Positive Solutions Bifurcating from Zero Solution in a Predator-Prey Reaction–Diffusion System∗

An elliptic system subject to the homogeneous Dirichlet boundary condition denoting the steady-state system of a two-species predator-prey reaction– diffusion system with the modified Leslie–Gower and Holling-type II schemes is considered. By using the Lyapunov–Schmidt reduction method, the bifurcation of the positive solution from the trivial solution is demonstrated and the approximated expressions of the positive solutions around the bifurcation point are also given according to the implicit function theorem. Finally, by applying the linearized method, the stability of the bifurcating positive solution is also investigated. The results obtained in the present paper improved the existing ones.


Introduction
This paper is concerned with the following elliptic system where ∆ is the Laplacian operator and Ω is a bounded domain in R n with smooth boundary ∂Ω. As a predator-prey model with the modified Leslie-Gower and Holling-type II schemes, the ODE model corresponding to system (1.1) was proposed and studied by Aziz-Alaoui and Okiye [1]. In model (1.1), the variables u and v define the population densities of prey and predator species in the habitat Ω; a, m, k 1 , b and k 2 are positive constants and the corresponding biological meaning can be referred to [1]. In addition, the homogeneous Dirichlet boundary condition in (1.1) implies that the exterior environment is hostile. Note that u and v are population densities, therefore the nonnegative solutions of system (1.1) are of importance and interest [7,11,12,13]. Suppose that q(x) ∈ C(Ω) and λ 1 (q) < λ 2 (q) ≤ λ 3 (q) ≤ · · · are the eigenvalues of the eigenvalue problem −∆w + q(x)w = λw, x ∈ Ω, w = 0, x ∈ ∂Ω.
Recently, Peng and Wang [11] investigated the positive solutions of system (1.1) in the case that the parameter m is large, and obtained a complete understanding of the existence, multiplicity and stability of positive solutions. In addition, Wang and Wang [14] studied the existence and stability of positive solutions of system (1.1) and they found that (1.1) has at least a positive solution when b > λ 1 and a > λ 1 (mθ b /k 1 ) by the fixed point index theory in a positive cone. Meanwhile, by regarding a as the bifurcation parameter, they obtained the connected component connecting the semitrivial solution (0, θ b ) with the unique positive solution of the limit equation of (1.1) as a → ∞ and the stability of positive solution of (1.1) close to (0, θ b ) was also investigated by the linearized stability theory [4,9]. In addition, when a, b > λ 1 and 0 < m 1 or k is big enough, they also gave the existence and stability of positive solutions bifurcating from the unique positive solution of (1.1) in the case when m = 0. However, the existence and stability of positive solutions of (1.1) bifurcating from the zero solution were not discussed in [14]. In this paper, we consider mainly the existence and stability of positive solutions of (1.1) bifurcating from the zero solution.
This paper is organized as follows. In Section 2, by applying Lyapunov-Schmidt reduction process, we demonstrate the existence of positive solutions of (1.1) bifurcating from the zero solution. In Section 3, according to the implicit function theorem, the asymptotic expression of positive solutions of (1.1) bifurcating from the zero solution is given. By analyzing the spectrum of the linearized operator of (1.1) at the positive solutions obtained in Section 3, we analyze the stability of the bifurcating positive solutions of (1.1) in Section 4.

Existence of Positive Solutions Bifurcating from the Zero Solution
In this section, we discuss the existence of positive solutions of system (1.1) bifurcating from zero solution according to the Lyapunov-Schmidt reduction method [6,8]. For system (1.1), we have the following result. Proof. Suppose that φ 1 is the eigenfunction corresponding to the eigenvalue λ 1 , that is, φ 1 satisfies the following boundary value problem Then from [5] we know that φ 1 > 0, x ∈ Ω. Let (u, v) be a positive solution of (1.1). It follows from the first equation of (1.1) that Thus the Green's identity, (2.1) and (2.2) imply that The positivity of u and φ 1 in Ω gives a > λ 1 . Similarly, one can obtain that b > λ 1 and thus the proof is complete.

Asymptotic Expression of Small Bifurcating Positive Solutions
In the previous section, we have obtained the existence of positive solutions of system (1.1) bifurcating from the zero solution by using the Lyapunov-Schmidt reduction method. In this section, by virtue of the implicit function theorem, we give a more accurate asymptotic expression for the small positive solutions of system (1.1) bifurcating from the zero solution. Suppose that 0 < a − λ 1 , b − λ 1 1 and let a − λ 1 = b − λ 1 =: r. If (u r , v r ) is a positive solution of (1.1) when 0 < r 1, then (u r , v r ) should be a solution of the following elliptic boundary value problem (3.1) Define the operator D by and let N(D) and R(D) denote the null space and the range of D, respectively. Then it is easy to see and X can be decomposed as holds, then α 0 > 0 and β 0 > 0. Now, we consider the following boundary value problem in X ∩ R(D): In virtue of the definitions of α 0 and β 0 , and notice that D is a bijective mapping from X ∩ R(D) to R(D), we can obtain easily the following result: (Ω) and H 2 0 (Ω) = {y ∈ L 2 (Ω): y , y ∈ L 2 (Ω), y = 0 on ∂Ω}. Theorem 2. If the condition (3.2) holds, then there exists a constant r * > 0 and a unique continuously differential mapping r → (ξ r , η r , α r , β r ) from [0, r * ] to (X ∩ R(D)) 2 × (R + ) 2 such that system (3.1) has a unique positive solution parameterized by r as u r = α r r(φ 1 + rξ r ), v r = β r r(φ 1 + rη r ), r ∈ (0, r * ], (3.4) and φ 1 , ξ r = φ 1 , η r = 0.
An easy calculation shows that (u r , v r ) given by (3.4) solves the boundary value problem (3.1) and this completes the proof.

Stability of Small Bifurcating Positive Solutions
Suppose that 0 < r 1 and (u r , v r ) is the positive solution of system (1.1) given by (3.4). Then the linearized system of system (1.1) at (u r , v r ) is .