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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 857134, 17 pages
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.
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.
Ecological systems are mainly characterized by the interaction between species and their surrounding natural environment . 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 [2–4]. The interaction mechanism of predators and their preys can be described as differential equations, such as Lotaka-Volterra models .
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) [6–8]. 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 [9–11].
It is worthy to note that, based on the Michaelis-Menten or Holling type II function, Arditi and Ginzburg  firstly proposed a ratio-dependent function of the form and a ratio-dependent predator-prey model of the form Here, and represent population densities of the prey and the predator at time , respectively. Parameters , , , , , and are positive constants in which is the carrying capacity of the prey, , , , , and 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 [12–19].
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 and represent the intrinsic growth and death rate of and , 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 and by respectively, where and are mutually independent Brownian motions and and represent the intensities of the white noises. As a result, (1.2) becomes a stochastic differential equation (SDE, in short): By the Itô formula, Ji et al.  showed that (1.4) is persistent or extinct in some conditions.
The predator-prey model describes a prey population that serves as food for a predator . 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 [22–26]. Therefore, Zhang and Hou  studied the following general ratio-dependent predator-prey model of the form: which is more realistic. Motivated by [3, 27], this paper is concerned with a stochastic ratio-dependent predator-prey model of the following form: where , , , , , and are positive bounded continuous functions on and , are bounded continuous functions on , and and are defined in (1.4). There would be some difficulties in studying this model since the parameters are changed by time . Under some suitable conditions, we obtain some results such as the stochastic permanence of (1.6).
Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let and be mutually independent Brownian motions, the positive cone in , , and .
For convenience and simplicity in the following discussion, we use the notation where is a bounded continuous function on .
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 and in (1.6) are population densities of the prey and the predator at time , 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 . 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 , 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 , there is a unique positive local solution to (1.6) on .
Proof. We first consider the equation on with initial value , . Since the coefficients of system (2.1) satisfy the local Lipschitz condition, there is a unique local solution on , where is the explosion time . Therefore, by the Itô formula, it is easy to see that , is the unique positive local solution of system (2.1) with initial value . Lemma 2.1 is finally proved.
Since the solution is positive, we have Let Then, is the unique solution of equation by the comparison theorem of stochastic equations. On the other hand, we have Similarly, is the unique solution of equation Consequently, Next, we consider the predator population . As the arguments above, we can get Let By using the comparison theorem of stochastic equations again, we obtain
From the representation of solutions , , , and , we can easily see that they exist on , that is, . Therefore, we get the following theorem.
Theorem 2.2. For any initial value , there is a unique positive solution to (1.6) on and the solution will remain in with probability 1, namely, for all . Moreover, there exist functions , , , and defined as above such that
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.
Now we show that the solution of system (1.6) with any positive initial value is uniformly bounded in mean.
Theorem 3.2. If and , then the solution of system (1.6) with any positive initial value has the following properties: that is, it is uniformly bounded in mean. Furthermore, if , then
Proof. Combining . with (3.1), it is easy to see that Next, we will show that is bounded in mean. Denote Calculating the time derivative of along system (1.6), we get Integrating it from 0 to yields which implies Obviously, the maximum value of is , so Thus, we get by the comparison theorem that Since the solution of system (1.6) is positive, it is clear that
Remark 3.4. If , , , , and are positive constant numbers, we will get Theorem 2.1 in .
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 4.2. The solutions of (1.6) are called stochastically ultimately bounded, if, for any , there exists a positive constant such that the solutions of (1.6) with any positive initial value have the property It is obvious that if a stochastic equation is stochastically permanent, its solutions must be stochastically ultimately bounded.
Lemma 4.3 (see ). One has
Theorem 4.4. If , and , then solutions of (1.6) are stochastically ultimately bounded.
Proof. Let be an arbitrary solution of the equation with positive initial. By Theorem 3.2, we know that Now, for any , let and . Then, by Chebyshev’s inequality, it follows that Taking , we have Hence, This completes the proof of Theorem 4.4.
Lemma 4.5. Let be the solution of (1.6) with any initial value . If , then where .
Theorem 4.6. Let be the solution of (1.6) with any initial value . If and , then, for any , there exist positive constants and such that
Proof. By Theorem 3.2, there exists a positive constant such that . Now, for any , let . Then, by Chebyshev’s inequality, we obtain which implies By Lemma 4.3, we know that For any , let ; then which yields This implies This completes the proof of Theorem 4.6
Remark 4.7. Theorem 4.6 shows that if we guarantee and , then the prey species must be permanent. Otherwise, the prey species may be extinct. Thus the predator species will be extinct too whose survival is absolutely dependent on . However, if becomes extinct, then will not turn to extinct when the noise intensities are sufficiently small in the sense that and .
Theorem 4.8. If , and , then (1.6) is stochastically permanent.
Proof. Assume that is an arbitrary solution of the equation with initial value . By Theorem 4.6, for any , there exists a positive constant such that
For any , we have by Theorem 4.4 that This completes the proof of Theorem 4.8
Lemma 4.10. Assume that is the solution of (1.6) with any initial value . If and , then where , .
Proof. By (2.9), it is easy to have Let be the unique solution of equation Then, by the comparison theorem of stochastic equations, we have So, Denote By Lemma 4.3 and Hölder’s inequality, it is easy to get that Combing with (2.7), it follows that It is easy to compute that By Hölder’s inequality again, Substituting (4.31) into (4.30) yields On the other hand, by (4.29) and (4.32), we get Finally, substituting (4.33) into (4.28) and noting from (4.24), we obtain the required assertion (4.21).
Theorem 4.11. Let be the solution of (1.6) with any initial value . If , , , , and , then, for any , there exist positive constants , such that
Remark 4.12. Theorem 4.11 shows that if , , , , and , then the predator species must be permanent in probability. This implies that species prey and (1.6) are permanent in probability. In other words, the predator species and species prey in (1.6) are both permanent in probability.
Remark 4.13. Obviously, system (1.4) is a special case of system (1.6). If , then, by Theorem 3.3 in , (1.4) is persistent in mean, but, by our Theorem 4.11, the predator species and species prey in (1.4) are both stochastically permanent.
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).
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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