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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 857134, 17 pages
http://dx.doi.org/10.1155/2012/857134
Research Article

Dynamical Behavior of a Stochastic Ratio-Dependent Predator-Prey System

School of Mathematical Science, Anhui University, Hefei 230039, China

Received 11 December 2011; Revised 10 February 2012; Accepted 17 February 2012

Academic Editor: Ying U. Hu

Copyright © 2012 Zheng Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with a stochastic ratio-dependent predator-prey model with varible coefficients. By the comparison theorem of stochastic equations and the Itô formula, the global existence of a unique positive solution of the ratio-dependent model is obtained. Besides, some results are established such as the stochastically ultimate boundedness and stochastic permanence for this model.

1. Introduction

Ecological systems are mainly characterized by the interaction between species and their surrounding natural environment [1]. Especially, the dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology, due to its universal existence and importance [24]. The interaction mechanism of predators and their preys can be described as differential equations, such as Lotaka-Volterra models [5].

Recently, many researchers pay much attention to functional and numerical responses over typical ecological timescales, which depend on the densities of both predators and their preys (most likely and simply on their ration) [68]. Such a functional response is called a ratio-dependent response function, and these hypotheses have been strongly supported by numerous and laboratory experiments and observations [911].

It is worthy to note that, based on the Michaelis-Menten or Holling type II function, Arditi and Ginzburg [6] firstly proposed a ratio-dependent function of the form𝑃𝑥𝑦=𝑐𝑥/𝑦=𝑚+𝑥/𝑦𝑐𝑥𝑚𝑦+𝑥(1.1) and a ratio-dependent predator-prey model of the forṁ𝑥(𝑡)=𝑥(𝑡)𝑎𝑏𝑥(𝑡)𝑐𝑦(𝑡),𝑚𝑦(𝑡)+𝑥(𝑡)̇𝑦(𝑡)=𝑦(𝑡)𝑑+𝑓𝑥(𝑡).𝑚𝑦(𝑡)+𝑥(𝑡)(1.2) Here, 𝑥(𝑡) and 𝑦(𝑡) represent population densities of the prey and the predator at time 𝑡, respectively. Parameters 𝑎, 𝑏, 𝑐, 𝑑, 𝑓, and 𝑚 are positive constants in which 𝑎/𝑏 is the carrying capacity of the prey, 𝑎, 𝑐, 𝑚, 𝑓, and 𝑑 stand for the prey intrinsic growth rate, capturing rate, half capturing saturation constant, conversion rate, and the predator death rate, respectively. In recent years, several authors have studied the ratio-dependent predator-prey model (1.2) and its extension, and they have obtained rich results [1219].

It is well known that population systems are often affected by environmental noise. Hence, stochastic differential equation models play a significant role in various branches of applied sciences including biology and population dynamics as they provide some additional degree of realism compared to their deterministic counterpart [20, 21]. Recall that the parameters 𝑎 and 𝑑 represent the intrinsic growth and death rate of 𝑥(𝑡) and 𝑦(𝑡), respectively. In practice we usually estimate them by an average value plus errors. In general, the errors follow normal distributions (by the well-known central limit theorem), but the standard deviations of the errors, known as the noise intensities, may depend on the population sizes. We may therefore replace the rates 𝑎 and 𝑑 bẏ𝐵𝑎𝑎+𝛼1̇𝐵(𝑡),𝑑(𝑑+𝛽)2(𝑡),(1.3) respectively, where 𝐵1(𝑡) and 𝐵2(𝑡) are mutually independent Brownian motions and 𝛼 and 𝛽 represent the intensities of the white noises. As a result, (1.2) becomes a stochastic differential equation (SDE, in short):𝑑𝑥(𝑡)=𝑥(𝑡)𝑎𝑏𝑥(𝑡)𝑐𝑦(𝑡)𝑚𝑦(𝑡)+𝑥(𝑡)𝑑𝑡+𝛼𝑥(𝑡)𝑑𝐵1(𝑡),𝑑𝑦(𝑡)=𝑦(𝑡)𝑑+𝑓𝑥(𝑡)𝑚𝑦(𝑡)+𝑥(𝑡)𝑑𝑡𝛽𝑦(𝑡)𝑑𝐵2(𝑡).(1.4) By the Itô formula, Ji et al. [3] showed that (1.4) is persistent or extinct in some conditions.

The predator-prey model describes a prey population 𝑥 that serves as food for a predator 𝑦. However, due to the varying of the effects of environment and such as weather, temperature, food supply, the prey intrinsic growth rate, capturing rate, half capturing saturation constant, conversion rate, and predator death rate are functions of time 𝑡 [2226]. Therefore, Zhang and Hou [27] studied the following general ratio-dependent predator-prey model of the form: ̇𝑥(𝑡)=𝑥(𝑡)𝑎(𝑡)𝑏(𝑡)𝑥(𝑡)𝑐(𝑡)𝑦(𝑡),𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡)̇𝑦(𝑡)=𝑦(𝑡)𝑑(𝑡)+𝑓(𝑡)𝑥(𝑡),𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡)(1.5) which is more realistic. Motivated by [3, 27], this paper is concerned with a stochastic ratio-dependent predator-prey model of the following form:𝑑𝑥(𝑡)=𝑥(𝑡)𝑎(𝑡)𝑏(𝑡)𝑥(𝑡)𝑐(𝑡)𝑦(𝑡)𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡)𝑑𝑡+𝛼(𝑡)𝑥(𝑡)𝑑𝐵1(𝑡),𝑑𝑦(𝑡)=𝑦(𝑡)𝑑(𝑡)+𝑓(𝑡)𝑥(𝑡)𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡)𝑑𝑡𝛽(𝑡)𝑦(𝑡)𝑑𝐵2(𝑡),(1.6) where 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑓(𝑡), and 𝑚(𝑡) are positive bounded continuous functions on [0,) and 𝛼(𝑡), 𝛽(𝑡) are bounded continuous functions on [0,), and 𝐵1(𝑡) and 𝐵2(𝑡) are defined in (1.4). There would be some difficulties in studying this model since the parameters are changed by time 𝑡. Under some suitable conditions, we obtain some results such as the stochastic permanence of (1.6).

Throughout this paper, unless otherwise specified, let (Ω,,{𝑡}𝑡0,𝑃) be a complete probability space with a filtration {𝑡}𝑡0 satisfying the usual conditions (i.e., it is right continuous and 0 contains all 𝑃-null sets). Let 𝐵1(𝑡) and 𝐵2(𝑡) be mutually independent Brownian motions, 𝑅2+ the positive cone in 𝑅2, 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡)), and |𝑋(𝑡)|=(𝑥2(𝑡)+𝑦2(𝑡))1/2.

For convenience and simplicity in the following discussion, we use the notation𝜑𝑢=sup[𝑡0,)𝜑(𝑡),𝜑𝑙=inf[𝑡0,)𝜑(𝑡),(1.7) where 𝜑(𝑡) is a bounded continuous function on [0,).

This paper is organized as follows. In Section 2, by the Itô formula and the comparison theorem of stochastic equations, the existence and uniqueness of the global positive solution are established for any given positive initial value. In Section 3, we find that both the prey population and predator population of (1.6) are bounded in mean. Finally, we give some conditions that guarantee that (1.6) is stochastically permanent.

2. Global Positive Solution

As 𝑥(𝑡) and 𝑦(𝑡) in (1.6) are population densities of the prey and the predator at time 𝑡, respectively, we are only interested in the positive solutions. Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of equation are generally required to satisfy the linear growth condition and local Lipschitz condition [28]. However, the coefficients of (1.6) satisfy neither the linear growth condition nor the local Lipschitz continuous. In this section, by making the change of variables and the comparison theorem of stochastic equations [29], we will show that there is a unique positive solution with positive initial value of system (1.6).

Lemma 2.1. For any given initial value 𝑋0𝑅2+, there is a unique positive local solution 𝑋(𝑡) to (1.6) on 𝑡[0,𝜏𝑒)a.s.

Proof. We first consider the equation 𝛼𝑑𝑢(𝑡)=𝑎(𝑡)2(𝑡)2𝑏(𝑡)𝑒𝑢(𝑡)𝑐(𝑡)𝑒𝑣(𝑡)𝑚(𝑡)𝑒𝑣(𝑡)+𝑒𝑢(𝑡)𝑑𝑡+𝛼(𝑡)𝑑𝐵1𝛽(𝑡),𝑑𝑣(𝑡)=𝑑(𝑡)2(𝑡)2+𝑓(𝑡)𝑒𝑢(𝑡)𝑚(𝑡)𝑒𝑣(𝑡)+𝑒𝑢(𝑡)𝑑𝑡𝛽(𝑡)𝑑𝐵2(𝑡)(2.1) on 𝑡0 with initial value 𝑢(0)=ln𝑥0, 𝑣(0)=ln𝑦0. Since the coefficients of system (2.1) satisfy the local Lipschitz condition, there is a unique local solution (𝑢(𝑡),𝑣(𝑡)) on 𝑡[0,𝜏𝑒), where 𝜏𝑒 is the explosion time [28]. Therefore, by the Itô formula, it is easy to see that 𝑥(𝑡)=𝑒𝑢(𝑡), 𝑦(𝑡)=𝑒𝑣(𝑡) is the unique positive local solution of system (2.1) with initial value 𝑋0=(𝑥0,𝑦0)𝑅2+. Lemma 2.1 is finally proved.

Lemma 2.1 only tells us that there is a unique positive local solution of system (1.6). Next, we show that this solution is global, that is, 𝜏𝑒=.

