# Drummond:PopGen

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- | == | + | 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> | ||

+ | With the total population size | ||

+ | :<math>n(t) = n_1(t) + n_2(t)</math> | ||

+ | we have the proportion of type 1 | ||

+ | :<math>p(t) = {n_1(t) \over n(t)}</math> | ||

+ | Define the fitness advantage | ||

+ | :<math>s \equiv s_{12} = r_1 - r_2\!</math> | ||

+ | |||

+ | Given our interest in understanding the change in gene frequencies, our goal is to compute the rate of change of <math>p(t)</math>. | ||

+ | :{| | ||

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

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

+ | |- | ||

+ | | | ||

+ | |<math>= {\partial n_1(t) \over \partial t}\left({1 \over n(t)}\right) + n_1(t){-1 \over n(t)^2}{\partial n(t) \over \partial t}</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= {\partial n_1(t) \over \partial t}\left({1 \over n(t)}\right) + n_1(t){-1 \over n(t)^2}\left({\partial n_1(t) \over \partial t} + {\partial n_2(t) \over \partial t}\right)</math> | ||

+ | |- | ||

+ | | | ||

+ | |<math>= {\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)</math> | ||

+ | |} | ||

+ | |||

+ | ==Diffusion approximation== | ||

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

## Revision as of 11:31, 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.

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)**