The spatially homogeneous Hopf bifurcation induced jointly by memory and general delays in a diffusive system

TL;DR

This paper analyzes how memory and general delays jointly induce stable, spatially homogeneous oscillations in a diffusive predator-prey system, providing a new algorithm for Hopf bifurcation analysis.

Contribution

It introduces a novel algorithm for calculating the normal form of Hopf bifurcation in systems with memory and delays, applied to a predator-prey model.

Findings

Supercritical, stable periodic solutions induced by delays.

Algorithm effectively predicts bifurcation direction and stability.

Numerical simulations confirm analytical results.

Abstract

In this paper, by incorporating the general delay to the reaction term in the memory-based diffusive system, we propose a diffusive system with memory delay and general delay (e.g., digestion, gestation, hunting, migration and maturation delays, etc.). We first derive an algorithm for calculating the normal form of Hopf bifurcation in the proposed system. The developed algorithm for calculating the normal form of Hopf bifurcation can be used to investigate the direction and stability of Hopf bifurcation. As a real application, we consider a diffusive predator-prey model with ratio-dependent Holling type-3 functional response, which includes with memory and gestation delays. The Hopf bifurcation analysis without gestation delay is first studied, then the Hopf bifurcation analysis with memory and gestation delays is studied. By using the developed algorithm for calculating the normal form of Hopf bifurcation, the supercritical and stable spatially homogeneous periodic solutions induced jointly by memory and general delays are found. The stable spatially homogeneous periodic solutions are also found by the numerical simulations which confirms our analytic result.