STABILITY ANALYSIS OF A RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH DIFFUSION AND STAGE STRUCTURE

Predator-prey models have been studied by many authors (see [6, 21]), but the stage structure of species has been ignored in the existing literature. In the natural world, however, there are many species whose individual members have a life history that take them through two stages: immature and mature (see [1–3, 7–9, 18–20]). In particular, we have in mind mammalian populations and some amphibious animals, which exhibit these two stages. In these models, the age to maturity is represented by a time delay, leading to a system of retarded functional differential equations. For general models one can see [11]. Specifically, the standard Lotka-Volterra type models, on which nearly all existing theories are built, assume that the per capita rate predation depends on the prey numbers only. An alternative assumption is that, as the numbers of predators change slowly (relative to prey change), there is often competition among the predators and the per capita rate of predation depends on the numbers of both preys and predators, most likely and simply on their ratio. A ratio-dependent predator-prey model has been investigated by [10]. Recently, a model of ratio-dependent two species predator-prey with stage structure was derived in [19]. The model takes the form


Introduction
Predator-prey models have been studied by many authors (see [6,21]), but the stage structure of species has been ignored in the existing literature.In the natural world, however, there are many species whose individual members have a life history that take them through two stages: immature and mature (see [1-3, 7-9, 18-20]).In particular, we have in mind mammalian populations and some amphibious animals, which exhibit these two stages.In these models, the age to maturity is represented by a time delay, leading to a system of retarded functional differential equations.For general models one can see [11].
Specifically, the standard Lotka-Volterra type models, on which nearly all existing theories are built, assume that the per capita rate predation depends on the prey numbers only.An alternative assumption is that, as the numbers of predators change slowly (relative to prey change), there is often competition among the predators and the per capita rate of predation depends on the numbers of both preys and predators, most likely and simply on their ratio.A ratio-dependent predator-prey model has been investigated by [10].
Recently, a model of ratio-dependent two species predator-prey with stage structure was derived in [19].The model takes the form 2 A ratio-dependent predator-prey system x 1 (0) > 0, y(0) > 0, x 2 (t) = ϕ(t) ≥ 0, −τ ≤ t ≤ 0, (1.1) where X 1 (t), X 2 (t) represent, respectively, the immature and mature prey populations densities; Y (t) represents the density of predator population; f > 0 is the transformation coefficient of mature predator population; αe −γτ X 2 (t − τ) represents the immatures who were born at time t − τ and survive at time t (with the immature death rate γ), and τ represents the transformation of immatures to matures; α > 0 is the birth rate of the immature prey population; γ > 0 is the death rate of the immature prey population; and β > 0 represents the mature death and overcrowding rate.The model is derived under the following assumptions.
(H1) The birth rate of the immature prey population is proportional to the existing mature population with a proportionality constant α > 0; the death rate of the immature prey population is proportional to the existing immature population with a proportionality constant γ > 0; we assume for the mature population that the death rate is of a logistic nature.(H2) In the absence of prey spaces, the population of the predator decreased, and d > 0, f > 0, m > 0. Note that the first equation of system (1.1) can be rewritten to so we have This suggests that if we know the properties of X 2 (t), then the properties of X 1 (t) can be obtained from X 2 (t) and Y (t).Therefore, in the following we need only to consider the following model: , (1.4) In [19], the effect of delay on the populations and the global asymptotic attractivity of the system (1.4) were considered, for detailed results we refer to [19].However, the diffusion of the species which is in addition to the species' natural tendency to diffuse to areas of smaller population concentration is not considered.For the details of diffusion in different areas, we can see [4,[12][13][14][15][16][17]22].In this paper, we study the system (1.1) with Xinyu Song et al. 3 diffusion terms, taking into account the diffusion of the species in different areas.The role of diffusion in the following system of nonlinear pdes with diffusion terms and stage structure will be studied: where ∂/∂n is differentiation in the direction of the outward unit normal to the boundary ∂Ω, we assume Ω Denote u 2 (x,t) and v(x,t) as u 1 (x,t) and u 2 (x,t), respectively, so we get the following subsystem of the system (1.5): Note that the quantities u 2 (x,t) and v(x,t) of the system (1.5) are independent of the quantity u 1 (x,t), so we may only consider the subsystem (1.7) to be easy to get the properties of the system (1.5).Before proceeding further, let us nondimensionalize the system (1.7) with the following scaling: 4 A ratio-dependent predator-prey system We obtain the following nondimensionless system: where The remaining part of this paper is organized as follows.The existence and uniqueness of the solutions of system (1.8) will be proved in Section 2. In Section 3, we obtain conditions for local asymptotic stability of the nonnegative equilibria of system (1.8).
In Section 4, we analyze the global asymptotic stability and obtain conditions for global asymptotic stability of the nonnegative equilibria of system (1.8).

