Impact of discontinuous harvesting on fishery dynamics in a stock-effort fishing model

We consider a stock-effort fishing model with discontinuous harvesting strategies. Under some reasonable assumptions on the discontinuous harvesting function, we are able to confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that the solutions of the fishing model can arrive at a sliding mode in finite time. A qualitative analysis shows that the goal of maintaining the system at a sustainable equilibrium and optimizing the harvesting can be achieved by introducing the discontinuous harvesting strategies. From the viewpoint of optimal harvesting, we can obtain that discontinuous harvesting strategies are superior to continuous harvesting strategies, which are usually adopted in previous literature. The main difficulty resides in the discontinuity of the model, and is conquered by exploiting the theory of differential equations with discontinuous righthand sides and variable structure system theory.