Since the solution is positive, we have[]𝑑𝑥(𝑡)𝑥(𝑡)𝑎(𝑡)𝑏(𝑡)𝑥(𝑡)𝑑𝑡+𝛼(𝑡)𝑥(𝑡)𝑑𝐵1(𝑡).(2.2) LetΦ(𝑡)=exp𝑡0𝛼𝑎(𝑠)2(𝑠)/2𝑑𝑠+𝑡0𝛼(𝑠)𝑑𝐵1(𝑠)𝑥01+𝑡0𝑏(𝑠)exp𝑠0𝛼𝑎(𝜏)2(𝜏)/2𝑑𝜏+𝑠0𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠.(2.3) Then, Φ(𝑡) is the unique solution of equation[]𝑑Φ(𝑡)=Φ(𝑡)𝑎(𝑡)𝑏(𝑡)Φ(𝑡)𝑑𝑡+𝛼(𝑡)Φ(𝑡)𝑑𝐵1Φ(𝑡),(0)=𝑥0,(2.4)𝑥(𝑡)Φ(𝑡)a.s.𝑡0,𝜏𝑒(2.5) by the comparison theorem of stochastic equations. On the other hand, we have𝑑𝑥(𝑡)𝑥(𝑡)𝑎(𝑡)𝑐(𝑡)𝑚(𝑡)𝑏(𝑡)𝑥(𝑡)𝑑𝑡+𝛼(𝑡)𝑥(𝑡)𝑑𝐵1(𝑡).(2.6) Similarly,𝜙(𝑡)=exp𝑡0𝛼𝑎(𝑠)(𝑐(𝑠)/𝑚(𝑠))2(𝑠)/2𝑑𝑠+𝑡0𝛼(𝑠)𝑑𝐵1(𝑠)𝑥01+𝑡0𝑏(𝑠)exp𝑠0𝛼𝑎(𝜏)(𝑐(𝜏)/𝑚(𝜏))2(𝜏)/2𝑑𝜏+𝑠0𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠(2.7) is the unique solution of equation𝑑𝜙(𝑡)=𝜙(𝑡)𝑎(𝑡)𝑐(𝑡)𝑚(𝑡)𝑏(𝑡)𝜙(𝑡)𝑑𝑡+𝛼(𝑡)𝜙(𝑡)𝑑𝐵1𝜙(𝑡),(0)=𝑥0,𝑥(𝑡)𝜙(𝑡)a.s.𝑡0,𝜏𝑒.(2.8) Consequently,𝜙(𝑡)𝑥(𝑡)Φ(𝑡)a.s.𝑡0,𝜏𝑒.(2.9) Next, we consider the predator population 𝑦(𝑡). As the arguments above, we can get[]𝑑𝑦(𝑡)𝑦(𝑡)𝑑(𝑡)+𝑓(𝑡)𝑑𝑡𝛽(𝑡)𝑦(𝑡)𝑑𝐵2(𝑡),𝑑𝑦(𝑡)𝑑(𝑡)𝑦(𝑡)𝑑𝑡𝛽(𝑡)𝑦(𝑡)𝑑𝐵2(𝑡).(2.10) Let𝑦(𝑡)=𝑦0exp𝑡0𝛽𝑑(𝑠)+2(𝑠)2𝑑𝑠𝑡0𝛽(𝑠)𝑑𝐵2,(𝑠)𝑦(𝑡)=𝑦0exp𝑡0𝛽𝑑(𝑠)+𝑓(𝑠)2(𝑠)2𝑑𝑠𝑡0𝛽(𝑠)𝑑𝐵2.(𝑠)(2.11) By using the comparison theorem of stochastic equations again, we obtain𝑦(𝑡)𝑦(𝑡)𝑦(𝑡)a.s.𝑡0,𝜏𝑒.(2.12)

From the representation of solutions 𝜙(𝑡), Φ(𝑡), 𝑦(𝑡), and 𝑦(𝑡), we can easily see that they exist on 𝑡[0,), that is, 𝜏𝑒=. Therefore, we get the following theorem.

Theorem 2.2. For any initial value 𝑋0𝑅2+, there is a unique positive solution 𝑋(𝑡) to (1.6) on 𝑡0 and the solution will remain in 𝑅2+ with probability 1, namely, 𝑋(𝑡)𝑅2+ for all 𝑡0a.s. Moreover, there exist functions 𝜙(𝑡), Φ(𝑡), 𝑦(𝑡), and 𝑦(𝑡) defined as above such that𝜙(𝑡)𝑥(𝑡)Φ(𝑡),𝑦(𝑡)𝑦(𝑡)𝑦(𝑡),a.s.𝑡0.(2.13)

3. Asymptotic Bounded Properties

In Section 2, we have shown that the solution of (1.6) is positive, which will not explode in any finite time. This nice positive property allows to further discuss asymptotic bounded properties for the solution of (1.6) in this section.

Lemma 3.1 (see [30]). Let Φ(𝑡) be a solution of system (2.4). If 𝑏𝑙>0, then limsup𝑡𝐸[]𝑎Φ(𝑡)𝑢𝑏𝑙.(3.1)

Now we show that the solution of system (1.6) with any positive initial value is uniformly bounded in mean.

