# Drummond:PopGen

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- | |<math>= | + | |<math>= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n_1(t) + r_2 n_2(t)\right)</math> |

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- | |<math>= | + | |<math>= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n_1(t) + (r_1-s)(n(t)-n_1(t))\right)</math> |

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- | |<math>= | + | |<math>= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n(t) -s n(t) + s n_1(t))\right)</math> |

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## Revision as of 09:36, 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,