Dynamics of an eco-epidemiological model with toxicity, treatment, time-varying incubation

In this paper, we considered an eco-epidemiological model including toxicity, treatment, time-varying incubation, and Holling Ⅱ functional response. First, for the model without time delay, the positivity and boundedness of solutions were investigated, and some conditions for local asymptotic stability of all possible equilibriums and condition for global stability of the positive equilibrium were also given. Then, the local stability around the positive equilibrium and conditions for the existence of Hopf bifurcation of such model with time delay were explored. Furthermore, by using the method of multiple scales, a control strategy based on time delay was obtained to suppress oscillation. Moreover, the global stability of the positive equilibrium of the model with time-varying delay was investigated. Finally, the theoretical findings were validated through numerical simulations. Those results demonstrated that time-varying delays can induce more complex dynamics in systems. By combining the method of multiple scale with periodic perturbations, the oscillatory behavior induced by Hopf bifurcation can be effectively suppressed, providing a novel approach for stability control in time-varying delay systems.