Homoclinic bifurcation of a ratio-dependent predator–prey system with impulsive harvesting

Abstract

In this paper, we investigate a ratio-dependent prey–predator model with state-dependent impulsive harvesting where the prey growth rate is subject to a strong Allee effect. The existence of order-1 homoclinic cycle is obtained, and choosing $$\alpha $$ as a control parameter, the existence, uniqueness and stability of order-1 periodic solution of the system are discussed by means of the geometry theory of semi-continuous dynamic system. We also investigate that system exhibits the phenomenon of homoclinic bifurcation about parameter $$\alpha $$ . Moreover, the numerical simulations are provided to show the main results. The used methods are intuitive to prove the existence of order-1 periodic solution and homoclinic bifurcation.