Periodic dynamics of a switching model with multiple stable states and linear harvest

Fishing moratoriums are vital for fishery management, restoring fish stocks, safeguarding ecosystems, and ensuring sustainable development. In this study, we explored and studied a canonical model with a Holling type-Ⅲ functional response consisting of two sub-equations switching each other. If the length of the closed season $\overline{T}\le T^{\ast }$, the trivial steady state is globally asymptotically stable. For $\overline{T}>T^{\ast }$, we provided a complete analysis of the existence of a unique periodic solution and three exact periodic solutions. We also provided sufficient conditions for the existence of a globally asymptotically stable periodic solution. Finally, numerical examples were provided to illustrate the theoretical results.