Stability and Hopf Bifurcation of a Multiple Delayed Predator-Prey System with Prey Refuge and Additive Allee Effect

This paper proposes a diffusive predator–prey model with double time delays and prey refuge, incorporating an additive Allee effect. First, we analyze the stability of boundary equilibrium, and the impact of Allee effect on the stability at boundary equilibrium is explored. Then we study the mechanisms by which the prey refuge influences the non-spatial system at positive equilibrium, revealing that under varying prey refuge coefficients, the system can exhibit stability, periodic oscillations or extinction. Subsequently, the occurrence conditions for the Hopf bifurcation are analyzed in a delayed system, and the direction and stability of Hopf bifurcation are obtained via the reaction–diffusion normal form theory. Finally, numerical simulations are carried out to verify our theoretical findings. Under the combined effect of maturation delay and digestion delay, spatio-temporal steady patterns and periodic patterns are observed in a reaction–diffusion system. Moreover, studies reveal that the Allee effect and prey refuge profoundly influence the stability in predator–prey systems.