₁𝑒^𝑡−1.(2.48) Letting 𝑘→+∞ yields 𝐸𝑥^𝑝(𝑡)≤𝐾₁+𝑒^−𝑡𝑥^𝑝0.(2.49) In other words, we have already shown that limsup_𝑡→+∞𝐸𝑥^𝑝≤𝐾₁.(2.50) Thus, for any given 𝜀>0, let 𝑀=𝐾^11/𝑝/𝜀^1/𝑝, by virtue of Chebyshevs inequality, we can derive that 𝑃{𝑥(𝑡)>𝑀}=𝑃{𝑥^𝑝(𝑡)>𝑀^𝑝}≤𝐸𝑥^𝑝(𝑡)𝑀^𝑝.(2.51) That is to say 𝑃^∗{𝑥(𝑡)>𝑀}≤𝜀. Consequently, 𝑃_∗{𝑥(𝑡)≤𝑀}≥1−𝜀.
Next, we claim that, for arbitrary 𝜀>0, there is constant 𝛽>0 such that liminf_𝑡→+∞𝑃{𝑥(𝑡)≥𝛽}≥1−𝜀.
Define 𝑉₁(𝑥)=1/𝑥² for 𝑥∈𝑅₊. Applying Itô formula to (1.2), we can obtain that 𝑑𝑉₁(𝑥(𝑡))=−2𝑥⁻³𝑑𝑥+3𝑥⁻⁴(𝑑𝑥)²=2𝑉₁(𝑥)1.5𝛼²(𝑡)𝑥−𝑥^∗²+𝑎(𝑡)𝑥^∗𝑥−𝑎(𝑡)𝑑𝑡−2𝛼(𝑡)𝑥−𝑥^∗𝑥²𝑑𝐵(𝑡).(2.52) Since liminf_𝑡→+∞{𝑎(𝑡)−(𝛼^𝑢)²(𝑥^∗)²/2}>0, we can choose a sufficient small constant 0<𝜃<1 and 𝜀>0 such that 𝑎_∗−(𝛼^𝑢)²(𝑥^∗)²/2−𝜃(𝛼^𝑢)²(𝑥^∗)²−𝜀>0.
Define 𝑉₂(𝑥)=1+𝑉₁(𝑥)^𝜃.(2.53) Making use of Itŏ’s formula again leads to 𝑑𝑉₂=𝜃1+𝑉₁(𝑥)^𝜃−1𝑑𝑉₁+0.5𝜃(𝜃−1)1+𝑉₁(𝑥)^𝜃−2𝑑𝑉₁²=𝜃1+𝑉₁(𝑥)^𝜃−21+𝑉¹(𝑥)2𝑉₁(𝑥)1.5𝛼²(𝑡)𝑥−𝑥^∗²+𝑎(𝑡)𝑥^∗+2𝛼₂𝑥−𝑎(𝑡)(𝑡)(𝜃−1)𝑥−𝑥^∗²𝑥⁴𝑑𝑡−𝜃1+𝑉¹(𝑥)^𝜃−12𝛼(𝑡)𝑥−𝑥^∗𝑥²=𝜃1+𝑉₁𝑑𝐵(𝑡)(𝑥)^𝜃−21+𝑉¹(𝑥)2𝑉₁(𝑥)1.5𝛼²(𝑡)𝑥²−2𝑥𝑥^∗+𝑥^∗²+𝑎(𝑡)𝑥^∗+2𝛼₂𝑥−𝑎(𝑡)−𝜃1+(𝑡)𝑉(₁𝜃−1)𝑥⁻²−2𝑥⁻³𝑥^∗+𝑥⁻⁴𝑥^∗²𝑑𝑡(𝑥)^𝜃−12𝛼(𝑡)𝑥−𝑥^∗𝑥²=𝜃1+𝑉₁𝑑𝐵(𝑡)(𝑥)^𝜃−2−2𝑎(𝑡)−(𝜃+0.5)𝛼²+2𝑎(𝑡)(𝑡)𝑥^∗²𝑉²¹(𝑥)𝑥^∗−2𝛼²(𝑡)𝑥^∗−4𝜃𝛼²(𝑡)𝑥^∗𝑉^11.5+−2𝑎(𝑡)+3𝛼₂(𝑥)+2𝑎(𝑡)(𝑡)𝑥^∗²+2𝛼²(𝑡)(𝜃+0.5)𝑉₁(𝑥)𝑥^∗−6𝛼²(𝑡)𝑥^∗²𝑉^10.5−𝜃1+𝑉₁(𝑥)+3𝛼²(𝑡)(𝑥)^𝜃−12𝛼(𝑡)𝑥−𝑥^∗𝑥²≤𝜃1+𝑉₁𝑑𝐵(𝑡)(𝑥)^𝜃−2−2𝑎^∗−(𝛼^𝑢)²𝑥^∗²2−𝜃(𝛼^𝑢+2𝑎(𝑡))²𝑥^∗²−𝜀𝑉²¹(𝑥)𝑥^∗−2𝛼²(𝑡)𝑥^∗−4𝜃𝛼²(𝑡)𝑥^∗𝑉^11.5+−2𝑎(𝑡)+3𝛼₂(𝑥)+2𝑎(𝑡)(𝑡)𝑥^∗²+2𝛼²(𝑡)(𝜃+0.5)𝑉₁(𝑥)𝑥^∗−6𝛼²(𝑡)𝑥^∗²𝑉^10.5−𝜃1+𝑉₁(𝑥)+3𝛼²(𝑡)(𝑥)^𝜃−12𝛼(𝑡)𝑥−𝑥^∗𝑥²𝑑𝐵(𝑡)(2.54) for sufficiently large 𝑡≥𝑇. Now, let 𝜂>0 be sufficiently small satisfying 0<𝜂2𝜃<𝑎^∗−(𝛼^𝑢)²𝑥^∗²2−𝜃(𝛼^𝑢)²𝑥^∗²−𝜀.(2.55)
