Homoclinic bifurcation of prey–predator model with impulsive state feedback control

In this paper, homoclinic bifurcations of a prey–predator system with state impulse are investigated. Firstly, the existence of order-1 homoclinic cycle of system (1.2) with δ=0δ=0 is investigated; Secondly, choosing q as a control parameter, the sufficient conditions of existence and stability of order-1 periodic solution of system (1.2) with δ=0δ=0 are obtained by means of the geometry theory of semi-continuous dynamic systems; Thirdly, on the basis of the theory of rotated vector fields, homoclinic bifurcation of system (1.2) about parameter δδ are also studied; Finally, some simulations are provided to prove the main results. The methods used in this paper are intuitive to prove the existence of order-1 homoclinic cycle and homoclinic bifurcations.