Attraction Region for the Classical Lotka−Volterra Predator−Prey model Caused by impulsive Effects

Abstract

Discontinuous phenomena appear in various research fields. Impulsive differential equations are often used to model such discontinuous dynamics. This study deals with the Lotka–Volterra predator–prey model dominated by impulsive effects. In the modeling, impulses are added considering the ratios of the interior equilibrium to the current populations of the prey and predator. In this case, there is no restriction of the time interval between the impulse and the next impulse being the same. By focusing on the time interval between impulses adjacent to one another as well as the impulsive effect amount, a simple sufficient condition for the interior equilibrium to become globally asymptotically stable and sufficient conditions that are useful for estimating an attraction region are provided. Our results reveal that impulse control for the Lotka–Volterra predator–prey model can reduce the variation in the population of the prey and predator and enable the stable coexistence of both.