Define 𝑉₃(𝑥)=𝑒^𝜂𝑡𝑉₂(𝑥). By virtue of Itŏ’s formula, 𝑑𝑉₃(𝑥(𝑡))≤=𝜂𝜃𝑒𝑒^𝜂𝑡_𝜂𝑡𝑉₂(𝑥)+𝑒^𝜂𝑡𝑑𝑉₂(𝑥)1+𝑉₁(𝑥)^𝜃−2𝜂1+𝑉¹(𝑥)²𝜃−2𝑎^∗−(𝛼^𝑢)²𝑥^∗²2−𝜃(𝛼^𝑢+2𝑏(𝑡)−2𝛼₂)²𝑥^∗²−𝜀𝑉²¹(𝑥)+−2𝑎(𝑡)+3𝛼(₂𝑡)𝑥^∗−4𝜃𝛼²(𝑡)𝑥^∗𝑉^11.5(𝑥)+2𝑏(𝑡)−6𝛼₂(𝑡)𝑥^∗²+2𝛼²(𝑡)(𝜃+0.5)𝑉₁(𝑥)−𝜃𝑒_𝜂𝑡(𝑡)𝑥^∗²𝑉^10.5(𝑥)+3𝛼²(𝑡)𝑑𝑡1+𝑉₁(𝑥)^𝜃−12𝛼(𝑡)𝑥−𝑥^∗𝑥²≤𝜃𝑒_𝜂𝑡𝑑𝐵(𝑡)1+𝑉₁(𝑥)^𝜃−2−2𝑎^∗−(𝛼^𝑢)²𝑥^∗²2−𝜃(𝛼^𝑢)²𝑥^∗²−𝜀−𝜂2𝜃𝑉²¹+2𝑏(𝑡)−2𝛼₂(𝑥)+−2𝑎(𝑡)+3𝛼(𝑡₂)𝑥^∗−4𝜃𝛼²(𝑡)𝑥^∗𝑉^11.5(𝑥)(𝑡)𝑥^∗²+2𝛼²(𝑡)(𝜃+0.5)+2𝜂𝜃𝑉¹+2𝑏(𝑡)−6𝛼₂(𝑥)(𝑡)𝑥^∗²𝑉^10.5(𝑥)+3𝛼²(𝑡)+𝜂−𝜃𝑒_𝜂𝑡𝜃𝑑𝑡1+𝑉₁(𝑥)^𝜃−12𝛼(𝑡)𝑥−𝑥^∗𝑥²=𝑒_𝜂𝑡𝑑𝐵(𝑡)𝐻(𝑥)𝑑𝑡−2𝜃𝑒^𝜂𝑡1+𝑉₁(𝑥)^𝜃−1𝛼(𝑡)𝑥−𝑥^∗𝑥²𝑑𝐵(𝑡)(2.56) for 𝑡≥𝑇. Note that 𝐻(𝑥) is upper bounded in 𝑅₊, namely, 𝐻=sup_𝑥∈𝑅+𝐻(𝑥)<+∞. Consequently, 𝑑𝑉₃(𝑥(𝑡))=𝐻𝑒^𝜂𝑡𝑑𝑡−2𝜃𝑒^𝜂𝑡1+𝑉₁(𝑥)^𝜃−1𝛼(𝑡)𝑥−𝑥^∗𝑥²𝑑𝐵(𝑡)(2.57) for sufficiently large 𝑡. Integrating both sides of the above inequality and then taking expectations give 𝐸𝑉₃(𝑥(𝑡))=𝐸𝑒^𝜂𝑡1+𝑉₁(𝑥(𝑡))^𝜃≤𝑒^𝜂𝑇1+𝑉₁(𝑥(𝑇))^𝜃+𝐻𝜂𝑒^𝜂𝑡−𝑒^𝜂𝑇.(2.58) That is to say limsup_𝑡→+∞𝐸𝑉^𝜃1(𝑥(𝑡))≤limsup_𝑡→+∞𝐸1+𝑉₁(𝑥(𝑡))^𝜃<𝐻𝜂.(2.59) In other words, we have already shown that limsup_𝑡→+∞𝐸1𝑥^2𝜃(𝑡)≤𝐻𝜂=𝑀⁴.(2.60) So, for any 𝜀>0, set 𝛽=𝜀^1/2𝜃/𝑀^41/2𝜃, by Chebyshev's inequality, one can derive that 𝑃{𝑥(𝑡)<𝛽}=𝑃1𝑥^2𝜃(𝑡)>𝛽1^2𝜃≤𝐸1/𝑥^2𝜃(𝑡)1/𝛽^2𝜃,(2.61) this is to say that limsup_𝑡→+∞{𝑥(𝑡)<𝛽}≤𝛽^2𝜃𝑀₄=𝜀.(2.62) Consequently, liminf_𝑡→+∞{𝑥(𝑡)≥𝛽}≥1−𝜀.(2.63) This completes the whole proof.

