Permanence and periodic solution for a modiﬁed Leslie-Gower type predator-prey model with diffusion and non constant coefﬁcients

—In this paper we study a predator-prey system, modeling the interaction of two species with diffusion and T -periodic environmental parameters. It is a Leslie-Gower type predator-prey model with Holling-type-II functional response. We establish some sufﬁcient conditions for the ultimate boundedness of solutions and permanence of this system. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Numerical simulations are presented to illustrate the results.


MODEL
The dynamical properties of the predator-prey models can be used to analyze the relations between the prey and predator and to predicate whether they can coexist. As we known, one of the earliest and also the best known predatorprey models is the Leslie-Gower model [16], [17], which is a modificiation of the Lotka-Volterra model [22]. The Leslie-Gower type model can be described by the following autonomous bidimensional system [16], [17]  where u is the population of the prey and v is the population of the predator. In (1) we assume the prey grows logistically with carrying capacity K = a b and intrinsic growth rate a in the absence of predation. The predation is assumed to be proportional to the population size of the prey. The predator grows logistically with intrinsic growth rate c and carrying capacity c β u(t) proportional to the population size of prey (or prey abundance).
The parameter β is a measure of the food quality that the prey provides for conversion into predator birth. The stability of the interior equilibrium is studied in [25] by numerical methods. Lindstrom [18] investigated the nonexistence, existence and limit cycles. Hsu and Huang [13] prove that all the solutions are bounded and positive if their initial values are in the first quadrant, and study the globally asymptotical stability of the interior equilibrium using Liapunov function and LaSalle's invariance principle. Aziz-Alaoui and Daher Okiye [3] argued that a suitable predator-prey model should incorporate some kind of functional response, while the predator species could have other food resource. Basing on those assumption, they proposed a predator-prey model with modified Leslie-Gower and Holling-type II schemes [12] as follow: where k 1 is the half-saturation constant in the Holling-type II functional response and k 2 is a measure of alternative prey densities in the environment, allowing the predator to persist when the prey population disappears. The authors investigated the boundedness and global stability of the system (2). Nindjin et al. [23] further incorporated the time delay to the system considered in [3], and they showed that time delay plays important role on the dynamic behaviors of the system. Yafia et al. [27] studied the limit cycle bifurcated from time delay. For more works on Leslie-Gower predator-prey model, one could refer to [1], [4], [7], [8], [14], [26], [28], [29] and the references cited therein. To achieve further understanding it is now essential to consider more general and hence more "difficult" models. We will focus here on the case in which the biological or environmental parameters are time-periodic, and will assume that the species are free to move at random throughout some bounded habitat. Under these assumptions, we model the species interaction via a system of reaction-diffusion equations of the form where the function u(t, x) and v(t, x) determine the densities of prey and predator, respectively, at a point x and time t. Here the equations are assumed to be satisfied in a cylinder x ∈Ω, 0 < t < ∞, where Ω is an open, bounded, smooth domain in R n . These equations are supplemented with homogeneous Neumann boundary conditions where n is the outward unit vector of the boundary ∂Ω which we assume is smooth, and the following nonnegative initial values The various coefficients on the right-hand side depend on both t and x modelling the fact that effects vary in both time and space. The periodicity of coefficients models seasonal fluctuations. d 1 and d 2 , are positive diffusion coefficients reflecting the non-homogeneous dispersion of populations. Many authors studied the qualitative properties of this system, but for the case for which the parameters are constant, using Neumann or Dirichlet boundary conditions, see [8], [24], [27], [29]. Motivated by the papers mentioned above, we deal here with the permanence and existence of periodic solutions of the diffusive system (3). The content of this paper is as follows. In section 2, we give conditions for the ultimate boundedness of solutions and permanence of the system. In Section 3, we establish conditions for the existence of a unique periodic solution of the system. Numerical simulations are presented in Section 4 to illustrate the feasibility of our results.

II. BOUNDEDNESS AND PERMANENCE
We analyze the permanence (dissipation and persistence) of system (3) with non-negative initial functions, this ensures the long-term survival (i.e., will not vanish in time) of all components of system (3), under some conditions. We first recall a well known result on the logistic equation.
Lemma 2.1: [30]. Assume that u(t, x) is defined by then, lim t→∞ u(t, x) = K. Throughout the paper we always assume that: are bounded positive-valued functions on R + ×Ω continuously differentiable in t and x, and are periodic in t with a period T > 0. For a bounded function φ(t, x), we denote A. Dissipation Proposition 2.2: All the solutions of (3) initiated in the positive octant are nonnegative and satisfy lim sup

Proof
The nonnegativity of the solutions of (3) is obvious since the initial value is nonnegative. We consider now the second part of the theorem. For convenience, we denote a = a(t, x), and similar meaning to b, c, α, β, k 1 and k 2 . From the first From the comparison principle of the parabolic equations [9], [11], it is easy to verify that . This implies, by using Then, for ε > 0 there exists T 1 > 0 such that where η 1 = a M b m + ε. Therefore, from the second equation of system (3) and (7) and using the same reasoning, we have Therefore, any positive solution of system (3) is ultimately bounded, which completes the proof.

Proof
From (3), (7) and (8), it follows that for t ≥ T 2 , then from the comparison principle of the parabolic equations, it is easy to verify that u(t, x) ≥ u(t), where u(t) is the spatially homogeneous solutions of where u * = min x∈Ω u(0, x). Thanks to lemma 2.1, we obtain, Hence, there exists T 3 > T 2 such that for any t > T 3 , where, From the predator equation, it follows that Hence, there exists T 4 > 0 such that for any t > T 4 v(t, x) > ξ 2 , where Therefore, from (11) and (12), we obtain,

III. PERIODIC SOLUTIONS
A very basic and important problem in the study of a population growth models with periodic environment is the global existence and stability of positive periodic solutions, which plays a similar role as a globally stable equilibrium for autonomous models [5], [11], [15], [19], [21]. In this section, we derive sufficient conditions that guarantee existence, uniqueness and global stability of a T -periodic positive solution of system (3). For this aim, we consider the matrix M , which reads as: 14) where ζ and η are the bounds of any non-zero arbitrary solution of system (3), initialing with non-negative function, given by Proposition 2.5. Proposition 3.1: Assume that condition (9) holds, that is system (3) where µ(M ) is the maximal eigenvalue of the matrix M . Then, system (3) has a unique globally asymptotic stable strictly positive T -periodic solution.

V. CONCLUSION
The interacting species play important roles in real ecosystem. In this paper, we have studied time-periodic Leslie-Gower type predatorprey model with diffusion and Holling-type-II functional response whose growth rates and interaction rates are periodic functions of time. We have obtained sufficient conditions for the persistence of (3) in Proposition 2.4. The conditions are given in term of parameters of the model. Biologically speaking, we may expect the coexistence when the predator growth rate is sufficiently small, or if the predation rate α is small enough.  Next, we have investigated sufficient conditions which ensure the existence of positive T -periodic solutions of (3) in Proposition 3.1. The conditions are given in term of the largest eigenvalue of certain matrix. Our study demonstrates how parameters of the model which are not constant but vary in response to environmental fluctuations, influence a species prosperity, and gives some valuable suggestions for saving the two species and regulating populations when the ecological and environmental parameters are affected by periodic factors such as the season switching. Numerical simulations are carried out to support our theoretical results.