2021/12/23 by Esser, Manuel, Kraut, Anna
#37N25 #60J27 #60J80 #92D15 #92D25 #FOS: Biological sciences #FOS: Mathematics #Populations and Evolution (q-bio.PE) #Probability (math.PR)
paper · doi:10.48550/arxiv.2112.12675
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 the works of Bovier, Coquille and Smadi (2019) and Coquille, Kraut and Smadi (2021), 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 multi-scale 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.