Per-generation and instantaneous growth rates
Let ni(t) be the number of organisms of type i at time t, and let R be the per-capita reproductive rate per generation. If t counts generations, then
Now we wish to move to the case where t is continuous and real-valued.
where the last simplification follows from L'Hôpital's rule. Explicitly, let ε = Δt. Then
The solution to the equation
Continuous rate of change
Let r1 and r2 be the instantaneous rates of increase of type 1 and type 2, respectively. Then
With the total population size
- n(t) = n1(t) + n2(t)
we have the proportion of type 1
Define the fitness advantage
Given our interest in understanding the change in gene frequencies, our goal is to compute the rate of change of p(t).
The logit function , which takes , induces a more natural space for considering changes in frequencies. In logit terms,