Drummond:PopGen

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Per-generation and instantaneous growth rates

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

n_i(t+1) = n_i(t)R\!
and
n_i(t) = n_i(0)R^t.\!

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

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

where the last simplification follows from L'Hôpital's rule. Explicitly, let ε = Δt. Then

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

The solution to the equation

\frac{d n_i(t)}{dt} = n_i(t) \ln R
is
n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!
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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