Time to extinction in deteriorating environments

Habitat degradation and destruction are the predominant drivers of population extinction, but there is little theory to guide the analysis of population viability in deteriorating environments. To address this gap, we investigated extinction times in time-varying, demographically stochastic versions of the logistic model for population dynamics. A property of these models is the extinction delay, a quantitative measure of the time lag in extinction created by species-specific extinction debt. For completeness, three models were constructed to represent the different demographic routes by which deterioration may affect population dynamics. Numerical analysis for two notional life histories indicated that the demographic response to environmental deterioration had a large effect on extinction delay, but a third analysis showed that the trajectory of the decline in carrying capacity ultimately characterized its magnitude. A concave decline in carrying capacity produced a large extinction delay while a small delay occurred with a convex decline. Furthermore, our results explore the non-monotonicity of extinction debt with respect to the speed of deterioration. A peak is present at low levels of deterioration, and the height of the peak and the asymptote of delay are affected by both life history parameterizations and the rate of change of the carrying capacity. The results suggest that population viability analyses must consider not only environmental deterioration, but also the effects of deterioration on the trajectory of the decline in carrying capacity.