Bifurcation analysis of a delayed predator–prey model with Holling-III functional response

In this paper, we mainly explore the dynamical behaviors of a delayed predator–prey system with Holling-III functional response. First, for the corresponding non-delayed model, we study some basic properties of its equilibria, and the effects of the parameter c on the unique positive equilibrium. Next, for the delayed system, the local stability and Hopf bifurcation of the positive equilibrium are analyzed by choosing the sum τ of two time delays as a bifurcation parameter. In particular, the delayed model with double delays can be converted to the delayed model with the single delay through variable transformations. Further, using the normal form method and center manifold theorem, we derive the explicit formulae for determining the direction of Hopf bifurcation. Finally, numerical simulations of all these findings are carried out for verifying our theoretical results.