Theorem 3.2. If 𝑏𝑙>0 and 𝑑𝑙>0, then the solution 𝑋(𝑡) of system (1.6) with any positive initial value has the following properties: limsup𝑡𝐸[]𝑎𝑥(𝑡)𝑢𝑏𝑙,limsup𝑡𝐸𝑐𝑥(𝑡)+𝑙𝑓𝑢𝑦(𝑡)(𝑎𝑢+𝑑𝑢)24𝑏𝑙𝑑𝑙,(3.2) that is, it is uniformly bounded in mean. Furthermore, if 𝑐𝑙>0, then limsup𝑡𝐸[𝑦]𝑓(𝑡)𝑢(𝑎𝑢+𝑑𝑢)24𝑏𝑙𝑐𝑙𝑑𝑙.(3.3)

Proof. Combining 𝑥(𝑡)Φ(𝑡)a.s. with (3.1), it is easy to see that limsup𝑡𝐸[]𝑎𝑥(𝑡)𝑢𝑏𝑙.(3.4) Next, we will show that 𝑦(𝑡) is bounded in mean. Denote 𝑐𝐺(𝑡)=𝑥(𝑡)+𝑙𝑓𝑢𝑦(𝑡).(3.5) Calculating the time derivative of 𝐺(𝑡) along system (1.6), we get 𝑑𝐺(𝑡)=𝑥(𝑡)𝑎(𝑡)𝑏(𝑡)𝑥(𝑡)𝑐(𝑡)𝑦(𝑡)𝑚(𝑠)𝑦(𝑡)+𝑥(𝑡)𝑑𝑡+𝛼(𝑡)𝑥(𝑡)𝑑𝐵1𝑐(𝑡)+𝑦(𝑡)𝑙𝑓𝑢𝑐𝑑(𝑡)+𝑙𝑓𝑢𝑓(𝑡)𝑥(𝑡)𝑐𝑚(𝑠)𝑦(𝑡)+𝑥(𝑡)𝑑𝑡𝑙𝑓𝑢𝛽(𝑡)𝑦(𝑡)𝑑𝐵2=[](𝑡)𝑎(𝑡)+𝑑(𝑡)𝑥(𝑡)𝑏(𝑡)𝑥2𝑐(𝑡)𝑑(𝑡)𝐺(𝑡)+𝑐(𝑡)+𝑙𝑓𝑢𝑓(𝑡)𝑥(𝑡)𝑦(𝑡)𝑚(𝑠)𝑦(𝑡)+𝑥(𝑡)𝑑𝑡+𝛼(𝑡)𝑥(𝑡)𝑑𝐵1𝑐(𝑡)𝑙𝑓𝑢𝛽(𝑡)𝑦(𝑡)𝑑𝐵2(𝑡).(3.6) Integrating it from 0 to 𝑡 yields 𝐺(𝑡)=𝐺(0)+𝑡0[]𝑎(𝑠)+𝑑(𝑠)𝑥(𝑠)𝑏(𝑠)𝑥2(+𝑐𝑠)𝑑(𝑠)𝐺(𝑠)𝑐(𝑠)+𝑙𝑓𝑢𝑓(𝑠)𝑥(𝑠)𝑦(𝑠)+𝑚(𝑠)𝑦(𝑠)+𝑥(𝑠)𝑑𝑠𝑡0𝛼(𝑠)𝑥(𝑠)𝑑𝐵1(𝑠)𝑡0𝑐𝑙𝑓𝑢𝛽(𝑠)𝑑𝐵2(𝑠),(3.7) which implies 𝐸[]𝐺(𝑡)=𝐺(0)+𝐸𝑡0[]𝑎(𝑠)+𝑑(𝑠)𝑥(𝑠)𝑏(𝑠)𝑥2(+𝑐𝑠)𝑑(𝑠)𝐺(𝑠)𝑐(𝑠)+𝑙𝑓𝑢𝑓(𝑠)𝑥(𝑠)𝑦(𝑠)[]𝑚(𝑠)𝑦(𝑠)+𝑥(𝑠)𝑑𝑠,𝑑𝐸𝐺(𝑡)=[]𝐸[]𝑥𝑑𝑡𝑎(𝑡)+𝑑(𝑡)𝑥(𝑡)𝑏(𝑡)𝐸2[]+𝑐(𝑡)𝑑(𝑡)𝐸𝐺(𝑡)𝑐(𝑡)+𝑙𝑓𝑢𝐸𝑓(𝑡)𝑥(𝑡)𝑦(𝑡)[]𝐸[]𝑥𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡)𝑎(𝑡)+𝑑(𝑡)𝑥(𝑡)𝑏(𝑡)𝐸2[](𝑡)𝑑(𝑡)𝐸𝐺(𝑡)(𝑎𝑢+𝑑𝑢[])𝐸𝑥(𝑡)𝑏𝑙[])(𝐸𝑥(𝑡)2𝑑𝑙𝐸[].𝐺(𝑡)(3.8) Obviously, the maximum value of (𝑎𝑢+𝑑𝑢)𝐸[𝑥(𝑡)]𝑏𝑙𝐸2[𝑥(𝑡)] is (𝑎𝑢+𝑑𝑢)2/4𝑏𝑙, so []𝑑𝐸𝐺(𝑡)𝑑𝑡(𝑎𝑢+𝑑𝑢)24𝑏𝑙𝑑𝑙𝐸[]𝐺(𝑡).(3.9) Thus, we get by the comparison theorem that 0limsup𝑡𝐸[]𝐺(𝑡)(𝑎𝑢+𝑑𝑢)24𝑏𝑙𝑑𝑙.(3.10) Since the solution of system (1.6) is positive, it is clear that limsup𝑡𝐸[𝑦]𝑓(𝑡)𝑢(𝑎𝑢+𝑑𝑢)24𝑏𝑙𝑐𝑙𝑑𝑙.(3.11)

Remark 3.3. Theorem 3.2 tells us that the solution of (1.6) is uniformly bounded in mean.

