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

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 2– 4 . 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 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 6 firstly proposed a ratio-dependent function of the form


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 2-4 .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 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 6 firstly proposed a ratio-dependent function of the form 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 by For convenience and simplicity in the following discussion, we use the notation 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.

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 X 0 ∈ R 2 , there is a unique positive local solution X t to 1.6 on t ∈ 0, τ e a.s.
Proof.We first consider the equation f t e u t m t e v t e u t dt − β t dB 2 t 2.1 on t ≥ 0 with initial value u 0 ln x 0 , v 0 ln y 0 .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, τ e , where τ e is the explosion time 28 .Therefore, by the It ô formula, it is easy to see that x t e u t , y t e v t is the unique positive local solution of system 2.1 with initial value X 0 x 0 , y 0 ∈ R 2 .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, τ e ∞.
Since the solution is positive, we have Then, Φ t is the unique solution of equation by the comparison theorem of stochastic equations.On the other hand, we have Similarly,

2.11
By using the comparison theorem of stochastic equations again, we obtain y t ≤ y t ≤ y t a.s.t ∈ 0, τ e .

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, τ e ∞.Therefore, we get the following theorem.Theorem 2.2.For any initial value X 0 ∈ R 2 , there is a unique positive solution X t to 1.6 on t ≥ 0 and the solution will remain in R 2 with probability 1, namely, X t ∈ R 2 for all t ≥ 0 a.s.Moreover, there exist functions φ t , Φ t , y t , and y t defined as above such that φ t ≤ x t ≤ Φ t , y t ≤ y t ≤ y t , a.s.t ≥ 0. 2.13

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 b l > 0, then Now we show that the solution of system 1.6 with any positive initial value is uniformly bounded in mean.Theorem 3.2.If b l > 0 and d l > 0, then the solution X t of system 1.6 with any positive initial value has the following properties: that is, it is uniformly bounded in mean.Furthermore, if c l > 0, then Next, we will show that y t is bounded in mean.Denote G t x t c l f u y t .

3.5
Calculating the time derivative of G t along system 1.Integrating it from 0 to t yields which implies

3.8
Obviously, the maximum value of

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.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 lim sup Now, for any ε > 0, let H 1 > a u /b l ε and H 2 > a u d u 2 f u /4b l c l d l ε.Then, by Chebyshev's inequality, it follows that

4.5
Taking H 3 max{H 1 , H 2 }, we have Hence, lim sup This completes the proof of Theorem 4.4.
Lemma 4.5.Let X t be the solution of 1.6 with any initial value where r t a t − c t /m t − α 2 t .
Proof.Combing 2.7 with Lemma 4.3, we have 4.9 From 2.9 , it has This completes the proof of Lemma 4.3.
Theorem 4.6.Let X t be the solution of 1.6 with any initial value X 0 ∈ R 2 .If b l > 0 and r l > 0, then, for any ε > 0, there exist positive constants δ δ ε and H H ε such that This completes the proof of Theorem 4.6 Remark 4.7.Theorem 4.6 shows that if we guarantee b l > 0 and r l > 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 b l > 0 and r l > 0.
Proof.Assume that X t is an arbitrary solution of the equation with initial value X 0 ∈ R 2 .By Theorem 4.6, for any ε > 0, there exists a positive constant δ such that lim inf For any ε > 0, we have by Theorem 4.4 that lim inf This completes the proof of Theorem 4.8 Remark 4.9.Theorem 4.8 shows that if we guarantee b l > 0, c l > 0, d l > 0, and r l > 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 X 0 ∈ R 2 .If ρ l > 0 and σ l > 0, then

4.23
Then, by the comparison theorem of stochastic equations, we have Theorem 4.11.Let X t be the solution of 1.6 with any initial value X 0 ∈ R 2 .If b l > 0, c l > 0, d l > 0, ρ l > 0, and σ l > 0, then, for any ε > 0, there exist positive constants δ δ ε , H H ε such that

4.34
Remark 4.12.Theorem 4.11 shows that if b l > 0, c l > 0, d l > 0, ρ l > 0, and σ l > 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.

Conclusions
In this paper, by the comparison theorem of stochastic equations and the It ô 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 .
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 results12-19 .
By the It ô formula,Ji et al. 3showed 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 22-26 ., 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 B 1 t and B 2 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, 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 F 0 contains all P -null sets .Let B 1 t and B 2 t be mutually independent t , 1.3 respectively, where B 1 t and B 2 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 : 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 .
If b l > 0, c l > 0, and d l > 0, then solutions of 1.6 are stochastically ultimately bounded.

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 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.