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