ABSTRACT Periodicity is a common phenomenon in biological systems. In this paper, we first characterize it with an abstract switching discrete system of reaction–diffusion equations. Under certain conditions, we obtain the critical thresholds for the time length before switching and temporal period, denoted by and , respectively. With the values of the other parameters being fixed, if the time length before switching or if the temporal period , then the system admits a unique globally asymptotically stable ‐periodic solution; conversely, if or if , then the trivial equilibrium is globally asymptotically stable. We further demonstrate the applicability of our theoretical framework with two biologically motivated examples: a mosquito population suppression model based on the Sterile Insect Technique and a harvesting model in fishery management. Numerical simulations are also conducted to validate our analytical findings.
Jiang et al. (Sun,) studied this question.