Sliding and oscillations in fisheries with on–off harvesting and different switching times

In this paper, we propose a fishery model with a discontinuous on–off harvesting policy, based on a very simple and well known rule: stop fishing when the resource is too scarce, i.e. whenever fish biomass is lower than a given threshold. The dynamics of the one-dimensional continuous time model, represented by a discontinuous piecewise-smooth ordinary differential equation, converges to the Schaefer equilibrium or to the threshold through a sliding process. We also consider the model with discrete time impulsive on–off switching that shows oscillations around the threshold value. Finally, a discrete-time version of the model is considered, where on–off harvesting switchings are decided with the same discrete time scale of non overlapping reproduction seasons of the harvested fish species. In this case the border collision bifurcations leading to the creations and destruction of periodic oscillations of the fish biomass are studied.