Remark 2.8. It is easy to see that, if 𝑥^∗=0, our Theorems 2.2–2.7 will become Theorem 2–5 in [18], respectively. At the same time, Theorem 2.7 improves and generalizes the work of Liu and Wang [17] and Jiang et al. [19] in some cases.

Remark 2.9. Generally, from the biological viewpoint, Theorem 2.2 means the population will go to extinction which is the worst case. Theorem 2.3 indicates the population is rare. Theorem 2.4 means the species will be survival, but it admits the case that liminf_𝑡→+∞𝑥(𝑡)=0, which implies that the population size is closed to zero even if the time is sufficiently large. That is to say the survival of species could be dangerous in reality. Theorem 2.7 is more desired than Theorems 2.2–2.4. Theorem 2.7 means that the population size will be neither too small nor too large with large probability if the time is sufficiently large. That is to say, with large probability, the population will stably exist, which is the most desired case. In other words, that is the reasons why liminf_𝑡→+∞{𝑎(𝑡)−𝛼²(𝑡)(𝑥^∗)²/2}>0 is used by Theorem 2.7 whereas 𝑎−liminf_𝑡→+∞⟨(𝑥^∗(𝛼^𝑢)²/2⟩>0 and 𝑎 are used in Theorem 2.2–2.3, respectively.

