Bifurcation Sequences in a Discontinuous Piecewise-Smooth Map Combining Constant-Catch and Threshold-Based Harvesting Strategies\ast

We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). We refer to this strategy as threshold constant-catch (TCC) harvesting. We provide analytical and numerical results when applying TCC to monotone population growth models. TCC remedies the tendency to fishery collapse of pure constant-catch harvesting and provides a buffer for quotas larger than the maximum sustainable yield. From a dynamical systems point of view, TCC gives rise to a piecewise-smooth map with a discontinuity at the threshold population size. The dynamical behavior includes border-collision bifurcations, basin boundary metamorphoses, and boundary-collision bifurcation. We further find Farey trees, a slightly modified truncated skew tent map scenario, and the bandcount incrementing scenario. Our results underline, on the one hand, the protective function of thresholds in harvest control rules. On the other hand, they highlight the dynamical complexities due to discontinuities that can arise naturally in threshold-based harvesting strategies.