The Complex Dynamics of a Stochastic Predator-Prey Model

and Applied Analysis 3 From the second equation of model 1.3 , we see that, for t > T , we have dy/dt ≤ ey 1 − bfy/ mb 1 . A standard comparison argument shows that lim sup t→∞ y t ≤ 1 bm bf . 2.2 Thus, we have the following conclusion. Lemma 2.1. Model 1.3 is dissipative. Lemma 2.2. If c < n, then model 1.3 is permanent. Proof. If c < n, from the first equation of model 1.3 , we have dx/dt ≥ ax 1 − bx − c/n . Therefore, by standard comparison argument, we have lim inf t→∞ x t ≥ n − c bn . 2.3 Hence, for any ε > 0 and large t, x t > n − c /bn − ε, and dy dt ≥ ey ( bmn nc − bnε − bfny bmn nc − bnε ) . 2.4 From the arbitrariness of ε > 0, we can get that lim inf t→∞ y t ≥ bmn n − c bfn . 2.5 2.2. Stability Analysis of the Equilibria In this section, we will focus on the existence of equilibria and their stabilities of model 1.3 . It is easy to find that model 1.3 always has three boundary equilibria E0 0, 0 , E1 1/b, 0 , E2 0, m/f . And the positive equilibria x, y satisfies the equations ax ( 1 − bx − cy x ny ) 0, ey ( 1 − fy m x ) 0, 2.6


Introduction
The dynamic relationship between predators and their prey 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 1 . In recent years, one of important predator-prey models is Leslie-Gower model 2, 3 , which has been extensively studied 4, 5 . And, more and more obvious evidences of biology and physiology show that in many conditions, especially when the predators have to search for food consequently, have to share or compete for food , a more realistic and general predator-prey system should rely on the theory of ratio-dependence, this is confirmed by lots of experimental results 4, 6 . A ratio-dependent Leslie-Gower predator-prey  where x x t , y y t stand for the population the density of the prey and the predator at time t, respectively, and p x/y is the predator functional response to prey. And we assumed that the prey grows logistically with growth rate a and carrying capacity 1/b in the absence of predation. The predator consumes the prey according to the functional response p x/y and grow logistically with growth rate e and carrying capacity x/f proportional to the population size of prey or prey abundance . The parameter f is a measure of the food quality that the prey provides for conversion into predator birth. The term fy/x of this equation is called the Leslie-Gower term.
On the other hand, the predator y can switch over to other population when the prey population is severely scarce, but its growth will be limited, because we cannot forget the fact that its most favorite food, the prey x, is not in abundance. In this situation, a positive constant m can be added to the denominator, m measures the extent to which the environment provides protection to the predator 8, 9 ,  Based on the above discussions, in the paper, we will focus on the following ratiodependent Leslie-Gower model: The rest of the paper is organized as follows. In Section 2, we give some theorems about the stability property of the equilibria of model 1.3 . In Section 3, we establish a stochastic model based on model 1.3 and focus on the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability of the stochastic model. In Section 4, we give some numerical examples and make a comparative analysis of the stability of the model system within deterministic and stochastic environments.

Stability Analysis of the Equilibria
In this section, we will focus on the existence of equilibria and their stabilities of model 1.3 . It is easy to find that model 1.3 always has three boundary equilibria E 0 0, 0 , then 2.7 can be rewritten as Lemma 2.3 existence of equilibria . a Suppose A 1 > 0 and B 1 > 0, in 2.9 , then one has a1 If C 2 1 > 0, then it has two positive roots given by

2.10
Therefore, model 1.3 has two positive equilibria E 3 x e1 , y e1 x e2 , y e2 x, y Next, we discuss the local stabilities of these equilibria. Easy to obtain the following results: ii if c > n, E 2 0, m/f is a stable node point. If c ≤ n, E 2 0, m/f is a saddle point.
Abstract and Applied Analysis 5 Following, let E x, y be an arbitrary positive equilibrium. The Jacobian matrix for E x, y is given by Then we can get the following:

