Collapse and recovery times in non-linear harvesting with demographic stochasticity

Recent collapses of many fisheries across the globe have challenged the mathematical approach to these systems through classic bioeconomic models. Decimated populations did not recover as fast as predicted by these models and depensatory effects were introduced to better fit the dynamics at low population abundances. Alternative to depensation, modeling captures by non-linear harvesting functions produces equivalent outcomes at small abundances, and the dynamics undergoes a bifurcation leading to population collapse and recovery once catching efforts are above or below certain thresholds, respectively. The time that a population takes to undergo these transitions has been mostly overlooked in bioeconomic contexts, though. In this work we quantify analytically and numerically the times associated to these collapse and recovery transitions in a model incorporating non-linear harvesting and immigration in the presence and absence of demographic stochasticity. Counterintuitively, although species at low abundances are prone to extinction due to demographic stochasticity, our results show that stochastic collapse and recovery times are upper bounded by their deterministic estimates. This occurs over the full range of immigration rates. Our work may have relevant quantitative implications in the context of fishery management and rebuilding.