Fear of predators is a well-recognized phenomenon that results from predation in a natural food-chain or food-web ecosystem. Wang et al. (2016) published the first food-chain model that included the predator's fear effect on prey growth rate. The indirect impact of fear has been studied through its inclusion in several food web and food chain models in various ecological contexts. Some recent studies and field experiments show that predators affect their prey not only by direct capture; they also induce fear in prey species, which reduces their reproduction rate. The dynamics of a three-species prey-predator model are discussed, with the assumption that the logistic growth rate of prey is reduced as a result of both predators' fears, and the reproduction rate of intermediate consumers is affected by the top predator. We carry out the feasibility, existence of steady states, local stability analysis, and bifurcation analysis. Through numerical simulation, we show that the system stays chaotic at a low cost of fear, but an increase in the fear factor results in stability. We conclude that the chaotic dynamics of the system is controlled by fear effects, i.e., the whole prey-predator system is driven by the fear effect of predators.
Mathematics Subject Classification (2020): 34D20, 34C23, 65P20