2.13
So the sign of det J E is determined by Proof. a At the point E 3 , we have the following: Abstract and Applied Analysis thus, det J E 3 > 0, and the nature of singularity E 3 depends on the trace given by Clearly, a1 If e > ax e1 −b cf m x e1 / fx e1 n m x e1 2 , then tr J E 3 < 0, and a2 If e < ax e1 −b cf m x e1 / fx e1 n m x e1 2 , then tr J E 3 > 0, and b At the point E 4 , we have the following: then det J E 4 < 0, and E 4 is a saddle point. Figure 1 shows the dynamics of model 1.3 . In this case, E 0 0, 0 is a nodal source, E 1 4, 0 is a saddle point, E 2 0, 071429 is a nodal sink, E 4 0.31772, 1.5846 is a saddle point, and E 3 0.96454, 2.3932 is locally asymptotically stable. There exists a separatrix curve determined by the stable manifold of E 4 , which divides the behavior of trajectories, that is, the stable manifold of saddle E 4 split the feasible region into two parts such that orbits initiating inside tend to the positive equilibrium E 3 , while orbits initiating outside tend to E 3 except for the stable manifolds of E 4 .
as F x > 0, then det J E > 0, and the nature of singularity E x, m x /f depends on the trace given by Clearly, as F x * > 0, then det J E * > 0, and the nature of singularity E * x * , m x * /f depends on the trace given by Clearly,

2.29
Proof. At the point x e , m x e /f , we have Hence we can conclude 2.28 and 2.29 .
Abstract and Applied Analysis 9 Theorem 2.9. If n < c < n bm 1 and fn n − c bmn n − c < 0 hold the boundary equilibria Proof. Since lim sup t → ∞ x t ≤ 1/b and lim inf t → ∞ y t ≥ bmn n − c /bfn, from the first equation of model 1.3 , for any μ > 0, there exists a T 1 > 0, for all t ≥ T 1 , we have

2.31
From the arbitrariness of μ > 0, we can get that As 1 − c bmn n − c /n bmn f n − c < 0, by standard comparison arguments, it follows that x t 0.

2.34
As a result, using the second equation of model 1.3 , one can easily know that lim t → ∞ y t m/f. The proof is complete. Figure 2 shows the dynamics of model 1.3 . In this case, E 0 0, 0 is a nodal source, E 1 4, 0 is a saddle point, and E 2 0, 1.1875 is globally asymptotically stable, that is, all orbits tend to the equilibrium E 2 for any initial values.
x * , y * is globally asymptotically stable, if the following conditions hold Proof. Define a Lyapunov function: so,

2.37
Abstract and Applied Analysis

2.41
Considering A 2 > 0, B 2 2 < 4A 2 C 2 , we obtain dV/dt < 0. This ends the proof. Figure 3 shows the dynamics of model 1.3 for the case of B1 < 0. In this case, E 0 0, 0 is a nodal source, E 1 4, 0 is a saddle point, E 2 0, 1.1875 is a saddle point, and E * 1.4761, 3.0326 is globally asymptotically stable, that is, all orbits tend to the equilibrium E * for any initial values.

The Stochastic Model
Those important and useful works on deterministic models provide a great insight into the effect of the pollution. In the real world, population dynamics is inevitably subjected to environmental noise see e.g., 10, 11 , which is an important component in an ecosystem. May 12 pointed out the fact that due to environmental noise, the birth rates, carrying capacity, competition coefficients, and other parameters involved in the system exhibit random fluctuation to a greater or lesser extent.
In this part, we focus on the stochastic stability analysis of model 1. where B i t , i 1, 2 are the 1-dimensional standard 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 F 0 contains all P-null sets andḂ 1 t ,Ḃ 2 t are, respectively, white noises with possible intensity α 2 , β 2 . Proof. Consider the equations

Existence of Global Positive Solutions
3.3 on t ≥ 0 with initial value u 0 ln x 0 , v 0 ln y 0 . It is easy to see that the coefficients of model 3.3 satisfy the local Lipschitz condition, then there is a unique local solution u t , v t on 0, τ e 13 . Therefore, by Itô formula, x t e u t , y t e v t are the unique positive local solutions to model 3.3 with initial value x 0 > 0, y 0 > 0. Lemma 3.1 only tells us that there has a unique local positive solution to model 3.2 . Next, we show this solution is global, that is, τ e ∞. Theorem 3.2. Consider model 3.2 , for any given initial value x 0 , y 0 ∈ R 2 , there is a unique solution x t , y t on t ≥ 0 and the solution will remain in R 2 with probability 1, where R 2 {x ∈ R 2 | x i > 0, i 1, 2}.
Proof. For convenience of statement, we introduce some notations. Define Let n 0 > 0 be sufficiently large for x 0 and y 0 lying within the interval 1/n 0 , n 0 . For each integer n > n 0 , define the stopping times τ n inf t ∈ 0, τ e : x t / ∈ 1 n , n or y t / ∈ 1 n , n .

