  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.
