Discrete vs Continuous Replicator Dynamics

Question

The replicator eqn in the case of discrete non-overlapping generations and asexual reproduction is given by the discrete replicator eqn: $$x_i(t+1) = x_i (t)\frac{f_i(t)}{\bar f (t)}$$ where $x_i$ is the frequency of the strategy $i$ and each individual playing strategy $i$ produces $f_i$ copies of itself in the next generation. $\bar f$ is the average fitness. Thus \begin{equation} \Delta x_i = x_i(t) \frac{(f_i(t) - \bar f (t))}{\bar f (t)} \end{equation} In the case in which $x_i$ is a continuous variable, we have the continuous replicator eqn given by $$\dot x_i = x_i (f_i - \bar f)$$

Question: I suppose we should be able to derive the continuous replicator eqn from the discrete replicator eqn by evaluating the discrete replicator eqn for $\lim_{\Delta t \to 0}$. How can I do this derivation?

Answer 1

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.