# Drummond:PopGen

### From OpenWetWare

(→Continuous rate of change) |
(→Continuous rate of change) |
||

Line 68: | Line 68: | ||

:<math>{dn_i(t) \over dt} = r_i n_i(t).</math> | :<math>{dn_i(t) \over dt} = r_i n_i(t).</math> | ||

With the total population size | With the total population size | ||

- | :<math>n(t) = n_1(t) + n_2(t)</math> | + | :<math>n(t) = n_1(t) + n_2(t)\!</math> |

we have the proportion of type 1 | we have the proportion of type 1 | ||

:<math>p(t) = {n_1(t) \over n(t)}</math> | :<math>p(t) = {n_1(t) \over n(t)}</math> |

## Revision as of 09:34, 15 July 2008

## Per-generation and instantaneous growth rates

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

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

where the last simplification follows from L'Hôpital's rule. Explicitly, let ε = Δ*t*. Then

The solution to the equation

*t*. We can define the

*instantaneous rate of increase*

*r*= ln

*R*for convenience.

## Continuous rate of change

Let *r*_{1} and *r*_{2} 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 *p*(*t*).

The logit function , which takes , induces a more natural space for considering changes in frequencies. In logit terms,