A general multi-scale description of metastable adaptive motion across fitness valleys

Abstract

We consider a stochastic individual-based model of adaptive dynamics on a finite trait graph G = (V, E). The evolution is driven by a linear birth rate, a density dependent logistic death rate an the possibility of mutations along the (possibly directed) edges in E. We study the limit of small mutation rates for a simultaneously diverging population size. Closing the gap between [8] and [14] we give a precise description of transitions between evolutionary stable conditions (ESC), where multiple mutations are needed to cross a valley in the fitness landscape. The system shows a metastable behaviour on several divergent time scales associated to a degree of stability. We develop the framework of a meta graph that is constituted of ESCs and possible metastable transitions between those. This allows for a concise description of the multiscale jump chain arising from concatenating several jumps. Finally, for each of the various time scale, we prove the convergence of the population process to a Markov jump process visiting only ESCs of sufficiently high stability.