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Per-generation and instantaneous growth rates

Let be the number of organisms of type at time , and let be the per-capita reproductive rate per generation. If counts generations, then

and

Now we wish to move to the case where is continuous and real-valued. As before,

but now

where the last simplification follows from L'Hôpital's rule. Explicitly, let . Then

The solution to the equation

is
Note that the continuous case and the original discrete-generation case agree for all values of . We can define the instantaneous rate of increase for convenience.

Continuous rate of change

Let and be the instantaneous rates of increase of type 1 and type 2, respectively. Then

With the total population size

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 .

Diffusion approximation

==Diffusion approximation==