# 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.

## Continuous rate of change

Let r1 and r2 be the instantaneous rates of increase of type 1 and type 2, respectively. Then

${dn_i(t) \over dt} = r_i n_i(t).$

With the total population size

n(t) = n1(t) + n2(t)

we have the proportion of type 1

$p(t) = {n_1(t) \over n(t)}$

$s \equiv s_{12} = r_1 - r_2\!$
 ${\partial p(t) \over \partial t}$ $= {\partial \over \partial t}\left({n_1(t) \over n(t)}\right)$ $= {\partial n_1(t) \over \partial t}\left({1 \over n(t)}\right) + n_1(t){-1 \over n(t)^2}{\partial n(t) \over \partial t}$ $= {\partial n_1(t) \over \partial t}\left({1 \over n(t)}\right) + n_1(t){-1 \over n(t)^2}\left({\partial n_1(t) \over \partial t} + {\partial n_2(t) \over \partial t}\right)$ Failed to parse (syntax error): = {\left({r_1 n_1(t) \over n(t)}\right) - {n_1(t) \over n(t)^2}\left({\partial n_1(t) \over \partial t} + {\partial n_2(t) \over \partial t}\right)