Stability and bifurcation analysis for a delayed Lotka–Volterra predator–prey system

The present paper deals with a delayed Lotka–Volterra predator–prey system. By linearizing the equations and by analyzing the locations on the complex plane of the roots of the characteristic equation, we find the necessary conditions that the parameters should verify in order to have the oscillations in the system. In addition, the normal form of the Hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these Hopf bifurcations. To verify the obtained conditions, a special numerical example is also included.