Drummond:PopGen: Difference between revisions
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==Notes on population genetics== | ==Notes on population genetics== | ||
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 | <p> | ||
<math>n_i(t+1) = n_i(t)R</math> and | 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 | ||
<math>n_i(t) = n_i(0)R^t</math> | :<math>n_i(t+1) = n_i(t)R</math> | ||
and | |||
:<math>n_i(t) = n_i(0)R^t</math>. | |||
</p> | |||
<p> | |||
Now we wish to move to the case where <math>t</math> is continuous and real-valued. | |||
As before,<br/> | |||
:<math>n_i(t+1) = n_i(t)R</math><br/> | |||
but now<br/> | |||
:<math>\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}</math> | |||
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/> | |||
:<math> | |||
\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} | |||
</math> | |||
</p> | |||
<p> | |||
The solution to the equation | |||
:<math>\frac{d n_i(t)}{dt} = n_i(t) \ln R</math> | |||
is | |||
:<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. | |||
</p> | |||
</div> | </div> | ||
{{Drummond_Bottom}} | {{Drummond_Bottom}} |
Revision as of 16:22, 4 July 2008
Notes on population genetics
Let [math]\displaystyle{ n_i(t) }[/math] be the number of organisms of type [math]\displaystyle{ i }[/math] at time [math]\displaystyle{ t }[/math], and let [math]\displaystyle{ R }[/math] be the per-capita reproductive rate per generation. If [math]\displaystyle{ t }[/math] counts generations, then
- [math]\displaystyle{ n_i(t+1) = n_i(t)R }[/math]
- [math]\displaystyle{ n_i(t) = n_i(0)R^t }[/math].
Now we wish to move to the case where [math]\displaystyle{ t }[/math] is continuous and real-valued.
As before,
- [math]\displaystyle{ n_i(t+1) = n_i(t)R }[/math]
- [math]\displaystyle{ \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} }[/math]
- [math]\displaystyle{ \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} }[/math]
The solution to the equation
- [math]\displaystyle{ \frac{d n_i(t)}{dt} = n_i(t) \ln R }[/math]
- [math]\displaystyle{ n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t} }[/math]