Replicator-mutator equations with quadratic fitness

Matthieu Alfaro (1) , Rémi Carles (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat equation with a harmonic potential, plus a specific nonlocal term. We give an explicit formula for the solution, thanks to which we prove that when the fitness is non-positive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is non-negative (inverted harmonic potential), solutions always become extinct in finite time.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Alfaro, Rémi Carles. Replicator-mutator equations with quadratic fitness. Proceedings of the American Mathematical Society, 2017, 145 (2), pp.5315-5327. ⟨10.1090/proc/13669⟩. ⟨hal-01399269⟩