What can Invasion Analyses Tell us about Evolution under Stochasticity in Finite Populations?

Abstract

The geometric mean of fitness is considered to be the main indicator of evolutionary change in stochastic models. However, this measure was initially derived for models with infinite population sizes, where the long-term evolutionary behavior can be described with certainty. In this paper we begin an exploration of the limitations and utility of this approach to evolution in finite populations and discuss alternate methods for predicting evolutionary dynamics. We reanalyze a model of lottery competition under environmental stochasticity by including population finiteness, and show that the geometric mean predictions do not always agree with those based on the fixation probability of rare alleles. Further, the fixation probability can be inserted into adaptive dynamics equations to derive the mean state of the population. We explore the effects of increasing population size on these conclusions through simulations. These simulations show that for small population sizes the fixation probability accurately predicts the course of evolution, but as population size becomes large the geometric mean predictions are upheld. The two approaches are reconciled because the time scale on which the fixation probability approach applies becomes very large as population size grows.