Counter-attack in a stage-structured model with adult predator-induced fear and competition

The role of non-consumptive effects due to fear induced by adult predators in two species systems with stage structure is very crucial in understanding the physiological behavior of the prey in ecological communities. In this regard, we propose and investigate a stage-structured predator-prey interaction where the prey counter-attacks the vulnerable juvenile predators due to fear of adult predator predation. In light of the counter-attacks from the prey, the adult predator population may hurt from density-dependent non-predatory deprivations, such as the edge to reproduce. We model this via a competition term. Conditions for the existence of equilibrium points are given, and their stability criteria are analyzed locally. We demonstrate the possible occurrence of Hopf and transcritical bifurcations as the prey attack rate is varied. Multiple Hopf bifurcations are observed as the level of fear induced by the adult predator is varied. It is interesting to note that varying the level of fear induced by the adult predator does not only stabilize the prey's population, but also leads to dangerously low adult and juvenile predators populations under certain biological restrictions. Furthermore, at low adult predator attack rate, the adult and juvenile predators can potentially go extinct due to insufficient depredation. This study provides useful insight into the conservation of critically endangered species. Numerical simulations and bifurcation diagrams are presented and interpreted biologically to validate our analytical results.