Stability and bifurcation analysis of a Holling–Tanner model with discontinuous harvesting action

Abstract

This work addresses the study of dynamics and bifurcations in a prey–predator model, known in the literature as the Holling–Tanner model, subject to a harvesting action of predators that is activated when the prey population is less than a certain threshold, and stopped otherwise. Such a model is represented by a piecewise smooth system with a switching boundary given by a straight line that is defined by the threshold established for the prey population. Under certain conditions on the system parameters, a pseudo-focus point appears at the switching boundary. Based on the Poincaré map defined in a neighborhood of the pseudo-focus, explicit conditions are given on the system parameters that determine its local stability, the occurrence of Hopf-like bifurcations and the emergence of crossing limit cycles. In addition to Hopf-like bifurcations, other local and global bifurcations such as the classical Hopf bifurcation, the Boundary Equilibrium bifurcations, the Saddle–Node bifurcation of periodic orbits and the Grazing bifurcation are also identified. A complete description of the existence and stability of equilibria and periodic orbits is provided based on the obtained two-parameter bifurcation set, from which the coexistence of four periodic orbits in the phase portrait of the system under study is proved.