Fixation in finite populations: discrete and continuous views
Max Oliveira de Souza (UFF)
We will present two different viewpoints on fixation: In the first part of the talk, we identify a general class of evolutive processes which share many features with the classical Moran and Wright-Fisher (WF) processes---and include both of them. We also identify when a process in this class will have a strictly increasing fixation, and show that (WF) processes may have a decreasing fixation, contrary to the Moran processes. We also show that WF is universal from the point of view of fixation: given almost any fixation vector, there is at least one WF process that realises it. In the second part, we show how to construct continuous approximations of the fixation probability for birth-death processes that are valid beyond the weak-selection limit. Using this approximation we give continuous restatements of two classical concepts in the discrete setting: (i) the ESS$_N$ and (ii) risk dominant strategies. In particular, we obtain an asymptotic definition in the quasi-neutral regime of the celebrated 1/3 law. This is joint work with FACC Chalub.