Appearance of chaos and bi-stability in a fear induced delayed predator–prey system: A mathematical modeling study

Since many years ago, it has been only a priori assumption that the diversity of ecological demography is largely influenced by the direct interactions of predator and prey species. But over the years, it has also been recognized that an indirect effect can also affect the system more strongly. The effect of fear on prey species is that it restricts physical activities and feeding times of prey thereby reducing interactions with their predators and leading to an intraspecific competition among predators. By these motives, we investigate a delay-induced model inducing fear of predators on the prey population. Existence of feasible steady points, well posedness of the solutions, local & global stability, subcritical & supercritical bifurcation are well presented. The results show that predator fear reduces both prey and predator populations, but does not affect the steady state of the system. As delay increases within a specific range, it changes the dynamics of the system in a stable–unstable–stable sequences. A small delay shifts an unstable interior equilibrium to a predator-free stable equilibrium. The main result of this study is that a large delay produces chaotic dynamics when cost of fear is high and prey birth is low, whereas a small delay creates bi-stable dynamics when cost of fear is low and prey birth is high. Furthermore, by increasing the competitiveness among the predators, the chaos can be completely controlled. We examine that small fear has a greater effect while relatively large fear has less effect. Delay parameter is responsible for appearing stability switch more than once. Extensive numerical simulations are also carried out for various parameters.