# Drummond:PopGen

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+ | ==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== | ==Per-generation and instantaneous growth rates== | ||

<p> | <p> | ||

- | Let <math>n_i(t)</math> be the number of organisms of type <math>i</math> at time <math>t</math>, and let <math>R</math> be the ''per-capita reproductive rate | + | What is the relationship between per-generation growth rates and the Malthusian parameter, the instantaneous rate of growth? |

+ | </p> | ||

+ | <p> | ||

+ | Let <math>n_i(t)</math> be the number of organisms of type <math>i</math> at time <math>t</math>, and let <math>R</math> be the ''per-capita reproductive rate per generation''. If <math>t</math> counts generations, then | ||

:<math>n_i(t+1) = n_i(t)R\!</math> | :<math>n_i(t+1) = n_i(t)R\!</math> | ||

and | and | ||

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is | is | ||

:<math>n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!</math> | :<math>n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!</math> | ||

- | Note that the continuous case and the original discrete-generation case agree for all values of <math>t</math>. We can define the ''instantaneous rate | + | Note that the continuous case and the original discrete-generation case agree for all values of <math>t</math>. We can define the ''instantaneous growth rate'' <math>r = \ln R</math> for convenience. |

</p> | </p> | ||

==Continuous rate of change== | ==Continuous rate of change== | ||

- | + | <p> | |

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

+ | </p> | ||

+ | <p> | ||

Let <math>r_1</math> and <math>r_2</math> be the instantaneous rates of increase of type 1 and type 2, respectively. Then | Let <math>r_1</math> and <math>r_2</math> be the instantaneous rates of increase of type 1 and type 2, respectively. Then | ||

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

## Revision as of 14:37, 15 July 2008

## 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*).

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