Species extinction and permanence of an impulsively controlled two-prey one-predator system with seasonal effects

Recently, the population dynamic systems with impulsive controls have been researched by many authors. However, most of them are reluctant to study the seasonal effects on prey. Thus, in this paper, an impulsively controlled two-prey one-predator system with the Beddington-DeAngelis type functional response and seasonal effects is investigated. By using the Floquet theory, the sufficient conditions for the existence of a globally asymptotically stable two-prey-free periodic solution are established. Further, it is proven that this system is permanent under some conditions via a comparison method involving multiple Lyapunov functions and meanwhile the conditions for extinction of one of the two prey and permanence of the remaining two species are given.