A continuous-time Crow-Kimura replicator-mutator equation with a non-local fitness function is shown to converge, in a weak pointwise sense, to a modified Zakai equation from nonlinear filtering, with the classical Zakai equation recovered for a parameter choice.
Replicator-mutator equations with quadratic fitness
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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.
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Connections between sequential Bayesian inference and evolutionary dynamics
A continuous-time Crow-Kimura replicator-mutator equation with a non-local fitness function is shown to converge, in a weak pointwise sense, to a modified Zakai equation from nonlinear filtering, with the classical Zakai equation recovered for a parameter choice.