# Drummond:PopGen

### From OpenWetWare

## Notes on population genetics

Let *n*_{i}(*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

*n*_{i}(*t*+ 1) =*n*_{i}(*t*)*R*

*n*_{i}(*t*) =*n*_{i}(0)*R*^{t}.

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

*n*_{i}(*t*+ 1) =*n*_{i}(*t*)*R*

*t*. Then

The solution to the equation

*n*_{i}(*t*) =*n*_{i}(0)*e*^{tlnR}=*n*_{i}(0)*R*^{t}

*t*. We can define the

*instantaneous rate of increase*

*r*= ln

*R*for convenience.