Drummond:PopGen

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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

[math]n_i(t+1) = n_i(t)R[/math]
and
[math]n_i(t) = n_i(0)R^t[/math].

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

[math]n_i(t+1) = n_i(t)R[/math]
but now
[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 L'Hopital's rule. Explicitly, let [math]\epsilon=\Delta t[/math]. Then
[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]

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.