Here I will treat some basic questions in population genetics. For personal reasons, I tend to include all the algebra.
Per-generation and instantaneous growth rates
What is the relationship between per-generation growth rates and the Malthusian parameter, the instantaneous rate of growth?
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
If two organisms grow at different rates, how do their proportions in the population change over time?
Let r1 and r2 be the instantaneous rates of increase of type 1 and type 2, respectively. Then
This result says that the proportion of type 1 p changes most rapidly when p = 0.5 and most slowly when p is very close to 0 or 1.
Evolution is linear on a log-odds scale
The logit function , which takes , induces a more natural space for considering changes in frequencies. Rather than tracking the proportion of type 1 or 2, we instead track their log odds. In logit terms, with ,
This differential equation Lp'(t) = s has the solution
showing that the log-odds of finding type 1 changes linearly in time, increasing if s > 0 and decreasing if s < 0.