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

SIAM Journal on Applied Mathematics, Volume 84, Issue 5, Page 1891-1909, October 2024.
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 [math] for the duration of the fishing ban ([math]) and establish the relationships between [math] and each of the downstream end [math], the advection rate [math], and the diffusion rate [math]. Under certain conditions, the trivial equilibrium point 0 is globally asymptotically stable if [math]. When [math], 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.