Crossing a fitness valley as a metastable transition in a stochastic population model

Anton Bovier (1) , Loren Coquille (2) , Charline Smadi (3)

1 Rheinische Friedrich-Wilhelms-Universität Bonn
2 IF - Institut Fourier
3 UR LISC - Laboratoire d'ingénierie pour les systèmes complexes

We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0,1,…,L} and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape exhibits a valley: mutant individuals with negative fitness have to be created in order for the population to reach a trait with positive fitness. We focus on the limit of large population and rare mutations at several speeds. In particular, when the mutation rate is low enough, metastability occurs: the exit time of the valley is an exponentially distributed random variable.

Mots clés: Eco-evolution; birth and death process with immigration; selective sweep; coupling; competitive Lotka–Volterra system with mutations

Domaines: Probabilités [math.PR]

Anton Bovier, Loren Coquille, Charline Smadi. Crossing a fitness valley as a metastable transition in a stochastic population model. The Annals of Applied Probability, 2019, 29 (6), pp.541-3589. ⟨10.1214/19-AAP1487⟩. ⟨hal-03230266⟩