Remark 3.4. If 𝑎, 𝑏, 𝑐, 𝑑, and 𝑓 are positive constant numbers, we will get Theorem 2.1 in [3].

4. Stochastic Permanence of (1.6)

For population systems, permanence is one of the most important and interesting characteristics, which mean that the population system will survive in the future. In this section, we firstly give two related definitions and some conditions that guarantee that (1.6) is stochastically permanent.

Definition 4.1. Equation (1.6) is said to be stochastically permanent if, for any 𝜀(0,1), there exist positive constants 𝐻=𝐻(𝜀),𝛿=𝛿(𝜀) such that liminf𝑡+𝑃||𝑋||(𝑡)𝐻1𝜀,liminf𝑡+𝑃||𝑋||(𝑡)𝛿1𝜀,(4.1) where 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡)) is the solution of (1.6) with any positive initial value.

Definition 4.2. The solutions of (1.6) are called stochastically ultimately bounded, if, for any 𝜀(0,1), there exists a positive constant 𝐻=𝐻(𝜀) such that the solutions of (1.6) with any positive initial value have the property limsup𝑡+𝑃||𝑋||(𝑡)>𝐻<𝜀.(4.2) It is obvious that if a stochastic equation is stochastically permanent, its solutions must be stochastically ultimately bounded.

Lemma 4.3 (see [30]). One has 𝐸exp𝑡𝑡01𝛼(𝑠)𝑑𝐵(𝑠)=exp2𝑡𝑡0𝛼2(𝑠)𝑑𝑠,0𝑡0𝑡.(4.3)

Theorem 4.4. If 𝑏𝑙>0,𝑐𝑙>0, and 𝑑𝑙>0, then solutions of (1.6) are stochastically ultimately bounded.

Proof. Let 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡)) be an arbitrary solution of the equation with positive initial. By Theorem 3.2, we know that limsup𝑡𝐸[]𝑎𝑥(𝑡)𝑢𝑏𝑙,limsup𝑡𝐸[]𝑓𝑦(𝑡)𝑙(𝑎𝑢+𝑑𝑢)24𝑏𝑙𝑐𝑙𝑑𝑙.(4.4) Now, for any 𝜀>0, let 𝐻1>𝑎𝑢/𝑏𝑙𝜀 and 𝐻2>(𝑎𝑢+𝑑𝑢)2𝑓𝑢/4𝑏𝑙𝑐𝑙𝑑𝑙𝜀. Then, by Chebyshev’s inequality, it follows that 𝑃𝑥(𝑡)>𝐻1𝐸[]𝑥(𝑡)𝐻1𝑃<𝜀,𝑦(𝑡)>𝐻2𝐸[]𝑦(𝑡)𝐻2<𝜀.(4.5) Taking 𝐻=3max{𝐻1,𝐻2}, we have 𝑃||||𝐸[]𝑋(𝑡)>𝐻𝑃{𝑥(𝑡)+𝑦(𝑡)>𝐻}𝑥(𝑡)+𝑦(𝑡)𝐻<23𝜀.(4.6) Hence, limsup𝑡𝑃||𝑋||(𝑡)>𝐻<𝜀.(4.7) This completes the proof of Theorem 4.4.

Lemma 4.5. Let 𝑋(𝑡) be the solution of (1.6) with any initial value 𝑋0𝑅2+. If 𝑟𝑙>0, then limsup𝑡+𝐸1𝑏𝑥(𝑡)𝑢𝑟𝑙,(4.8) where 𝑟(𝑡)=𝑎(𝑡)𝑐(𝑡)/𝑚(𝑡)𝛼2(𝑡).

Proof. Combing (2.7) with Lemma 4.3, we have 𝐸1𝜙(𝑡)=𝑥01𝐸exp𝑡0𝑎(𝑠)𝑐(𝑠)𝛼𝑚(𝑠)2(𝑠)2𝑑𝑠𝑡0𝛼(𝑠)𝑑𝐵1(𝑠)+𝐸𝑡0𝑏(𝑠)exp𝑡𝑠𝑎𝑐(𝜏)(𝜏)𝛼𝑚(𝜏)2(𝜏)2𝑑𝜏𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠=𝑥01exp𝑡0𝑎(𝑠)𝑐(𝑠)𝛼𝑚(𝑠)2(𝑠)2𝐸𝑑𝑠exp𝑡0𝛼(𝑠)𝑑𝐵1+(𝑠)𝑡0𝑏(𝑠)exp𝑡𝑠𝑎(𝜏)𝑐(𝜏)𝛼𝑚(𝜏)2(𝜏)2𝐸𝑑𝜏exp𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠=𝑥01exp𝑡0+𝑟(𝑠)𝑑𝑠𝑡0𝑏(𝑠)exp𝑡𝑠𝑟(𝜏)𝑑𝜏𝑑𝑠𝑥01𝑒𝑟𝑙𝑡+𝑏𝑢𝑡0𝑒𝑟𝑙(𝑡𝑠)𝑑𝑠𝑥01𝑒𝑟𝑙𝑡+𝑏𝑢𝑟𝑙.(4.9) From (2.9), it has 𝐸11𝑥(𝑡)𝐸𝜙(𝑡)𝑥01𝑒𝑟𝑙𝑡.+𝑏𝑢𝑟𝑙.(4.10) This completes the proof of Lemma 4.3.

