Bautin bifurcation in delayed reaction-diffusion systems with application to the Segel-Jackson model

TL;DR

This paper develops an algorithm to analyze Bautin bifurcations in delayed reaction-diffusion systems, applying it to the Segel-Jackson model to identify complex bifurcation phenomena and solution behaviors.

Contribution

The paper introduces a method for deriving normal forms of Bautin bifurcations in delayed reaction-diffusion systems with boundary conditions, with explicit Lyapunov coefficient calculations.

Findings

Existence of fold bifurcation of periodic orbits near the bifurcation point

Presence of subcritical and supercritical Hopf bifurcations

Solutions with small initial conditions converge to stable orbits, large ones diverge

Abstract

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).