Drummond:PopGen

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(Notes on population genetics)
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==Notes on population genetics==
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==Per-generation and instantaneous growth rates==
<p>
<p>
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
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+1) = n_i(t)R</math>
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:<math>n_i(t+1) = n_i(t)R\!</math>
and
and
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:<math>n_i(t) = n_i(0)R^t</math>.
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:<math>n_i(t) = n_i(0)R^t.\!</math>
</p>
</p>
<p>
<p>
Now we wish to move to the case where <math>t</math> is continuous and real-valued.
Now we wish to move to the case where <math>t</math> is continuous and real-valued.
As before,<br/>
As before,<br/>
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:<math>n_i(t+1) = n_i(t)R</math><br/>
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:<math>n_i(t+1) = n_i(t)R\!</math>
but now<br/>
but now<br/>
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:<math>\begin{matrix}
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:{|
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n_i(t+\Delta t) &=& n_i(t)R^{\Delta t}\\
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|align="right" |<math>n_i(t+\Delta t)\!</math>
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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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|<math>=n_i(t)R^{\Delta t}\!</math>
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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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|-
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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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|align="right" |<math>n_i(t+\Delta t) - n_i(t)\!</math>
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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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|<math>= n_i(t)R^{\Delta t} - n_i(t)\!</math>
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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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|-
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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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|align="right" |<math>\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}</math>
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\frac{d n_i(t)}{dt} &=& n_i(t) \ln R\\
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|<math>=\frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}</math>
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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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|align="right" |<math>\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}</math>
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:<math>
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|<math>=n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}</math>
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\begin{matrix}
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|-
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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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|align="right" |<math>\lim_{\Delta t \to 0} \left[{n_i(t+\Delta t) - n_i(t) \over \Delta t}\right]</math>
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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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|<math>=\lim_{\Delta t \to 0} \left[ n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\right]</math>
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&=& \lim_{\epsilon \to 0} \frac{R^{\epsilon}\ln R}{1}\\
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|-
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&=& \ln R \lim_{\epsilon \to 0} \frac{R^{\epsilon}}{1}\\
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|align="right" |<math>\frac{d n_i(t)}{dt}</math>
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&=& \ln R.
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|<math>=n_i(t) \lim_{\Delta t \to 0} \left[\frac{R^{\Delta t} - 1}{\Delta t}\right]</math>
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\end{matrix}
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|-
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</math>
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|align="right" |<math>\frac{d n_i(t)}{dt}</math>
 +
|<math>=n_i(t) \ln R\!</math>
 +
|}
 +
where the last simplification follows from [http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule L'Hôpital's rule].  Explicitly, let <math>\epsilon=\Delta t</math>.  Then<br/>
 +
:{|
 +
|-
 +
|<math>\lim_{\Delta t \to 0} \left[{R^{\Delta t} - 1 \over \Delta t}\right]</math>
 +
|<math>= \lim_{\epsilon \to 0} \left[\frac{R^{\epsilon} - 1}{\epsilon}\right]</math>
 +
|-
 +
|
 +
|<math>=\lim_{\epsilon \to 0} \left[\frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\right]</math>
 +
|-
 +
|
 +
|<math>=\lim_{\epsilon \to 0} \left[\frac{R^{\epsilon}\ln R}{1}\right]</math>
 +
|-
 +
|
 +
|<math>=\ln R \lim_{\epsilon \to 0} \left[R^{\epsilon}\right]</math>
 +
|-
 +
|
 +
|<math>=\ln R\!</math>
 +
|}
</p>
</p>
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:<math>\frac{d n_i(t)}{dt} = n_i(t) \ln R</math>
:<math>\frac{d n_i(t)}{dt} = n_i(t) \ln R</math>
is
is
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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 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.
</p>
</p>
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Revision as of 02:40, 5 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.

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