# Drummond:PopGen

the drummond lab

## 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.