# Drummond:PopGen

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- ==Notes on population genetics== + ==Per-generation and instantaneous growth rates==

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

Now we wish to move to the case where $t$ is continuous and real-valued. Now we wish to move to the case where $t$ is continuous and real-valued. As before,
As before,
- :$n_i(t+1) = n_i(t)R + :[itex]n_i(t+1) = n_i(t)R\!$ but now
but now
- :\begin{matrix} + :{| - n_i(t+\Delta t) &=& n_i(t)R^{\Delta t}\\ + |align="right" |[itex]n_i(t+\Delta t)\! - n_i(t+\Delta t) &=& n_i(t)R^{\Delta t} + n_i(t) - n_i(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}\\ + |align="right" |$n_i(t+\Delta t) - n_i(t)\!$ - \frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} &=& n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\\ + |$= n_i(t)R^{\Delta t} - n_i(t)\!$ - \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}\\ + |align="right" |$\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}$ - \frac{d n_i(t)}{dt} &=& n_i(t) \ln R\\ + |$=\frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t}$ - \end{matrix}[/itex] + |- - 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
+ |align="right" |$\frac{n_i(t+\Delta t) - n_i(t)}{\Delta t}$ - :$+ |[itex]=n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}$ - \begin{matrix} + |- - \lim_{\Delta t \to 0} \frac{R^{\Delta t} - 1}{\Delta t} &=& \lim_{\epsilon \to 0} \frac{R^{\epsilon} - 1}{\epsilon}\\ + |align="right" |$\lim_{\Delta t \to 0} \left[{n_i(t+\Delta t) - n_i(t) \over \Delta t}\right]$ - &=& \lim_{\epsilon \to 0} \frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\\ + |$=\lim_{\Delta t \to 0} \left[ n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\right]$ - &=& \lim_{\epsilon \to 0} \frac{R^{\epsilon}\ln R}{1}\\ + |- - &=& \ln R \lim_{\epsilon \to 0} \frac{R^{\epsilon}}{1}\\ + |align="right" |$\frac{d n_i(t)}{dt}$ - &=& \ln R. + |$=n_i(t) \lim_{\Delta t \to 0} \left[\frac{R^{\Delta t} - 1}{\Delta t}\right]$ - \end{matrix} + |- - [/itex] + |align="right" |$\frac{d n_i(t)}{dt}$ + |$=n_i(t) \ln R\!$ + |} + where the last simplification follows from [http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule L'Hôpital's rule].  Explicitly, let $\epsilon=\Delta t$.  Then
+ :{| + |- + |$\lim_{\Delta t \to 0} \left[{R^{\Delta t} - 1 \over \Delta t}\right]$ + |$= \lim_{\epsilon \to 0} \left[\frac{R^{\epsilon} - 1}{\epsilon}\right]$ + |- + | + |$=\lim_{\epsilon \to 0} \left[\frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\right]$ + |- + | + |$=\lim_{\epsilon \to 0} \left[\frac{R^{\epsilon}\ln R}{1}\right]$ + |- + | + |$=\ln R \lim_{\epsilon \to 0} \left[R^{\epsilon}\right]$ + |- + | + |$=\ln R\!$ + |}

Line 39: Line 59: :$\frac{d n_i(t)}{dt} = n_i(t) \ln R$ :$\frac{d n_i(t)}{dt} = n_i(t) \ln R$ is is - :$n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}$ + :$n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\!$ 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. 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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## 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.