Feedback stabilization of predator-prey systems for impulsive biological control

In agricultural applications, the introduction of predators is used to control the prey/pests level so that crop damage is limited; however, though a satisfactory equilibrium between predators and prey may exist, it may be unstable. In order to achieve pest control, its stabilization should then be achieved and, since predator releases occur at discrete time instants, we model them through impulses upon which we will build a positive impulsive feedback control law. Our controller is based on the measurement of the state every $T$ time instants followed by the introduction, when necessary, of predators into the system to instantaneously bring their population up to a prescribed level. The proposed controller is shown to lead to the global asymptotic stability of the desired equilibrium for $T$ small; local stability is shown for some larger periods, and some numerical asymptotic analysis is performed.