Abstract
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 Poincaré–Andronov–Hopf bifurcation, is named after the French mathematician Jules Henri Poincaré, 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.