ANALYSIS OF A STOCHASTIC TWO-PREDATORS ONE-PREY SYSTEM WITH MODIFIED LESLIE-GOWER AND HOLLING-TYPE II SCHEMES∗

In this paper, we consider a stochastic two-predators one-prey model with modified Leslie-Gower and Holling-type II schemes. Analytically, we completely classify the parameter space into eight categories containing eleven cases. In each case, we show that every population is either stable in time average or extinct, depending on the parameters of the model. Finally, we work out some simulation figures to illustrate the theoretical results.


Introduction
The famous predator-prey model with modified Leslie-Gower and Holling-type II schemes can be denoted as follows (Aziz-Alaoui and Daher Okiye [1]): , dy(t) dt = y(t) r 2 − f y(t) h + x(t) , (1.1) where x(t) and y(t) represent the population sizes of the prey and the predator respectively. r 1 , r 2 , a, c, f and h are positive constants. r 1 and r 2 is the growth rates of the prey and the the predator respectively, a represents the competitive strength among individuals of the prey, c stands for the per capita reduction rate, h describes the protection of the environment, the meaning of f is similar to c. Recently, many authors have paid attention to model (1.1) and its generalized forms, see e.g. [2, 6-8, 11, 12, 16, 22-28]. Aziz-Alaoui and Okiye [1] considered the boundedness and stability of the positive equilibrium of model (1.1). Nindjin et al. [23] introduced time delay into Eq. (1.1) and studied the stability of the positive equilibrium of their model; Eq. (1.1) with impulse was investigated by Guo and Song [8], Song and Li [24] and Nie et al. [22]; Ji et al. [11,12] considered model (1.1) with white noise and studied the persistence, extinction, and stationary distribution to the corresponding system. System (1.1) with reaction-diffusion was explored in [2,7,28]. The above studies have focused on two-species models. However, in the nature world it is a common phenomenon that several predators compete for a prey. At the same time, the growth of population is the real world is inevitably affected by environmental fluctuations ( [21]). And several authors have revealed that the environmental fluctuations may change the properties of population models greatly. For example, Mao, Marion and Renshaw [20] revealed that the environmental fluctuations can suppress a potential population explosion. Therefore it is useful to study how the environmental fluctuations affects the multi-predators one-prey model with modified Leslie-Gower and Holling-type II schemes. However, to the best of our knowledge, no result of this aspect has previously been reported. Suppose that the growth rate r i is affected by white noise (see, e.g., [3-5, 11-19, 29]), with r i → r i + σ iẆi (t), then we obtain the following stochastic two-predators one-prey system with modified Leslie-Gower and Holling-type II schemes: where W i (t) is a standard Wiener process and σ 2 i stands for the intensity of the noise.
For model (1.2), some interesting and important problems arise naturally.
(Q1): System (1.2) is a population model, then when the populations will be extinct and when will be not? The aim of this paper is to consider these questions. In Section 2, the almost complete parameters analysis is carried out. In each case, it is shown that every population is either stable in time average or extinct, depending on the coefficients of model (1.2), especially, depending on σ 2 1 , σ 2 2 , σ 2 3 , the intensities of the white noises. In Section 3, we work out some figures to support the theoretical findings. Section 4 gives to the concluding remarks.

