Bimatrix Replicator Dynamics with Periodic Impulses

This paper investigates the bimatrix replicator dynamics with periodic impulses. In biological system, impulsive perturbations may due to the occurrence of an unfavorable physical environment, or due to the seasonal life history effects in the physiological and reproductive mechanisms of the population. We show that impulsive perturbations can lead to complicated dynamical behaviors. On the one hand, the system can have multiple \(\tau \)-periodic solutions, where the lower bound of the number of solutions is increasing linearly in the impulsive period \(\tau \). On the other hand, for shorter impulsive period, we provide a differential approximation for the impulsive dynamical system. By analyzing the resulting differential dynamics, we show that the interior equilibrium (which corresponds to a \(\tau \)-periodic solution) must be globally stable if it exists. Furthermore, when the impulsive effect is weak, all interior trajectories of the impulsive dynamics evolve to a small neighborhood of the interior equilibrium of the bimatrix replicator dynamics. In summary, longer impulsive period causes an increase in the complexity of the evolutionary process and shorter period promotes evolutionary stability of the interior equilibrium where multiple strategies can coexist.