# Drummond:PopGen

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- | |<math>= {\partial \over \partial t}\left(\ln e^{st}\right)</math> | + | |<math>= {\partial \over \partial t}\left(\ln {n_1(0) \over n_2(0)} e^{st}\right)</math> |

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## Revision as of 19:39, 2 April 2009

## Introduction

Here I will treat some basic questions in population genetics. For personal reasons, I tend to include all the algebra.

## Per-generation and instantaneous growth rates

What is the relationship between per-generation growth rates and the Malthusian parameter, the instantaneous rate of growth?

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 growth rate*

*r*= ln

*R*for convenience.

## Continuous rate of change

If two organisms grow at different rates, how do their proportions in the population change over time?

Let *r*_{1} and *r*_{2} be the instantaneous rates of increase of type 1 and type 2, respectively. Then

*p*(

*t*).

This result says that the proportion of type 1 *p* changes most rapidly when *p* = 0.5 and most slowly when *p* is very close to 0 or 1.

## Evolution is linear on a log-odds scale

The logit function , which takes , induces a more natural space for considering changes in frequencies. Rather than tracking the proportion of type 1 or 2, we instead track their log odds. In logit terms, with ,

This differential equation *L*_{p}'(*t*) = *s* has the solution

showing that the log-odds of finding type 1 changes linearly in time, increasing if *s* > 0 and decreasing if *s* < 0.