Drummond:PopGen

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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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