Abstract

An elliptic system subject to the homogeneous Dirichlet boundary con- dition 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 con- sidered. By using the Lyapunov–Schmidt reduction method, the bifurcation of the positive solution from the trivial solution is demonstrated and the approximated ex- pressions of the positive solutions around the bifurcation point are also given accord- ing 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.

Highlights

  • This paper is concerned with the following elliptic system mv−∆u = u a−u− k1 + u, x ∈ Ω, v −∆v = v b −k2 + u u(x) = v(x) = 0, x ∈ ∂Ω, (1.1)where ∆ is the Laplacian operator and Ω is a bounded domain in Rn 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]

  • We discuss the existence of positive solutions of system (1.1) bifurcating from zero solution according to the Lyapunov–Schmidt reduction method [6, 8]

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Summary

Introduction

Where ∆ is the Laplacian operator and Ω is a bounded domain in Rn with smooth boundary ∂Ω. When a, b > λ1 and 0 < m 1 or k is big enough, they gave the existence and stability of positive solutions bifurcating from the unique positive solution of (1.1) in the case when m = 0. The existence and stability of positive solutions of (1.1) bifurcating from the zero solution were not discussed in [14]. We consider mainly the existence and stability of positive solutions of (1.1) bifurcating from the zero solution.

Existence of Positive Solutions Bifurcating from the Zero Solution
Asymptotic Expression of Small Bifurcating Positive Solutions
Stability of Small Bifurcating Positive Solutions

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