3. Global Stability

In this section, we suppose that the equilibrium 𝑥^∗=𝑎(𝑡)/𝑏(𝑡) is a positive constant. When studying biologic dynamical system, one important topic is when the population will survive forever. Since model (1.4) is the perturbation system of model (1.2) which has a positive equilibrium 𝑥^∗2, it seems reasonable to consider that the population will have chance to survive forever if the solution of model (1.4) is going around 𝑥^∗ at the most time. We get following results.

Theorem 3.1. If inf_𝑡∈𝑅{𝑎(𝑡)−𝛼²(𝑡)(𝑥^∗)²/2}>0, then 𝑥^∗ in (1.4) is global asymptotical stability almost surely (a.s.), that is, lim_𝑡→+∞𝑥(𝑡)=𝑥^∗ a.s., almost surely.

Proof. From the stability theory of stochastic functional differential equations, we only need to find a Lyapunov function 𝑉(𝑥,𝑡) satisfying 𝐿𝑉(𝑥,𝑡)≤0, and the identity holds if and only if 𝑧=𝑧^∗(see, e.g., [21, 23]), where 𝑧=𝑧(𝑡) is the solution of the one-dimensional stochastic functional differential equation: 𝑑𝑧(𝑡)=𝑓(𝑧(𝑡),𝑡)𝑑𝑡+𝑔(𝑧(𝑡),𝑡)𝑑𝐵(𝑡),𝑡≥0.(3.1) Here, let 𝑓∶𝑅×𝑅₊→𝑅 and 𝑔∶𝑅×𝑅₊→𝑅. 𝐵(𝑡) be a one-dimensional Brownian motion defined on the complete probability space (Ω,ℱ,𝒫). 𝑧^∗ is the positive equilibrium position of (3.1) and 𝐿𝑉(𝑡)=𝑉_𝑡+𝑉_𝑧(𝑧)𝑓+0.5trace𝑔^𝑇𝑉_𝑧𝑧(𝑧)𝑔.(3.2) For 𝑡∈𝑅₊, define Lyapunov functions: 𝑉(𝑡)=𝑥(𝑡)−𝑥^∗−𝑥^∗ln𝑥(𝑡)𝑥^∗.(3.3) Applying Itŏ’s formula leads to 𝐿𝑉(𝑡)=𝑥(𝑡)−𝑥^∗𝑎(𝑡)−𝑎(𝑡)𝑥^∗𝑥(𝑡)𝑑𝑡+𝛼²(𝑡)𝑥^∗𝑥(𝑡)−𝑥^∗²=𝑥(𝑡)−𝑥_∗2−𝑎𝑥(_∗𝑡)𝑥(𝑡)−𝑥^∗+𝛼²(𝑡)𝑥^∗𝑥(𝑡)−𝑥^∗²=−𝑎(𝑡)2𝑑𝑡𝑥^∗+𝛼²(𝑡)𝑥^∗2𝑥(𝑡)−𝑥^∗²≤−inf_𝑡∈𝑅𝑎(𝑡)−𝛼²(𝑡)𝑥^∗²2𝑥(𝑡)−𝑥^∗²𝑥^∗.(3.4) The assumption of inf_𝑡∈𝑅{𝑎(𝑡)−𝛼²(𝑡)(𝑥^∗)²/2}>0 implies that 𝐿𝑉(𝑥,𝑡)<0 along all trajectories in 𝑅₊ except 𝑥^∗. Then, the desired assertion follows immediately.
Now, let us return back to system (1.2).

Corollary 3.2. If inf_𝑡∈𝑅𝑎(𝑡)>0, then 𝑥^∗ in (1.2) is global asymptotic stability.

Remark 3.3. By comparing Theorem 3.1 with Corollary 3.2, we can find that if the positive equilibrium of the deterministic model is global asymptotic stability, then the stochastic system will keep this nice property provided the noise is not very large.

4. Examples and Numerical Simulations

In order to conform to the results above, we numerically simulate the solution of system (1.4). By the Milstein scheme mentioned in [24], we consider the discretized equation: 𝑥_𝑘+1=𝑥_𝑘+𝑥_𝑘𝑎(𝑘Δ𝑡)−𝑏(𝑘Δ𝑡)𝑥_𝑘Δ𝑡+𝛼(𝑘Δ𝑡)𝑥_𝑘𝑥_𝑘−𝑥^∗2√Δ𝑡𝜉_𝑘+0.5𝛼²(𝑘Δ𝑡)𝑥_𝑘−𝑥^∗2^𝑘𝜉^2𝑘−1Δ𝑡,(4.1) where 𝜉_𝑘 are Gaussian random variable that follows 𝑁(0,1).