Main results
For the sake of convenience, we define some notations.
Before we state and prove our main results, we prepare some lemmas.
(II) If there exist three positive constants T, τ and τ 0 such that for all t ≥ T , Proof. Without loss of generality, we only prove the case b 2 > 0. Let T be sufficiently large satisfying 0.5 exp{b 1 t} ≥ 1 for t ≥ T . Hence for t ≥ T , it follows from (2.5) that Clearly, Then we obtain When this inequality is used in (2.7), we can derive that where When these identities are used in (2.11), we can observe that lim sup To complete the proof, it suffices to show lim inf t→+∞ t −1 ln y 1 (t) ≥ 0, a.s. An appli- In other words, Clearly, for arbitrary ε > 0, there exists T > 0 such that for t ≥ T , Substituting this inequality into (2.12) yields that for t ≥ T , (2.14) We can choose ε be sufficiently small satisfying b 2 − ε > 0. Now applying (I) and (II) in Lemma 2.2 to (2.13) and (2.14) respectively, one can derive that It therefore follows from the arbitrariness of ε that When this identity is used in (2.12), then by lim t→+∞ t −1 ln y 1 (0) = 0 and lim t→+∞ W 2 (t)/t = 0, we obtain lim t→+∞ t −1 ln ψ 1 (t) = 0, a.s. In view of (2.4), one can see that (ii) If b 1 < 0, b 2 > 0 and b 3 < 0, then x and y 2 go to extinction and y 1 is stable in time average, i.e., (iii) If b 1 < 0, b 2 < 0 and b 3 > 0, then x and y 1 go to extinction and y 2 is stable in time average, i.e., (iv) If b 1 < 0, b 2 > 0 and b 3 > 0, then x goes to extinction and both y 1 and y 2 are stable in time average: then y 1 and y 2 go to extinction and x is stable in time average: then y 2 goes to extinction and moreover (a) If b 1 < c1 f1 b 2 , then x goes to extinction and y 1 is stable in time average: (vii) If b 1 > 0, b 2 < 0 and b 3 > 0, then y 1 goes to extinction and moreover (c) If b 1 < c2 f2 b 3 , then x goes to extinction and y 2 is stable in time average: , then x goes to extinction and both y 1 and y 2 are stable in time average: Proof. Applying Itô's formula to model (1.2) leads to That is to say The proof of (i): by virtue of (2.19), Since (2.23) Applying of (I) and (II) in Lemma 2.2 to (2.22) and (2.23) respectively, one can see that In view of the arbitrariness of ε, we obtain lim t→+∞ t −1 t 0 y 1 (s)ds = h1b2 f1 , a.s. The proof of (iii) is similar to (ii) and hence is omitted. (iv): Since b 1 < 0, it then follows from (i) that lim t→+∞ x(t) = 0. The proof of (2.16) is similar to (ii) and hence is omitted.
We are in the position to show (vi). Clearly, b 3 < 0 ⇒ lim t→+∞ y 2 (t) = 0. (a): (2.24) According to (2.10), for arbitrary ε > 0, there exists T > 0 such that for t ≥ T , Substituting these inequalities into (2.24), we can observe that for t ≥ T It then follows from b1 c1 < b2 f1 that we can let ε be sufficiently small such that In view of (2.10) and lim t→+∞ t −1 W 2 (t) = 0, we can see that At the same time, by virtue of (2.19), It therefore follows from lim t→+∞ y 2 (t) = 0 and (2.26) that for arbitrary ε > 0, there exists T > 0 such that for t ≥ T , When these inequalities are used in (2.27), one can derive that for t ≥ T , Let ε be sufficiently small such that b 1 − c1b2 f1 − ε > 0, and then using (I) and (II) in Lemma 2.2 to (2.28) and (2.29) respectively, one derives According to the arbitrariness of ε, we have This completes the proof of (vi). The proof of (vii) is similar to (vi) and hence is omitted.
To completes the proof, we need only to show (viii). (e): By (2.24)×f 2 − (2.21) × c 1 f 1 , (2.30) It follows from (2.10) that for arbitrary ε > 0, there exists T > 0 such that for t ≥ T , When these inequalities are used in (2.30), we obtain that for t ≥ T is similar to that of (ii) and hence is omitted.
(f): Similar to (2.26), we can show that It then follows from (2.27) that for sufficiently large t, Then using Lemma 2.2 and the arbitrariness of ε, we can derive the desired assertion (2.18).

Numerical simulations
In this section, we shall work out some figures to validate our analytical results by using the famous Milstein method (see e.g. [9]). Without loss of generality, we suppose that b 2 = r 2 − 0.5σ 2 2 > b 3 = r 3 − 0.5σ 2 3 . Consider the discretization equation: where ξ (k) , η See Fig.1(b).

Concluding remarks
This paper is devoted to the asymptotic properties of a stochastic two-predators one-prey model with modified Leslie-Gower and Holling-type II schemes. We have carried out the almost complete parameters analysis of the model. From these results, one can see that the stochastic noise play a key role in determining the extinction and stability in time average of the species. Some interesting topics deserve further investigation. It is interesting to consider more realistic but complex models, for example, Markovian-switching (see e.g. [29]) or Lévy jumps (see e.g. [3]). The motivation is that the growth of population in the natural world often suffer sudden-environmental shocks, e.g., epidemics, waterflood, drought, etc. Moreover, it is interesting to study the stochastic one-predator twopreys model.