Stability and Hopf Bifurcation of a Delayed Prey-Predator System with Fear, Hunting Cooperative, and Allee Effect

Abstract

The effect of fear, the hunting cooperative process, and Allee's impact on the behavior of an ecological system are investigated and discussed. The impact of the delay of the prey's response to the predation risk is included. The Leslie-Gower growth is used to describe the growth of the predator population. Firstly, the solutions' existence, positivity, and boundedness within the limits of a suitable region in the parametric space for all time are studied. The stability of all equilibrium points under the surrounding environmental effects is established. Moreover, the occurrence of a Hopf bifurcation is discovered. The stability of the bifurcating periodic dynamic and their dynamical properties are studied. Finally, the obtained theoretical results are confirmed and validated utilizing numerical simulation. It is observed that the system possesses a bi-stable behavior and a Hopf bifurcation.