Dynamics of a modified Leslie–Gower eco-epidemiological model with fear effect, carry-over effect, and time-varying delay

In this study, a modified Leslie-Gower time-varying delay eco-epidemiological model with fear effect, fear carry-over effect, and cooperative hunting is investigated. First, the positivity and boundedness of solutions for the delay-free system, as well as the stability of equilibria are discussed, and sufficient conditions for the global asymptotic stability of the positive equilibrium are obtained. Then, taking the fear effect parameter and the fear carry-over effect parameter as bifurcation parameters, the Hopf bifurcation near the positive equilibrium is studied. Furthermore, the effects of time delay on system stability are analyzed, and the conditions for the occurrence of delay-induced Hopf bifurcation are derived. Next, to suppress the periodic oscillations caused by time delay, a time-varying delay perturbation is introduced as a control strategy, and the critical condition for oscillation suppression is obtained using the method of multiple scales. Finally, the parameter sensitivity analysis is conducted. The numerical results show that changes in the fear effect and fear carry-over effect can drive the system from a stable coexistence state to a periodic oscillatory state. Once the time delay exceeds a critical threshold, the system loses stability and exhibits sustained oscillations, however a suitable time-varying delay perturbation can effectively suppress these oscillations and restore the system to a stable state.