Demographic and environmental stochasticity both influence extinction risk, but because they act at different population sizes, it is difficult to incorporate both stochastic forces into a single model. Extinction risk predictions are also frequently hampered by insufficient data: estimates of parameters such as birth and death rates are unavailable for many endangered species, which are rare and often poorly studied. Body mass allometries offer one method of estimating parameters when data are scarce. Allometries also provide a means of investigating how body mass influences the time to extinction.
I will present an analytical stochastic model of extinction time that incorporates both demographic and environmental stochasticity. Using a novel method, we have incorporated environmental stochasticity into a birth and death process model of demographic stochastic dynamics. For this presentation, I will focus on a stochastic analogue of the logistic equation, but the modeling approach is completely general and can be extended to other ecological interactions. We incorporate environmental noise into both r and K (intrinsic rate of increase and carrying capacity) and we derive stochastic differential and Fokker-Planck equations for population size and the probability distribution of population size. We then derive a potential function (similar to the potential functions employed in physics and chemistry, e.g. for activation energy) that is used to generate an analytical approximation for the mean, median and quartiles of extinction time. Finally, to investigate how body mass affects the time to extinction, we express the model parameters, r and K, in terms of body mass via allometries. Utilizing the Ecological Archives database for body masses of world mammals and published estimates for r and K body mass allometries, we compare predicted extinction times and body masses of extant and extinct mammals.
Results/Conclusions
The model agrees closely with explicit simulations of stochastic population dynamics. In addition, our analytical approximation of mean extinction time agrees well with more precise numerical calculations based on backward Fokker-Planck equations. For some parameter ranges, the model predicts that demographic stochasticity is important at larger population sizes than previously thought. The model also predicts that extinction time is a steeply decreasing function of body size, providing an explanation why so many large animals are threatened. Predicted extinction times for extinct mammals (which were larger, on average, than their extant counterparts) were also shorter than those for extant mammals.