Theorem 4.6. Let 𝑋(𝑡) be the solution of (1.6) with any initial value 𝑋0𝑅2+. If 𝑏𝑙>0 and 𝑟𝑙>0, then, for any 𝜀>0, there exist positive constants 𝛿=𝛿(𝜀) and 𝐻=𝐻(𝜀) such that liminf𝑡+𝑃{𝑥(𝑡)𝐻}1𝜀,liminf𝑡+𝑃{𝑥(𝑡)𝛿}1𝜀.(4.11)

Proof. By Theorem 3.2, there exists a positive constant 𝑀 such that 𝐸[𝑥(𝑡)]𝑀. Now, for any 𝜀>0, let 𝐻=𝑀/𝜀. Then, by Chebyshev’s inequality, we obtain 𝐸[]𝑃{𝑥(𝑡)>𝐻}𝑥(𝑡)𝐻𝜀,(4.12) which implies 𝑃{𝑥(𝑡)𝐻}1𝜀.(4.13) By Lemma 4.3, we know that limsup𝑡+𝐸1𝑏𝑥(𝑡)𝑢𝑟𝑙.(4.14) For any 𝜀>0, let 𝛿=𝜀𝑟𝑙/𝑏𝑢; then 1𝑃{𝑥(𝑡)<𝛿}=𝑃>1𝑥(𝑡)𝛿𝐸[]1/𝑥(𝑡)[]1/𝛿𝛿𝐸1/𝑥(𝑡),(4.15) which yields limsup𝑡+𝑃[]𝑥(𝑡)<𝛿𝛿𝑏𝑢𝑟𝑙=𝜀.(4.16) This implies liminf𝑡+𝑃[]𝑥(𝑡)𝛿1𝜀.(4.17) This completes the proof of Theorem 4.6

Remark 4.7. Theorem 4.6 shows that if we guarantee 𝑏𝑙>0 and 𝑟𝑙>0, then the prey species 𝑥 must be permanent. Otherwise, the prey species 𝑥 may be extinct. Thus the predator species 𝑦 will be extinct too whose survival is absolutely dependent on 𝑥. However, if 𝑦 becomes extinct, then 𝑥 will not turn to extinct when the noise intensities 𝛼(𝑡) are sufficiently small in the sense that 𝑏𝑙>0 and 𝑟𝑙>0.

Theorem 4.8. If 𝑏𝑙>0,𝑐𝑙>0,𝑑𝑙>0, and 𝑟𝑙>0, then (1.6) is stochastically permanent.

Proof. Assume that 𝑋(𝑡) is an arbitrary solution of the equation with initial value 𝑋0𝑅2+. By Theorem 4.6, for any 𝜀>0, there exists a positive constant 𝛿 such that liminf𝑡+𝑃{𝑥(𝑡)𝛿}1𝜀.(4.18) Hence, liminf𝑡+𝑃||𝑋||(𝑡)𝛿liminf𝑡+𝑃{𝑥(𝑡)𝛿}1𝜀.(4.19)
For any 𝜀>0, we have by Theorem 4.4 that liminf𝑡+𝑃||𝑋||(𝑡)𝐻1𝜀.(4.20) This completes the proof of Theorem 4.8

Remark 4.9. Theorem 4.8 shows that if we guarantee 𝑏𝑙>0, 𝑐𝑙>0, 𝑑𝑙>0, and 𝑟𝑙>0, (1.6) is permanent in probability, that is, the total number of predators and their preys is bounded in probability.

Lemma 4.10. Assume that 𝑋(𝑡) is the solution of (1.6) with any initial value 𝑋0𝑅2+. If 𝜌𝑙>0 and 𝜎𝑙>0, then limsup𝑡+𝐸1𝑦(𝑡)𝑦01+𝑓𝑢𝑚𝑢2𝑥01+2(𝑏𝑢)2𝜌𝑙21/2,(4.21) where 𝜌(𝑡)=𝑎(𝑡)𝑐(𝑡)/𝑚(𝑡)3/2𝛼2(𝑡), 𝜎(𝑡)=𝑓(𝑡)𝑑(𝑡)3/2𝛽2(𝑡).

