DYNAMICAL PROPERTIES OF A STOCHASTIC PREDATOR-PREY MODEL WITH FUNCTIONAL RESPONSE

A stochastic prey-predator model with functional response is investigated in this paper. A complete threshold analysis of coexistence and extinction is obtained. Moreover, we point out that the stochastic predatorprey model undergoes a stochastic Hopf bifurcation from the viewpoint of numerical simulations. Some numerical simulations are carried out to support our results.


Introduction
Predator-prey dynamics is one of the dominant fields in both theoretical and applied ecology, which has encouraged numerous researchers to develop various mathematical models to better understand it over the last few decades [2,22,25]. In population dynamics, the functional response is one of the nonlinear components in biological systems, which describes the feeding rate of prey consumption by predators, and plays a key role in understanding the dynamical complexity of the systems [16,18].
A predator-prey model with the Crowley-Martin functional response is described as follows: x, growth rate of prey, c represents the growth rate of predator when it's positive and the death rate when it's negative. f stands for the conversion rate of nutrients into predator production, while a, b measure the competition strength among individuals of prey and predator respectively. In recent years, there were some relevant predator-prey models with this type of functional response [3,20,24,26,32].
As a matter of fact, environmental noises play an inevitable role in population dynamics and always contribute to random fluctuations on parameters appearing in ecosystems [9,10,21]. Therefore, we take the influence of randomly fluctuating environment into account. After incorporating white noise into the system (1.1), we consider the following stochastic system: where B 1 (t), B 2 (t) are mutually independent Brownian motions, σ 2 1 and σ 2 2 represent the intensities of white noise.
As this kind of stochastic model accommodates interference among predators and preys and is a better fit to the experimental data, we believe it deserves further attention. Some literatures used the corresponding stochastic model to describe the dynamic properties [18,19,27,28]. Liu et al [18] studied stochastic boundedness, stochastic permanence and extinction for a corresponding stochastic system with Crowley-Martin functional response. Zhang et al [28] showed the existence, boundedness and uniform continuity of the positive solution for a stochastic population system with this kind of functional response.
The threshold analysis of strong stochastic persistence and extinction is given for some stochastic population models [29][30][31]. However, to the best of our knowledge, literatures on the threshold analysis of coexistence and extinction, stochastic Hopf bifurcation for the stochastic predator-prey system (1.2) have not yet appeared. The Crowley-Martin functional response is a generalization of Holling type and Beddington-DeAngelis functional responses. And the parameter c may be positive or negative. If c > 0, the species y has extra source of food except x, however, if c < 0, the species y has no extra source of food except x. Both the two cases are considered in this paper. The aim of this paper is to investigate these issues for the system (1.2).
In Section 2, we obtain a complete threshold analysis of coexistence and extinction. Section 3 considers stochastic Hopf bifurcation of the stochastic predator-prey model (1.2) from the viewpoint of numerical simulations. A final discussion concludes the paper in Section 4.
To illustrate the asymptotic behaviors of the sample paths of the solution discussed above clearly, we show them in Table 1.
exists, then the discussions are similar to those appearing in (A2) and (A3) of Case A. ).
We only need to discuss signs of λ 1 (µ y ).
If λ 1 (µ y ) > 0, then there exists a uniquely ergodic stationary distribution π(·) in the interior of the first quadrant.
If λ 1 (µ y ) < 0, thenΠ t (·) converges to µ y (·) for any initial value (x 0 , y 0 ) ∈ R 2 + almost surely, and x(t) converges to 0 almost surely. Similarly, we show the discussions in Table 2 (Blue parts stand for those who have been deduced by the previous conditions). , the readers may refer to the reference [12]. And we give the subsequent corrections in our next work for the revisions of the stochastic predator-prey model with response function [J1] .

Simulations of persistence and extinction
Three examples are introduced to illustrate Table 1: Figure 1 shows that both x and y go extinct.
And six examples are listed to demonstrate Table 2: Under the conditions, both x and y go extinct, see Figure 4. Meanwhile, In view of mathematical software, we compute that λ 2 (µ x ) < 0, henceΠ t (·) → µ x (·). Figure 5 supports the result.
It is obvious that λ 1 (µ y ) < 0, henceΠ t (·) → µ y (·). We can see from Figure 7 that the prey x goes extinct, however the predator y is persistent. This support the point that y has extra source of food besides x.

Stochastic Hopf bifurcation
It follows from Section 2 that when t is sufficiently large, the statistical properties of sample paths can be used to replace spatial ones. Therefore, in this section, we will use numerical simulations of sample paths to study stochastic Hopf bifurcation of the system (1.2). Now, we will use numerical simulations to illustrate that the stochastic predatorprey model can undergo a stochastic Hopf bifurcation phenomenon. Let r = 1, a = 1, w = 10, α 1 = 2.1, α 2 = 1. ydt + 0.03ydB 2 (t), ydt + 0.03ydB 2 (t). (3. 2) The deterministic system for the system (3.1) becomes ydt. (3. 3) The deterministic system (3.3) exists a stable limit cycle according to the literature [26] (see . Here, Figures 10-12 are given in comparison with the stochastic system (3.1). The deterministic system for the system (3.2) becomes ydt.      [26] (see . Here, Figures 13-15 are given in comparison with stochastic system (3.2). It is observed that the deterministic system exists Hopf bifurcation phenomenon. Figure 16 is the stationary distribution of the system (3.1) in the phase space. Figure 17 shows a stochastic limit cycle for the system (3.1) in three-dimensional space introducing time axis. Figure 18 implies that there is a crater-like stationary distribution for the stochastic system (3.1). Figure 19 is the stationary distribution of the system (3.2) in the phase space. Figure 20 shows the stochastic solution for the system (3.2) in three-dimensional space introducing time axis. Figure 21 implies that there is a peak-like stationary distribution for the stochastic system (3.2). Now, from the viewpoint of numerical simulations, Figures 16-18 show the stochastic system (1.2) exists a crater-like stationary distribution, and Figures 19-21 show the stochastic system (1.2) exists a peak-like stationary distribution. Overall, the shapes of stationary distributions change from crater-like to peak-like. Therefore, the stochastic model (1.2) undergoes a stochastic Hopf-bifurcation phenomenon [4,7,14,15,17,23,34].

Concluding remarks
Here, we consider a stochastic predator-prey model with Crowley-Martin functional response. The main results are as follows: • We obtain the complete threshold analysis of coexistence and extinction of the stochastic system (1.2). Moreover, numerical simulations are introduced to support each conclusion in Table 1 and Table 2.
• From the perspective of numerical simulations, the stochastic model (1.2) exists peak-like stationary distribution and crater-like stationary distribution, that is, it undergoes a stochastic Hopf bifurcation.
Some interesting topics deserve further investigation. It will be interesting to study the stochastic high-order nonlinear systems. We will discuss these issues in the near future.