Description:
<label>Abstract</label>
<p>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 nonpositive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is nonnegative (inverted harmonic potential), solutions always become extinct in finite time.</p>