# Drummond:PopGen

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

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## Revision as of 22:49, 23 January 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*).

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