Replicator-mutator equations with quadratic fitness

TL;DR

This paper analyzes the replicator-mutator equation with quadratic fitness, providing explicit solutions and revealing long-term behaviors such as convergence to Gaussian distributions or finite-time extinction depending on the fitness sign.

Contribution

It offers explicit formulas for solutions and characterizes their asymptotic behavior based on the quadratic fitness sign, advancing understanding in evolutionary genetics models.

Findings

Solutions converge to Gaussian distributions when fitness is non-positive.

Solutions become extinct in finite time when fitness is non-negative.

Explicit solution formulas are derived for the quadratic fitness case.

Abstract

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.