Interplay between geometry and randomness in fitness landscapes for expanding populations

The course of evolution is a constant conflict shaped by the interplay between the deterministic forces of selective advantage, characterised by the geometry of a fitness landscape, and the randomness of mutation and reproduction. A central question in this context is predictability: given the geometry of the fitness landscape, can we predict the trajectory of evolution? Two regimes are very well understood, from both biological and mathematical perspectives, but both these regimes are essentially deterministic in the limit; once we fix the parameters and the fitness landscape, evolution is pre-determined. The focus of our proposal is on cases where mutations are sufficiently frequent and the number of local optima in the fitness landscape is sufficiently large, so that the population does not necessarily move as one and several species or quasispecies may exist concurrently. There is a significant lack of rigorous mathematical work in this situation, although simulations have shown intriguing results. The aim of this proposal is to develop new rigorous mathematical tools to gain insight into the possible random pathways taken by evolutionary processes.