Double Hopf bifurcation analysis in the memory-based diffusion system

TL;DR

This paper develops an algorithm to analyze double Hopf bifurcations in memory-based diffusion systems and applies it to a predator-prey model, revealing complex spatially inhomogeneous periodic solutions.

Contribution

It introduces a novel method for calculating the normal form of double Hopf bifurcations considering memory effects in diffusion systems.

Findings

Existence of stable spatially inhomogeneous periodic solutions.

Transition between different types of periodic solutions.

Coexistence of multiple periodic solutions with distinct spatial profiles.

Abstract

In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay.