  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.
