Drummond:PopGen

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==Notes on population genetics==
==Notes on population genetics==
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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
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<p>
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<math>n_i(t+1) = n_i(t)R</math> and
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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
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<math>n_i(t) = n_i(0)R^t</math>
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:<math>n_i(t+1) = n_i(t)R</math>
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and
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:<math>n_i(t) = n_i(0)R^t</math>.
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</p>
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<p>
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Now we wish to move to the case where <math>t</math> is continuous and real-valued.
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As before,<br/>
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:<math>n_i(t+1) = n_i(t)R</math><br/>
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but now<br/>
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:<math>\begin{matrix}
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n_i(t+\Delta t) &=& n_i(t)R^{\Delta t}\\
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n_i(t+\Delta t) &=& n_i(t)R^{\Delta t} + n_i(t) - n_i(t)\\
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n_i(t+\Delta t) - n_i(t) &=& n_i(t)R^{\Delta t} - n_i(t)\\
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\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} &=& \frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}\\
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\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} &=& n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\\
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\lim_{\Delta t \to 0} \left[\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}\right] &=& \lim_{\Delta t \to 0} n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\\
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\frac{d n_i(t)}{dt} &=& n_i(t) \lim_{\Delta t \to 0} \frac{R^{\Delta t} - 1}{\Delta t}\\
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\frac{d n_i(t)}{dt} &=& n_i(t) \ln R\\
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\end{matrix}</math>
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where the last simplification follows from [http://en.wikipedia.org/wiki/L%27Hopital%27s_rule L'Hopital's rule].  Explicitly, let <math>\epsilon=\Delta t</math>.  Then<br/>
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:<math>
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\begin{matrix}
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\lim_{\Delta t \to 0} \frac{R^{\Delta t} - 1}{\Delta t} &=& \lim_{\epsilon \to 0} \frac{R^{\epsilon} - 1}{\epsilon}\\
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&=& \lim_{\epsilon \to 0} \frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\\
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&=& \lim_{\epsilon \to 0} \frac{R^{\epsilon}\ln R}{1}\\
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&=& \ln R \lim_{\epsilon \to 0} \frac{R^{\epsilon}}{1}\\
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&=& \ln R.
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\end{matrix}
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</math>
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</p>
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<p>
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The solution to the equation
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:<math>\frac{d n_i(t)}{dt} = n_i(t) \ln R</math>
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is
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:<math>n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}</math>
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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.
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</p>
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Revision as of 19:22, 4 July 2008

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Notes on population genetics

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

ni(t + 1) = ni(t)R
and
ni(t) = ni(0)Rt.

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

ni(t + 1) = ni(t)R
but now
\begin{matrix}
n_i(t+\Delta t) &=& n_i(t)R^{\Delta t}\\
n_i(t+\Delta t) &=& n_i(t)R^{\Delta t} + n_i(t) - n_i(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[\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}\right] &=& \lim_{\Delta t \to 0} n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\\
\frac{d n_i(t)}{dt} &=& n_i(t) \lim_{\Delta t \to 0} \frac{R^{\Delta t} - 1}{\Delta t}\\
\frac{d n_i(t)}{dt} &=& n_i(t) \ln R\\
\end{matrix}
where the last simplification follows from L'Hopital's rule. Explicitly, let ε = Δt. Then

\begin{matrix}
\lim_{\Delta t \to 0} \frac{R^{\Delta t} - 1}{\Delta t} &=& \lim_{\epsilon \to 0} \frac{R^{\epsilon} - 1}{\epsilon}\\
&=& \lim_{\epsilon \to 0} \frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\\
&=& \lim_{\epsilon \to 0} \frac{R^{\epsilon}\ln R}{1}\\
&=& \ln R \lim_{\epsilon \to 0} \frac{R^{\epsilon}}{1}\\
&=& \ln R.
\end{matrix}

The solution to the equation

\frac{d n_i(t)}{dt} = n_i(t) \ln R
is
ni(t) = ni(0)etlnR = ni(0)Rt
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.


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