The predation strategy for predators and the avoidance strategy of prey are important topics in ecology and evolutionary biology. Both prey and predators adjust their behaviours in order to gain the maximal benefits and to increase their biomass for each. In the present paper, we consider a modified Leslie-Gower predator-prey model where predators cooperate during hunting and due to fear of predation risk, prey populations show anti-predator behaviour. We investigate step by step the impact of hunting cooperation and fear effect on the dynamics of the system. We observe that in the absence of fear effect, hunting cooperation can induce both supercritical and subcritical Hopf- bifurcations. It is also observed that fear factor can stabilize the predator-prey system by excluding the existence of periodic solutions and makes the system more robust compared to hunting cooperation. Moreover, the system shows two different types of bi-stabilities behaviour: one is between coexisting equilibrium and limit cycle oscillation, and another is between prey-free equilibrium and coexisting equilibrium. We also observe generalized Hopf-bifurcation and Bogdanov-Takens bifurcation in two parameter bifurcation analysis. We perform extensive numerical simulations for supporting evidence of our analytical findings.

In the present article, we investigate the impact of fear effect in a predator–prey model, where predator–prey interaction follows Beddington–DeAngelis functional response. We consider that due to fear of predator the birth rate of prey population reduces. Mathematical properties, such as persistence, equilibria analysis, local and global stability analysis, and bifurcation analysis, have been investigated. We observe that an increase in the cost of fear destabilizes the system and produces periodic solutions via supercritical Hopf bifurcation. However, with further increase in the strength of fear, system undergoes another Hopf bifurcation and becomes stable. The stability of the Hopf-bifurcating periodic solutions is obtained by computing the first Lyapunov coefficient. Our results suggest that fear of predation risk can have both stabilizing and destabilizing effects.