# Drummond:PopGen

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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 of increase'' <math>r = \ln R</math> for convenience. | 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 of increase'' <math>r = \ln R</math> for convenience. | ||

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

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 |

## Revision as of 19:39, 14 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

*n*(*t*) =*n*_{1}(*t*) +*n*_{2}(*t*)

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*).

**Failed to parse (syntax error): = {\left({r_1 n_1(t) \over n(t)}\right) - {n_1(t) \over n(t)^2}\left({\partial n_1(t) \over \partial t} + {\partial n_2(t) \over \partial t}\right)**