Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

TL;DR

This paper develops a new method to analyze double Hopf bifurcations in nonlocal reaction-diffusion systems with spatial averaging, revealing complex oscillatory behaviors in predator-prey models.

Contribution

It introduces a novel algorithm for computing normal forms of double Hopf bifurcations in nonlocal reaction-diffusion equations with spatial average kernels.

Findings

Identification of conditions for complex oscillations

Application to predator-prey models showing nonhomogeneous oscillations

Development of a new analytical framework for nonlocal bifurcations

Abstract

In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of the center manifold reduction technique and normal form theory, we present a new algorithm for computing normal forms associated with the codimension-two double Hopf bifurcation of nonlocal reaction-diffusion equations. The theoretical results are applied to a predator-prey model, and complex dynamic behaviors such as spatially nonhomogeneous periodic oscillations and spatially nonhomogeneous quasi-periodic oscillations could occur.