Evolutionary Branching via Replicator–Mutator Equations

Mario Veruete (1) , Matthieu Alfaro (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

We consider a class of non-local reaction-diffusion problems , referred to as replicator-mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schrödinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation rate for evolutionary branching to occur.

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Analyse fonctionnelle [math.FA]; Physique mathématique [math-ph]; Théorie spectrale [math.SP]; Automatique / Robotique; Traitement du signal et de l'image [eess.SP]

Mario Veruete, Matthieu Alfaro. Evolutionary Branching via Replicator–Mutator Equations. Journal of Dynamics and Differential Equations, 2019, 31, pp.2029-2052. ⟨10.1007/s10884-018-9692-9⟩. ⟨hal-01699053⟩