Complex Dynamics of a Predator–Prey Interaction with Fear Effect in Deterministic and Fluctuating Environments

Abstract

Many ecological models have received much attention in the past few years. In particular, predator–prey interactions have been examined from many angles to capture and explain various environmental phenomena meaningfully. Although the consumption of prey directly by the predator is a well-known ecological phenomenon, theoretical biologists suggest that the impact of anti-predator behavior due to the fear of predators (felt by prey) can be even more crucial in shaping prey demography. In this article, we develop a predator–prey model that considers the effects of fear on prey reproduction and on environmental carrying capacity of prey species. We also include two delays: prey species birth delay influenced by fear of the predator and predator gestation delay. The global stability of each equilibrium point and its basic dynamical features have been investigated. Furthermore, the “paradox of enrichment” is shown to exist in our system. By analysing our system of nonlinear delay differential equations, we gain some insights into how fear and delays affect on population dynamics. To demonstrate our findings, we also perform some numerical computations and simulations. Finally, to evaluate the influence of a fluctuating environment, we compare our proposed system to a stochastic model with Gaussian white noise terms.