Drummond:PopGen

Per-generation and instantaneous growth rates

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]

and

[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]

but now

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

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

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

The solution to the equation

[math]\displaystyle{ \frac{d n_i(t)}{dt} = n_i(t) \ln R }[/math]

is

[math]\displaystyle{ 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]\displaystyle{ t }[/math]. We can define the instantaneous rate of increase [math]\displaystyle{ r = \ln R }[/math] for convenience.