Dynamical study of a predator–prey system with Michaelis–Menten type predator-harvesting

Abstract

The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity. Hunting cooperation in predators is one of the most remarkable features of the predation process, which benefits the predators by developing fear upon their prey. This study investigates the dynamical behavior of a modified LV-type predator–prey system with Michaelis–Menten-type harvesting of predators where predators adopt cooperation strategy during hunting. The ecologically feasible steady states of the system and their asymptotic stabilities are explored. The local codimension one bifurcations, viz. transcritical, saddle-node and Hopf bifurcations, that emerge in the system are investigated. Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation. A backward Hopf-bifurcation is detected when the harvesting effort is increased, which destabilizes the system by generating periodic solutions. The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient. Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations. Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased. We have also explored codimension two bifurcations viz. the generalized Hopf and the Bogdanov–Takens bifurcation exhibit by the system. To visualize the dynamical behavior of the system, numerical simulations are conducted using an ecologically plausible parameter set. The existence of the bionomic equilibrium of the model is analyzed. Moreover, an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.