Spatio-temporal patterns in a diffusive model with non-local delay effect

摘要

Steady states and periodic oscillations are two important solution forms in reaction-diffusion equations. In this article, we deal with a diffusive Lotka-Volterra system with non-local delay effect and Dirichlet boundary condition. Firstly, the existence and multiplicity of spatially non-homogeneous synchronous/mirror-reflecting steady-state solutions are investigated by means of Lyapunov-Schmidt reduction. Then using centre manifold reduction, normal form analysis and equivariant bifurcation theory, we show that each of the standard Hopf bifurcations gives rise to only one branch of spatially non-homogeneous time-periodic synchronous waves, whereas each of the equivariant Hopf bifurcations gives rise to eight branches of periodic orbits, including two phase-locked oscillations, three mirror-reflecting waves and three standing waves. In particular, we derive the formula for determining the Hopf bifurcation direction and stability of these Hopf bifurcating periodic orbits. These theoretical results give an accurate picture of the bifurcation structure, in the neighbourhood of the bifurcation point, for any set of kernel functions.