Extinction and Permanence of a Three-Species Lotka-Volterra System with Impulsive Control Strategies

A three-species Lotka-Volterra system with impulsive control strategies containing the biological control the constant impulse and the chemical control the proportional impulse with the same period, but not simultaneously, is investigated. By applying the Floquet theory of impulsive differential equation and small amplitude perturbation techniques to the system, we find conditions for local and global stabilities of a lower-level prey and top-predator free periodic solution of the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. We also give a numerical example that seems to indicate the existence of chaotic behavior.