The Replicator Equation

The replicator equation models rates of change in population shares through relative fitness: dxᵢ/dt = xᵢ(fᵢ − f̄), where f̄ is the population-weighted mean fitness. Fitness may depend on the current mix.

The continuous replicator equation is dxᵢ/dt = xᵢ(fᵢ − f̄), with f̄ = Σxⱼfⱼ. It describes a type's change in share, not necessarily its change in absolute population. Subtracting the weighted average makes all share changes sum to zero, preserving total share under the model.

An equilibrium requires each present type to have no fitness advantage over the average, but this condition alone does not establish stability. An absent type has zero change under the basic equation even if it would be advantageous if introduced. Entry, mutation, finite-population fluctuations, and structured contact require additional assumptions or terms.

When to use it

When competitive dynamics need mathematical formalization; when predicting which strategies will grow or decline based on relative fitness; when understanding why below-average strategies are eliminated even if they're 'good enough' in absolute terms; when evolutionary game theory would illuminate competitive coexistence.

How it can help

Specify the reproduction or copying mechanism and fitness functions, verify conservation of shares, and analyze stability as well as equilibrium. Add entry and mutation when relevant.