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

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.

Keywords:

• predator-prey reaction–diffusion system
• positive solution
• existence
• steady state bifurcation
• stability

AMS Subject Classification::

• 35B32
• 35B35
• 35J25