Drummond:PopGen

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Per-generation and instantaneous growth rates

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]n_i(t+\Delta t)\![/math] [math]=n_i(t)R^{\Delta t}\![/math]
[math]n_i(t+\Delta t) - n_i(t)\![/math] [math]= n_i(t)R^{\Delta t} - n_i(t)\![/math]
[math]\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}[/math] [math]=\frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}[/math]
[math]\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}[/math] [math]=n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}[/math]
[math]\lim_{\Delta t \to 0} \left[{n_i(t+\Delta t) - n_i(t) \over \Delta t}\right][/math] [math]=\lim_{\Delta t \to 0} \left[ n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\right][/math]
[math]\frac{d n_i(t)}{dt}[/math] [math]=n_i(t) \lim_{\Delta t \to 0} \left[\frac{R^{\Delta t} - 1}{\Delta t}\right][/math]
[math]\frac{d n_i(t)}{dt}[/math] [math]=n_i(t) \ln R\![/math]

where the last simplification follows from L'Hôpital's rule. Explicitly, let [math]\epsilon=\Delta t[/math]. Then

[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]

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.