Discrete vs Continuous Replicator Dynamics
To make this derivation, it is better to focus on the population size of a strategy as opposed to the frequency of a strategy. Let $N_i$ be the population size of the strategy $i$. Strategies in the replicator dynamics change exponentially. The equation for continuous exponential growth is $\dot N_i = f_i N_i$. Since $x_i=N_i/N$ and $\dot N = \bar f N$, calculating $\dot x_i$ by using the quotient rule gives us the continuous replicator eqn.