The Hopf Bifurcation Analysis of the Boissonade Model with Time Delay and Diffusion

This paper investigates the dynamical characteristics of the Boissonade model in a class of reaction–diffusion chemical systems with time delays, analyzing the system’s Hopf bifurcation with time delay as the parameter under both diffusion-free and diffusion-included conditions. First, the stability of the positive equilibrium solution is examined in the absence of diffusion, with stability criteria derived for different parameter ranges. This analysis confirms that a Hopf bifurcation occurs near the positive equilibrium, revealing that the system exhibits periodic oscillations once the time delay exceeds a critical threshold. Subsequently, the impact of the diffusion term on the Hopf bifurcation is investigated, and the critical threshold for its occurrence is determined. Finally, numerical simulations are conducted, providing comprehensive numerical validation for the theoretical findings.