A prey‐predator model with two discrete delays incorporating Beddington‐DeAngelis functional response is investigated. The positivity and boundedness of the solution of the delayed system have been discussed. Further, permanence of the system has been established under sufficient conditions which proves long‐term survival of prey and predator population. The occurrence of periodic solutions in the system is confirmed via phenomenon of Hopf bifurcation with respect to both discrete delays. The properties of periodic solutions are determined by using central manifold and normal form theory. The supercritical Hopf bifurcation occurs with respect to one delay parameter. The global stability of the system has been established for different combinations of delays. The numerical computation has also been performed to verify analytical results. The presence of delays in the system causes a wide range of complex dynamics, viz., limit cycles, quasi‐periodicity, and chaos.

This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points.