# Drummond:PopGen

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==Notes on population genetics== ==Notes on population genetics== - Let $n_i(t)$ 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 + Let $n_i(t)$ 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 - $n_i(t) = n_i(0)R^t$ + :$n_i(t+1) = n_i(t)R$ + 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
+ :$\begin{matrix} + n_i(t+\Delta t) &=& n_i(t)R^{\Delta t}\\ + n_i(t+\Delta t) &=& n_i(t)R^{\Delta t} + n_i(t) - n_i(t)\\ + n_i(t+\Delta t) - n_i(t) &=& n_i(t)R^{\Delta t} - n_i(t)\\ + \frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} &=& \frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}\\ + \frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} &=& n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\\ + \lim_{\Delta t \to 0} \left[\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}\right] &=& \lim_{\Delta t \to 0} n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\\ + \frac{d n_i(t)}{dt} &=& n_i(t) \lim_{\Delta t \to 0} \frac{R^{\Delta t} - 1}{\Delta t}\\ + \frac{d n_i(t)}{dt} &=& n_i(t) \ln R\\ + \end{matrix}$ + where the last simplification follows from [http://en.wikipedia.org/wiki/L%27Hopital%27s_rule L'Hopital's rule].  Explicitly, let $\epsilon=\Delta t$.  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 + :$n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}$ + 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'' $r = \ln R$ for convenience. +

+
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## 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.