Periodic Dynamics of a Reaction-Diffusion-Advection Model with Michaelis–Menten Type Harvesting in Heterogeneous Environments

Abstract. Organisms inhabit streams, rivers, and estuaries where they are constantly subject to drift and overfishing. Consequently, these organisms often confront the risk of extinction. Can a reasonable fishing ban satisfy the human need for sufficient aquatic proteins without depleting fishery resources? We propose a reaction-diffusion-advection model to answer this question. The model consists of two subequations, which are constantly switched to describe closed seasons and open seasons with Michaelis–Menten type harvesting. We define a threshold value [Formula: see text] for the duration of the fishing ban ([Formula: see text]) and establish the relationships between [Formula: see text] and each of the downstream end [Formula: see text], the advection rate [Formula: see text], and the diffusion rate [Formula: see text]. Under certain conditions, the trivial equilibrium point 0 is globally asymptotically stable if [Formula: see text]. When [Formula: see text], we obtain sufficient conditions on the existence of a globally asymptotically stable periodic solution based on the thresholds in all parameter settings. Finally, some discussions on our findings are provided.