Bifurcation Analysis in A Diffusive Mussel-Algae Model with Delay

TL;DR

This paper analyzes the stability and bifurcation behavior of a delayed reaction-diffusion mussel-algae model, revealing conditions for stability, Hopf bifurcation, and periodic solutions through theoretical and numerical methods.

Contribution

It introduces a comprehensive bifurcation analysis for a delayed mussel-algae system, including stability criteria and bifurcation direction determination using advanced mathematical techniques.

Findings

Existence of positive solutions when delay is zero.

Stability of steady states for non-zero delay.

Identification of Hopf bifurcation and periodic solutions.

Abstract

In this paper, we consider the dynamics of a delayed reaction-diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.