Bifurcation analysis and hybrid control strategy in a three-dimensional prey-predator system with two time delays

On the basis of previous research, a new three-dimensional two time-delays predator-prey system is established in this paper, one time delay is the gestation delay of the predator and the other time delay is the time delay for the predator to supplement food. First, the existence, uniqueness, non-negativity and boundedness of the solution of the predator-prey model are investigated. Second, the asymptotic stability and the existence Hopf bifurcation related to two time-delays are determined by studying the characteristic equation of the system. Moreover, the Hopf bifurcation control of the predator-prey model is investigated by using extended hybrid control technique and hybrid control technique. The results show that: (i) The two time-delays in the system can change the stability state of the system. (ii) By designing two suitable hybrid controllers, we have successfully changed the occurrence time of the Hopf bifurcation and stability region of the system. Finally, Matlab simulations are carried out to verify the correctness of theoretical results. The results presented in this research have important theoretical significance in maintaining the density of three species.