Mathematical Scrutiny of Singular Predator-Prey Model with Stage-Structure of Prey

Abstract

In this paper, a stage structured predator-prey model with Holling type II functional response is formulated, considering the juvenile prey as the favorite food for the generalist predator. Further, the existence of predators is ensured by the sufficient amount of alternative food available in the habitat. The proportional harvesting of the adult prey is incorporated with the assumption that only the adult prey are of economic worth. The existence and local stability of the distinct equilibrium points of the system are investigated. The bifurcation from origin to predator free equilibrium state is obtained for the bifurcation parameter-effort on harvesting. The occurrence of Hopf bifurcation about the interior equilibrium state is established for arbitrary model parameters and the supercritical nature of this bifurcation is proved by fixing these parametric values. An algebraic equation is included to this modified model to analyze the economic benefits resulted from the harvesting of adult prey. The singularity-induced bifurcation (SIB) about the coexisting equilibrium state of differential-algebraic system is deduced along the parameter \(v\) at \(v=0\), \(v\) being the profit/loss due to harvesting. The state feedback controller is recommended to eliminate the SIB about the coexisting equilibrium state for the differential algebraic system. Adopting an appropriate feedback control would ensure the stability of co-existence interior equilibrium state along with economic profit from harvesting. Numerical examples are used to elaborate the analytical results obtained.