Replicator equations
The replicator equation describes continuous change in strategy frequencies under frequency-dependent selection.
If \(x_i\) is the frequency of strategy \(i\),
\[\frac{dx_i}{dt}=x_i\left(f_i(x)-\bar f(x)\right),\]where \(f_i\) is the fitness of strategy \(i\) and
\[\bar f=\sum_jx_jf_j.\]Interpretation
If \(f_i>\bar f\), strategy \(i\) increases in frequency; if \(f_i<\bar f\), it decreases.
The frequencies remain constrained by \(x_i\ge0\) and \(\sum_i x_i=1\).
Key idea. Replicator dynamics compare each strategy's fitness with the current population-average fitness.