Existence and uniqueness of the solutions
In order to solve the problem and prove theorems, we devote to some preliminaries.We rewrite system (1.8) to where Definition 2.1.Suppose ϕ 1 (x,t), ϕ 2 (x,t), ψ(x,t) be Hölder continuous, call ( u 1 , u 2 ), ( u 1 , u 2 ) to be a pair of strong upper and lower solutions, if u 1 , u 1 , u 2 , and and Similar to Definition 2.1, the definition of a pair of strong upper and lower solutions of the elliptic system corresponding to system (2.1) is easy to be given. ) , and c i j ≥ 0 for i, j = 1,2, and τ 2 = 0. Then From Lemma 2.2 we easily get the following lemma.

Local asymptotic stability of the equilibria
In this section, we discuss local asymptotic stability of the nonnegative equilibria by linearization method and analyzing the so-called characteristic equation of the equilibrium.It is obvious that system (2.1)only has three nonnegative equilibria: the equilibrium E 1 (0,0), the equilibrium E 2 (a,0), and the positive equilibrium We will point out that E 1 (0,0) cannot be linearized though it is defined for system (2.1), so the local stability of E 1 (0,0) will be studied in another paper. Let be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition, and let E(μ i ) be the eigenfunction space corresponding to μ i in C 1 (Ω).It is well known that μ 1 = 0 and the corresponding eigenfunction φ 1 (x) > 0. Let {φ i j | j = 1,2,..., dimE(μ i )} be an orthogonal basis of 8 A ratio-dependent predator-prey system , where c * 1 and c * 2 are both not zero.We still make u 1 (x,t), u 2 (x,t) corresponding to u * 1 (x,t), u * 2 (x,t), so the linearized equation of the system (2.1) at (c * 1 ,c * 2 ) is From [5], we know that the characteristic equation for the system (3.2) is equivalent to That is (3.4)

3.1.
Local asymptotic stability of the equilibrium E 2 (a,0).From (3.4), it follows that at the equilibrium E 2 (a,0), From the first factor of (3.5), we see Now we will determine that all roots of (3.7) satisfy Reλ < 0. Suppose that there exists λ 0 such that Re λ 0 ≥ 0. From (3.7), we deduce that Xinyu Song et al. 9 This implies that λ 0 is in the circle in the complex plane centered at (−(μ k D 1 + 2a),0) and of radius a.However, as for given μ k and D 1 , it follows for ever that μ k D 1 + 2a > a, therefore, Re λ < 0. (3.9) By the second factor of (3.5), we have If f > d, by taking k = 1(μ 1 = 0), from (3.10), we obtain that there at least exists a root λ 0 of (3.5) such that Reλ 0 > 0. Therefore, E 2 (a,0) is unstable if the condition f > d holds.
From the above discussion, we can conclude the following.

Global asymptotic stability of the equilibria
Note that F 1 (u 1 (x,t),u 2 (x,t),u 1 (x,t − τ)) and F 2 (u 1 (x,t),u 2 (x,t)), with respect to u 1 , u 2 , are continuous and mixed quasimonotone in Σ × Σ * , where Σ, Σ * are fixed and bounded subsets of R 2 .Thus there exist K i ≥ 0 (i = 1,2) such that In order to investigate the dynamics of the system (2.1) we define two sequences of constant vectors m=1 satisfying the following relation: where ( c, c) is a pair of coupled upper and lower solutions of system (2.1).It is easy to prove the following lemma.
Xinyu Song et al. 17 Hence, ( c 1 , c 2 ), ( c 1 , c 2 ) are a pair of coupled upper and lower solutions of the system (2.1).
By Theorem 4.2, we obtain that c and c satisfy