Stability and bifurcation of a delayed eco-epidemic model with nonlinear prey refuge and additional food

In this paper, we considered an eco-epidemic model incorporating additional food, time delay, and nonlinear prey refuge. First, the analysis began with the model without time delay, focusing on the positivity and boundedness of its solutions. Following this, we established criteria for the local asymptotic stability of all possible equilibria, and obtained some conditions of the global asymptotic stability at the positive equilibrium of model with time-varying delay. Subsequently, we investigated the conditions under which Hopf bifurcation occured in the model with discrete time delay. In addition, we employed the method of multiple time scales to analyze the delay differential system, thereby deriving a time-delay-based control strategy. Specifically, time-varying perturbation was introduced to the delay to suppress oscillation. Finally, the theoretical findings were validated through numerical simulations.