Proof. By (2.9), it is easy to have 𝑑𝑦(𝑡)=𝑦(𝑡)𝑑(𝑡)+𝑓(𝑡)𝑓(𝑡)𝑚(𝑡)𝑦(𝑡)𝑚(𝑡)𝑦(𝑡)+𝑥(𝑡)𝑑𝑡𝛽(𝑡)𝑦(𝑡)𝑑𝐵2(𝑡)𝑦(𝑡)𝑑(𝑡)+𝑓(𝑡)𝑓(𝑡)𝑚(𝑡)𝑦(𝑡)𝑥(𝑡)𝑑𝑡𝛽(𝑡)𝑦(𝑡)𝑑𝐵2(𝑡)𝑦(𝑡)𝑑(𝑡)+𝑓(𝑡)𝑓(𝑡)𝑚(𝑡)𝑦(𝑡)𝜙(𝑡)𝑑𝑡𝛽(𝑡)𝑦(𝑡)𝑑𝐵2(𝑡).(4.22) Let Ψ(𝑡) be the unique solution of equation 𝑑Ψ(𝑡)=Ψ(𝑡)𝑑(𝑡)+𝑓(𝑡)𝑓(𝑡)𝑚(𝑡)𝜙(𝑡)Ψ(𝑡)𝑑𝑡𝛽(𝑡)Ψ(𝑡)𝑑𝐵2Ψ(𝑡),(0)=𝑦0.(4.23) Then, by the comparison theorem of stochastic equations, we have 𝑦(𝑡)Ψ(𝑡),(4.24)Ψ(𝑡)=exp𝑡0𝑑(𝑠)+𝑓(𝑠)(1/2)𝛽2(𝑠)𝑑𝑠𝑡0𝛽(𝑠)𝑑𝐵2(𝑠)𝑦01+𝑡0(𝑓(𝑠)𝑚(𝑠)/𝜙(𝑠))exp𝑠0𝑑(𝜏)+𝑓(𝜏)(1/2)𝛽2(𝜏)𝑑𝜏𝑠0𝛽(𝜏)𝑑𝐵2.(𝜏)𝑑𝑠(4.25) So, Ψ1(𝑡)=𝑦01exp𝑡01𝑑(𝑠)𝑓(𝑠)+2𝑡0𝛽2(𝑠)𝑑𝑠+𝛽(𝑠)𝑑𝐵2(+𝑠)𝑡0𝑓(𝑠)𝑚(𝑠)𝜙(𝑠)exp𝑡𝑠1𝑑(𝜏)𝑓(𝜏)+2𝛽2(𝜏)𝑑𝜏+𝑡𝑠𝛽(𝜏)𝑑𝐵2(𝜏)𝑑𝑠.(4.26) Denote 1𝜆(𝑡)=𝑑(𝑡)𝑓(𝑡)+2𝛽2𝛼(𝑡),𝜈(𝑡)=𝑎(𝑡)2(𝑡)2𝑐(𝑡)𝑚(𝑡).(4.27) By Lemma 4.3 and Hölder’s inequality, it is easy to get that 𝐸Ψ1(𝑡)=𝑦01exp𝑡0𝑑(𝑠)𝑓(𝑠)+𝛽2(+𝑠)𝑑𝑠𝑡0𝑓(𝑠)𝑚(𝑠)exp𝑡𝑠𝐸𝜙𝜆(𝜏)𝑑𝜏1(𝑠)exp𝑡𝑠𝛽(𝜏)𝑑𝐵2(𝜏)𝑑𝑠𝑦01exp𝑡0𝑑(𝑠)𝑓(𝑠)+𝛽2+(𝑠)𝑑𝑠𝑡0𝑓(𝑠)𝑚(𝑠)exp𝑡𝑠𝜆𝐸𝜙(𝜏)𝑑𝜏2𝐸2(𝑠)exp𝑡𝑠𝛽(𝜏)𝑑𝐵2(𝜏)1/2𝑑𝑠𝑦01exp𝑡0𝑑(𝑠)𝑓(𝑠)+𝛽2+(𝑠)𝑑𝑠𝑡0𝑓(𝑠)𝑚(𝑠)exp𝑡𝑠3𝑑(𝜏)𝑓(𝜏)+2𝛽2(𝐸𝜙𝜏)𝑑𝜏2(𝑠)1/2𝑑𝑠.(4.28) Combing (𝑎+𝑏)22(𝑎2+𝑏2) with (2.7), it follows that 𝐸𝜙2(𝑥𝑡)=𝐸01exp𝑡0𝜈(𝑠)𝑑𝑠𝑡0𝛼(𝑠)𝑑𝐵1(+𝑠)𝑡0𝑏(𝑠)exp𝑡𝑠𝜈(𝜏)𝑑𝜏𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠22𝑥02𝐸exp2𝑡0𝜈(𝑠)𝑑𝑠2𝑡0𝛼(𝑠)𝑑𝐵1(𝑠)+2𝐸𝑡0𝑏(𝑠)exp𝑡𝑠𝜈(𝜏)𝑑𝜏𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠2.(4.29) It is easy to compute that 𝐸𝑡0𝑏(𝑠)exp𝑡𝑠𝜈(𝜏)𝑑𝜏𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠2=𝐸𝑡0𝑏(𝑠)𝑏(𝑢)exp𝑡𝑠𝜈(𝜏)𝑑𝜏𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)exp𝑡𝑢𝜈(𝜏)𝑑𝜏𝑡𝑢𝛼(𝜏)𝑑𝐵1=(𝜏)𝑑𝑢𝑑𝑠𝑡0𝑏(𝑠)𝑏(𝑢)exp𝑡𝑠𝜈(𝜏)𝑑𝜏exp𝑡𝑢𝜈(𝜏)𝑑𝜏𝐸exp𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)exp𝑡𝑢𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑢𝑑𝑠.(4.30) By Hölder’s inequality again, 𝐸exp𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)exp𝑡𝑢𝛼(𝜏)𝑑𝐵1(𝐸𝜏)exp2𝑡𝑠𝛼(𝜏)𝑑𝐵1𝐸(𝜏)exp2𝑡𝑢𝛼(𝜏)𝑑𝐵1(𝜏)1/2=exp𝑡𝑠𝛼2(𝜏)𝑑𝜏exp𝑡𝑢𝛼2.(𝜏)𝑑𝜏(4.31) Substituting (4.31) into (4.30) yields 𝐸𝑡0𝑏(𝑠)exp𝑡𝑠𝜈(𝜏)𝑑𝜏𝑡𝑠𝛼(𝜏)𝑑𝐵1(𝜏)𝑑𝑠2=𝑡0𝑏(𝑠)exp𝑡𝑠𝜈(𝜏)𝑑𝜏+𝑡𝑠𝛼2(𝜏)𝑑𝜏𝑑𝑠2=𝑡0𝑏(𝑠)exp𝑡𝑠𝑎(𝜏)𝑐(𝜏)3𝑚(𝜏)2𝛼2(𝜏)𝑑𝜏𝑑𝑠2.(4.32) On the other hand, by (4.29) and (4.32), we get 𝐸𝜙2(𝑡)2𝑥02exp2𝑡0𝜌(𝑠)𝑑𝑠+2𝑡0𝑏(𝑠)exp𝑡𝑠𝜌(𝜏)𝑑𝜏𝑑𝑠22𝑥01exp2𝜌𝑙𝑡𝑏+2𝑢𝜌𝑙22𝑥01𝑏+2𝑢𝜌𝑙2.(4.33) Finally, substituting (4.33) into (4.28) and noting from (4.24), we obtain the required assertion (4.21).

