Species lifetime distribution for simple models of ecologies

Interpretation of empirical results based on a taxa's lifetime distribution shows apparently conflicting results. Species' lifetime is reported to be exponentially distributed, whereas higher order taxa, such as families or genera, follow a broader distribution, compatible with power law decay. We show that both these evidences are consistent with a simple evolutionary model that does not require specific assumptions on species interaction. The model provides a zero-order description of the dynamics of ecological communities and its species lifetime distribution can be computed exactly. Different behaviors are found: an initial $t^{-3/2}$ power law, emerging from a random walk type of dynamics, which crosses over to a steeper $t^{-2}$ branching process-like regime and finally is cutoff by an exponential decay which becomes weaker and weaker as the total population increases. Sampling effects can also be taken into account and shown to be relevant: if species in the fossil record were sampled according to the Fisher log-series distribution, lifetime should be distributed according to a $t^{-1}$ power law. Such variability of behaviors in a simple model, combined with the scarcity of data available, cast serious doubts on the possibility to validate theories of evolution on the basis of species lifetime data.