The analysis of persistence times and species coexistence is traditionally focused on variations in the log-abundances of species, with the mean logarithmic growth rate when rare, E[r], used as a metric for persistence and invasibility. However, invasion probabilities and the times to extinction are not single-valued functions of E[r]. When random environmental variations act to stabilize competitor coexistence (storage effect) these quantities may even decrease as E[r] increases.
We present alternative and more reliable metrics of stability that one may utilize in the analysis of empirical data and/or well-calibrated models. To that aim we implement analytical and numerical techniques for measuring stability. The analytical techniques are based on the diffusion approximation for small magnitudes of environmental variations, while for larger magnitudes, we introduce a WKB-based approach. The numerical technique is based on the method of transition path sampling.
Results/Conclusions
When the diffusion approximation holds, we show through explicit analytical expressions that it is the ratio between the growth rate and the strength of abundance variations which controls the persistence properties. In this limit, this ratio acts as an appropriate metric for stability.
When the diffusion approximation breaks down, different persistence modes appear and large deviations (WKB) analysis is required. As an example, we analyze the two-species Chesson-Warner lottery model when environmental fluctuations are strong, taking into account both environmental and demographic stochasticity. We calculate analytical expressions for the probability of a species to increase or decrease its abundance to any number. These expressions allow us to implement our WKB technique and thereby calculate analytically the invasibility of a species in this model in different regimes of the amplitude of the environmental stochasticity. The analytical results in the different regimes agree well with the outcomes of Monte-Carlo simulations of the model.