Let 𝑎(𝑡)=−0.03,𝛼(𝑡)=0.1+0.01cos𝑡, and 𝑥^∗=−0.04. Then, the conditions of Theorem 2.2 are satisfied, which means that the population 𝑥(𝑡) by (1.4) will be extinction (see Figure 1).

Figure 1 

Stochastic extinction. The horizontal axis and the vertical axis in this and following figures represent the time 𝑡 and the populations size 𝑥(𝑡), step size Δ𝑡=0.001 and 𝑥(0)=0.8.

Let 𝑎(𝑡)=0,𝛼(𝑡)=0.4+0.2cos𝑡, and 𝑥^∗=0. Then, the conditions of Theorem 2.3 hold, which implies that the population 𝑥(𝑡) by (1.4) will be nonpersistent in the mean (see Figure 2).

Figure 2 

Stochastically nonpersistent in the mean, step size Δ𝑡=0.001 and 𝑥(0)=1.6.

Let 𝑎(𝑡)=0.15,𝛼(𝑡)=0.03+0.01sin𝑡, and 𝑥^∗=0.2. Then, the conditions of Theorem 2.4 are satisfied. One can see that 𝑥(𝑡) by (1.4) will be weakly persistent (see Figure 3).

Figure 3 

Stochastically weakly persistent, step size Δ𝑡=0.001 and 𝑥(0)=1.1.

Let 𝑎(𝑡)=0.96,𝛼(𝑡)=0.05+0.01cos𝑡, and 𝑥^∗=1.3. Then, the conditions of Theorem 2.7 are satisfied. That is to say, the population 𝑥(𝑡) by (1.4) will be stochastic permanent (see Figure 4).

Figure 4 

Seen from the two-dimensional spaces, the population 𝑥(𝑡) will be stochastic permanent, step size Δ𝑡=0.001 and 𝑥(0)=1.3.

Let 𝑎(𝑡)=1.2 and 𝑥^∗=1. In Figure 5, we consider 𝛼(𝑡)=0.2. Then the corresponding conditions of Theorem 3.1 are satisfied, which means that the positive equilibrium 𝑥^∗=1 in (1.4) is global asymptotic stability almost surely. In Figure 6, the parameters are same as in Figure 5 except 𝛼(𝑡)=0. Then the conditions of Corollary 3.2 hold, which shows that the positive equilibrium 𝑥^∗=1 of (1.2) is global asymptotic stability. By comparing Figure 5 with Figure 6, one can see that if the positive equilibrium of the deterministic model is asymptotically stable, then the stochastic system will keep this nice property provided the noise is sufficiently small.

Figure 5 

Global asymptotical stability of model (1.2), step size Δ𝑡=0.001 and 𝑥(0)=1.5.

Figure 6 

Global asymptotical stability of model (1.1), step size Δ𝑡=0.001 and 𝑥(0)=1.5.

5. Conclusions and Future Directions

In the real world, the natural growth of population is inevitably affected by random disturbances. In this paper, we are concerned with the effects of white noise on the survival analysis of logistic model. Firstly, we show that the system has a unique positive global solution. Afterward, sufficient criteria for extinction, nonpersistence in the mean, weak persistence, stochastic permanence, and global asymptotic stability of a positive equilibrium are established. Further, the threshold between weak persistence and extinction is obtained.

Some interesting topics deserve further investigation. One may propose some realistic but complex models. An example is to incorporate the colored noise, such as continuous-time Markov chain, into the system. The motivation is that the population may suffer sudden environmental changes, for example, rain falls and changes in nutrition or food resources, and so forth. Frequently, the switching among different environments is memoryless and the waiting time for the next switch is exponentially distributed, then the sudden environmental changes can be modeled by a continuous-time Markov chain (see, e.g., [25–27]), and these investigations are in progress.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (10671047) and the foundation of HITC (200713).