Impact of fear in a delay-induced predator–prey system with intraspecific competition within predator species

In theoretical ecology, predator–prey interaction is a natural phenomenon that significantly contributes for shaping the community structure and maintaining the ecological diversity. In almost every ecological model, the prey species is curtailed by the direct attacking of predator species. However, from different experimental shreds of evidence, it has been observed that fear (felt by prey) for predators can change the physiological behaviour of prey individuals and greatly reduces their reproduction rate as well as enhances their mortality rate. In this current work, we develop and explore a predator–prey model incorporating the cost of perceived fear into the birth and death rates of prey species with Holling type-II functional response. In addition, the intraspecific competition within predator species and a gestation delay are introduced in the model to obtain more realistic and natural dynamics. Feasibility of all the steady states and their stability conditions are analysed in terms of the model parameters. We show that only existence of an interior equilibrium point is sufficient to prevent the extinction of predator species. In this case, either both species can exist together or oscillate around that interior equilibrium point. We can also recognize the parametric region where the system produces multiple coexistence equilibria in which different initial biomass of populations may produce different long-term outcomes. The basic bifurcation analyses of the system exhibit that a higher level of fear or higher intraspecific competition rate helps the population to survive in a coexistence state. For a suitable choice of parametric values, the proposed model may produce the bi-stable phenomenon between two coexistence steady states. We obtain a parametric condition for which the model system experiences a Hopf bifurcation if the delay parameter exceeds some threshold value. All of these theoretical findings are verified by various numerical examples.