Drummond:PopGen

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(Continuous-time approximation)
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==Continuous-time approximation==
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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
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:<math>{dn_i(t) \over dt} = r_i n_i(t).</math>
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With the total population size
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:<math>n(t) = n_1(t) + n_2(t)</math>
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we have the proportion of type 1
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:<math>p(t) = {n_1(t) \over n(t)}</math>
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Define the fitness advantage
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:<math>s \equiv s_{12} = r_1 - r_2\!</math>
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Given our interest in understanding the change in gene frequencies, our goal is to compute the rate of change of <math>p(t)</math>.
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:{|
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|<math>{\partial p(t) \over \partial t}</math>
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|<math>= {\partial  \over \partial t}\left({n_1(t) \over n(t)}\right)</math>
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|-
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|
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|<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>
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|-
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|
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|<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>
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|-
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|<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>
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|}
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==Diffusion approximation==
==Diffusion approximation==
==Diffusion approximation==

Revision as of 11:31, 14 July 2008

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Per-generation and instantaneous growth rates

Let ni(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

n_i(t+1) = n_i(t)R\!
and
n_i(t) = n_i(0)R^t.\!

Now we wish to move to the case where t is continuous and real-valued. As before,

n_i(t+1) = n_i(t)R\!
but now
n_i(t+\Delta t)\! =n_i(t)R^{\Delta t}\!
n_i(t+\Delta t) - n_i(t)\! = n_i(t)R^{\Delta t} - n_i(t)\!
\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} =\frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}
\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} =n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}
\lim_{\Delta t \to 0} \left[{n_i(t+\Delta t) - n_i(t) \over \Delta t}\right] =\lim_{\Delta t \to 0} \left[ n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\right]
\frac{d n_i(t)}{dt} =n_i(t) \lim_{\Delta t \to 0} \left[\frac{R^{\Delta t} - 1}{\Delta t}\right]
\frac{d n_i(t)}{dt} =n_i(t) \ln R\!

where the last simplification follows from L'Hôpital's rule. Explicitly, let ε = Δt. Then

\lim_{\Delta t \to 0} \left[{R^{\Delta t} - 1 \over \Delta t}\right] = \lim_{\epsilon \to 0} \left[\frac{R^{\epsilon} - 1}{\epsilon}\right]
=\lim_{\epsilon \to 0} \left[\frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\right]
=\lim_{\epsilon \to 0} \left[\frac{R^{\epsilon}\ln R}{1}\right]
=\ln R \lim_{\epsilon \to 0} \left[R^{\epsilon}\right]
=\ln R\!

The solution to the equation

\frac{d n_i(t)}{dt} = n_i(t) \ln R
is
n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!
Note that the continuous case and the original discrete-generation case agree for all values of t. We can define the instantaneous rate of increase r = lnR for convenience.

Let r1 and r2 be the instantaneous rates of increase of type 1 and type 2, respectively. Then

{dn_i(t) \over dt} = r_i n_i(t).

With the total population size

n(t) = n1(t) + n2(t)

we have the proportion of type 1

p(t) = {n_1(t) \over n(t)}

Define the fitness advantage

s \equiv s_{12} = r_1 - r_2\!

Given our interest in understanding the change in gene frequencies, our goal is to compute the rate of change of p(t).

{\partial p(t) \over \partial t} = {\partial  \over \partial t}\left({n_1(t) \over n(t)}\right)
= {\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}
= {\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)
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)

Diffusion approximation

==Diffusion approximation==
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