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