Dynamical Analysis of a Delay Two-Prey–One-Predator Model Incorporating Fear Effect in the Growth Rate of Preys

In an ecological point of view, fears from predator cause physiological changes of prey population and these physiological changes may reduce the reproduction of prey. This paper deals with studying the effect of fear that is incorporated in the growth rate of prey on the dynamics of a delay ecological model consisting of two logistically growing prey species, namely, preyXand preyY, and a predator species. The predators take energy and biomass on predating on prey X as well as prey Y with extended Holling Type II functional response. We conduct a thorough analytical examination of the system, demonstrating the positivity and boundedness of the solutions. Criteria for the persistence of the model are founded. Seven biologically possible steady states are determined, and the local dynamics around the steady states based on the parameters are also investigated. Next, we investigated the occurrence of Hopf bifurcation around the prey X– and Y–free steady state, by taking the value of the time delay as a bifurcation parameter. Finally, we perform extensive numerical simulations to support the evidence of our analytical results and reveal the occurrence of Hopf bifurcation within the proposed model.