Forecasting Extinction in Ecosystems with Coevolving Species

Standard ecological models predict how populations change in time, given fixed interspecies interactions. However, as any phenotype, variations in interaction patterns should lead to their evolution. To incorporate ecological evolution into population dynamics, we start with the multi-species Lotka-Volterra equations and occasionally introduce mutants that have slight variations in their interactions with other species. Those favorable interaction patterns will fixate, subsequently causing ecological shifts. We find that the incremental shifts driven by successive fixations gradually accumulate, inevitably leading to a singularity, signaling the extinction of one or more species. Using a simple mean-field approach we obtain analytical upper and lower bounds for the time to soonest extinction. Furthermore, for two kind of mutation models (random, and zero-sum) we determine probability distributions governing the lifetime of species as well as the ecosystem as a whole.