Jump to content

User:Algebraic

From Wikipedia, the free encyclopedia

You can call me Al.

Per-generation and instantaneous growth rates

[edit]

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.

Change in relative frequencies

[edit]