By Theorem 3.2 and Lemma 4.10, similar to the proof of Theorem 4.6, we obtain the following result.

Theorem 4.11. Let 𝑋(𝑡) be the solution of (1.6) with any initial value 𝑋0𝑅2+. If 𝑏𝑙>0, 𝑐𝑙>0, 𝑑𝑙>0, 𝜌𝑙>0, and 𝜎𝑙>0, then, for any 𝜀>0, there exist positive constants 𝛿=𝛿(𝜀), 𝐻=𝐻(𝜀) such that liminf𝑡+𝑃{𝑦(𝑡)𝐻}1𝜀,liminf𝑡+𝑃{𝑦(𝑡)𝛿}1𝜀.(4.34)

Remark 4.12. Theorem 4.11 shows that if 𝑏𝑙>0, 𝑐𝑙>0, 𝑑𝑙>0, 𝜌𝑙>0, and 𝜎𝑙>0, then the predator species 𝑦 must be permanent in probability. This implies that species prey 𝑥 and (1.6) are permanent in probability. In other words, the predator species 𝑦 and species prey 𝑥 in (1.6) are both permanent in probability.

Remark 4.13. Obviously, system (1.4) is a special case of system (1.6). If 𝑎(3/2)𝛼2(𝑐/𝑚)>0and𝑓𝑑(3/2)𝛽2>0, then, by Theorem  3.3 in [3], (1.4) is persistent in mean, but, by our Theorem 4.11, the predator species 𝑦 and species prey 𝑥 in (1.4) are both stochastically permanent.

5. Conclusions

In this paper, by the comparison theorem of stochastic equations and the Itô formula, some results are established such as the stochastically ultimate boundedness and stochastic permanence for a stochastic ratio-dependent predator-prey model with variable coefficients. It is seen that several results in this paper extend and improve the earlier publications (see Remark 3.4).

Acknowledgments

The authors are grateful to the Editor Professor Ying U. Hu and anonymous referees for their helpful comments and suggestions that have improved the quality of this paper. This work is supported by the Natural Science Foundation of China (no. 10771001), Research Fund for Doctor Station of Ministry of Education of China (no. 20103401120002, no. 20113401110001), TIAN YUAN Series of Natural Science Foundation of China (no. 11126177), Key Natural Science Foundation (no. KJ2009A49), and Talent Foundation (no. 05025104) of Anhui Province Education Department, 211 Project of Anhui University (no. KJJQ1101), Anhui Provincial Nature Science Foundation (no. 090416237, no. 1208085QA15), Foundation for Young Talents in College of Anhui Province (no. 2012SQRL021).

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