Mathematical description of continuous time and space replicator-mutator equations for quadratic fitness landscapes

TL;DR

This paper derives analytical solutions for the continuous-time, continuous-trait replicator-mutator equation with quadratic fitness, explaining key evolutionary phenomena without simulations.

Contribution

It provides the first analytical solutions for these equations under quadratic fitness, enabling direct explanation of complex evolutionary behaviors.

Findings

Explains the flying kite effect

Quantifies survival of the flattest

Describes populations tracking dynamic optima

Abstract

The replicator-mutator equation is a model for populations of individuals carrying different traits, with a fitness function mediating their ability to replicate, and a stochastic model for mutation. We derive analytical solutions for the replicator-mutator equation in continuous time and for continuous traits for a quadratic fitness function. Using these results we can explain and quantify (without the need for numerical in-silico simulations) a series of evolutionary phenomena, in particular the flying kite effect, survival of the flattest, and the ability of a population to sustain itself while tracking an optimal feature which may be fixed, moving with bounded velocity in trait space, oscillating, or randomly fluctuating.