# Drummond:PopGen

### From OpenWetWare

(→Continuous rate of change) |
(→Continuous rate of change: logit) |
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|<math>= s p(t)(1-p(t))\!</math> | |<math>= s p(t)(1-p(t))\!</math> | ||

|} | |} | ||

+ | This result says that the proportion of type 1 <math>p</math> changes most rapidly when <math>p=0.5</math> and most slowly when <math>p</math> is very close to 0 or 1. | ||

+ | ==Evolution is linear on a log-odds scale== | ||

The logit function <math>\mathrm{logit} (p) = \ln {p \over 1-p}</math>, which takes <math>p \in [0,1] \to \mathbb{R}</math>, 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 <math>L_p(t) \equiv \mathrm{logit} (p(t))</math>, | The logit function <math>\mathrm{logit} (p) = \ln {p \over 1-p}</math>, which takes <math>p \in [0,1] \to \mathbb{R}</math>, 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 <math>L_p(t) \equiv \mathrm{logit} (p(t))</math>, | ||

- | <math>{\partial L_p(t) \over \partial t} = {\partial \over \partial t}\left(\ln {p(t) \over 1-p(t)}\right)</math> | + | :{| |

+ | |<math>{\partial L_p(t) \over \partial t} </math> | ||

+ | |<math>= {\partial \over \partial t}\left(\ln {p(t) \over 1-p(t)}\right)</math> | ||

+ | |- | ||

+ | | | ||

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

+ | |- | ||

+ | | | ||

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

+ | |- | ||

+ | | | ||

+ | |<math>= s. \!</math> | ||

+ | |} | ||

- | <math> | + | That is, <math>L_p(t)</math>, the log-odds of finding type 1 in a random draw from the population, changes linearly in time with slope <math>s</math>. This differential equation has the solution |

- | + | <math>L_p(t) = L_p(0)e^{st}\!</math> | |

+ | |||

+ | showing that the log-odds of finding type 1 changes exponentially in time, increasing if <math>s>0</math> and decreasing if <math>s<0</math>. | ||

==Diffusion approximation== | ==Diffusion approximation== | ||

Insert math here. | Insert math here. |

## Revision as of 09:45, 2 April 2009

## Contents |

## 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 ,

That is, *L*_{p}(*t*), the log-odds of finding type 1 in a random draw from the population, changes linearly in time with slope *s*. This differential equation has the solution

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