The existence of Hopf bifurcation for a delayed Holling-Tanner type predator-prey model

Abstract

In this study, a Holling-Tanner type predator-prey model with a discrete time delay is investigated, where the functional response of the predator dynamics is ratio-dependent. We first analyze the local stability of the equilibrium point and examine the existence of Hopf bifurcations. The Hopf bifurcation, also known as the Poincar & eacute;-Andronov-Hopf bifurcation, is named after the French mathematician Jules Henri Poincar & eacute;, the Russian mathematician Alexander A. Andronov, and the German mathematician Heinz Hopf, whose fundamental contributions laid the foundation of this theory. By treating the delay parameter $\tau $ as the bifurcation parameter, we show that a Hopf bifurcation occurs when the delay crosses certain critical values. Finally, numerical simulations are carried out to support and illustrate our theoretical results.