Finite populations with frequency-dependent selection: a genealogical approach

Evolutionary models for populations of constant size are frequently studied using the Moran model, the Wright–Fisher model, or their diffusion limits. When evolution is neutral, a random genealogy given through Kingman's coalescent is used in order to understand basic properties of such models. Here, we address the use of a genealogical perspective for models with weak frequency-dependent selection, i.e. N s =: α is small, where s is the fitness advantage of a fit individual and N is the population size. When computing fixation probabilities, this leads either to the approach proposed by Rousset (2003), who argues how to use Kingman's coalescent in order to study weak selection, or to extensions of the ancestral selection graph of Neuhauser and Krone (1997) and Neuhauser (1999). As an application of this genealogical approach, we re-derive the one-third rule of evolutionary game theory (Nowak et al., 2004). In addition, we provide the approximate distribution of the genealogical distance of two randomly sampled individuals under linear frequency-dependence.