Hopf bifurcation in a delayed Lokta–Volterra predator–prey system

Abstract

In this paper, we consider a delayed Lotka–Volterra predator–prey system with a single delay. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the linear stability of the system is investigated and Hopf bifurcations are demonstrated. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.