3.5
Assume inf ∅ ∞ as usual ∅ the empty set . Clearly, τ n is increasing as n → ∞. Let τ ∞ lim n → ∞ τ n , then τ ∞ ≤ τ e a.s. If we can show that τ ∞ ∞ a.s., then τ e ∞ a.s., and N t ∈ R a.s., for all t ≥ 0. In other words, to complete the proof all we need to prove that τ ∞ ∞ a.s. If this statement is false, then there is a pair of constants T > 0 and ε ∈ 0, 1 such that P{τ ∞ ≤ T } > ε. There is an integer n 1 ≥ n 0 , such that P{τ n ≤ T } ≥ ε, n ≥ n 1 . 3.6

14
Abstract and Applied Analysis Define a C 2 function V : Obviously, V x, y is nonnegative. And y G x, y dt 3.8 where k 1 , k 2 are positive numbers. Integrating both sides of the above inequality from 0 to τ n ∧ T and then taking expectations, yields Set Ω n {τ n ≤ T } for n ≥ n 1 and by 3.6 , we have P Ω n ≥ ε. Note that for every ω ∈ Ω n , there is some i such that x i τ n , ω equals either n or 1/n for i 1, 2, hence V x τ n , ω , y τ n , ω is no less than min{ √ n − 1 − ln n/2 , 1/n − 1 − ln 1/n /2 }. It then follows from 3.9 that

3.10
where I Ω n is the indicator function of Ω n . Let n → ∞, then This completes the proof.
Abstract and Applied Analysis 15

Stochastic Boundedness
Let us now recall the definition of stochastically ultimate boundedness 14-16 .
Definition 3. 3. The solution of model 3.2 is said to be stochastically ultimately bounded if for any ε ∈ 0, 1 , there is a positive constant χ χ ω , such that for any initial data x 0 , y 0 ∈ R 2 , the solution of model 3.2 has the property that Lemma 3.4. For any θ ∈ 0, 1 , there is a positive constant H H θ > 0, which is independent of the initial data x 0 , y 0 ∈ R 2 , such that the solution of model 3.2 has the property that Proof. Define V x, y x θ y θ , x, y ∈ R 2 . 3.14 For the sake of discussion, we rewrite the above as By using the Itô formula, we have Abstract and Applied Analysis as 0 < θ < 1, we have

3.17
here H 1 is an integer. So Now, by using the Itô formula again, we have d e t V x, y e t V x, y dt dV x, y ≤ e t H 1 dt e t θαx θ 1 − bx − F x, y dB 1 t e t θβ y θ 1 − G x, y dB 2 t .

3.19
According to the above, we can easily get This ends the proof.
Next, we study the asymptotic properties of the moment solutions of model 3.2 .
Theorem 3.6. For any given θ ∈ 0, 1 , there is a K K θ > 0, let x t , y t be the solution of model 3.2 with any initial value x 0 , y 0 , then Proof. Set V x, y : R 2 → R , from Lemma 3.4, we have 3.27 so, we get

3.28
here M is a positive number, so

3.30
That is,

Stochastic Asymptotic Stability
Note that a solution of model 1.3 is also a solution of model 3.2 , so, in the following, we will focus on stochastic asymptotic stability of the positive equilibria of model 3.2 . As an example, we only give the proof of the unique positive equilibrium E * x * , y * of model 3.2 .  α 0.8, β 0.4, starting with a homogeneous state E * 1.4761, 3.0326 , the random white noise leads to a slight oscillations, and the later random noise makes the oscillations decay, ending with the time-independent stability. Comparing Figure 4 a with Figure 4 b , one can realize that, if the white noise is not strong, the stochastic perturbation does not cause sharp changes of the dynamics of the system. However, in Figure 5, we choose that α 2 and β 4, which violates condition 3.34 , we find that the stochastic model 3.2 is not permanent. This shows that strong white noise might make a permanent system be nonpersistent.
We note that, when the noise is not large, the stochastic model preserves the property of the global stability, that is to say, when the noise is not sufficiently large, the populations may be stochastic permanence and stochastic persistent in mean. In this case, we can ignore the noise and use the deterministic model to describe the population dynamics. But, when the noise is sufficiently large, the noise can force the population to become extinct. In this case, we cannot ignore the effect of the noise. That's to say, in the case of sufficiently large noise, we cannot use deterministic model but stochastic model to describe the population dynamics. Our complete analysis of the model will give new suggestions to the studies of the population dynamics.