Drummond:PopGen
Per-generation and instantaneous growth rates
Let [math]\displaystyle{ n_i(t) }[/math] be the number of organisms of type [math]\displaystyle{ i }[/math] at time [math]\displaystyle{ t }[/math], and let [math]\displaystyle{ R }[/math] be the per-capita reproductive rate per generation. If [math]\displaystyle{ t }[/math] counts generations, then
- [math]\displaystyle{ n_i(t+1) = n_i(t)R\! }[/math]
- [math]\displaystyle{ n_i(t) = n_i(0)R^t.\! }[/math]
Now we wish to move to the case where [math]\displaystyle{ t }[/math] is continuous and real-valued.
As before,
- [math]\displaystyle{ n_i(t+1) = n_i(t)R\! }[/math]
[math]\displaystyle{ n_i(t+\Delta t)\! }[/math] [math]\displaystyle{ =n_i(t)R^{\Delta t}\! }[/math] [math]\displaystyle{ n_i(t+\Delta t) - n_i(t)\! }[/math] [math]\displaystyle{ = n_i(t)R^{\Delta t} - n_i(t)\! }[/math] [math]\displaystyle{ \frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} }[/math] [math]\displaystyle{ =\frac{n_i(t)R^{\Delta t} - n_i(t)}{\Delta t} }[/math] [math]\displaystyle{ \frac{n_i(t+\Delta t) - n_i(t)}{\Delta t} }[/math] [math]\displaystyle{ =n_i(t) \frac{R^{\Delta t} - 1}{\Delta t} }[/math] [math]\displaystyle{ \lim_{\Delta t \to 0} \left[{n_i(t+\Delta t) - n_i(t) \over \Delta t}\right] }[/math] [math]\displaystyle{ =\lim_{\Delta t \to 0} \left[ n_i(t) \frac{R^{\Delta t} - 1}{\Delta t}\right] }[/math] [math]\displaystyle{ \frac{d n_i(t)}{dt} }[/math] [math]\displaystyle{ =n_i(t) \lim_{\Delta t \to 0} \left[\frac{R^{\Delta t} - 1}{\Delta t}\right] }[/math] [math]\displaystyle{ \frac{d n_i(t)}{dt} }[/math] [math]\displaystyle{ =n_i(t) \ln R\! }[/math]
where the last simplification follows from L'Hôpital's rule. Explicitly, let [math]\displaystyle{ \epsilon=\Delta t }[/math]. Then
[math]\displaystyle{ \lim_{\Delta t \to 0} \left[{R^{\Delta t} - 1 \over \Delta t}\right] }[/math] [math]\displaystyle{ = \lim_{\epsilon \to 0} \left[\frac{R^{\epsilon} - 1}{\epsilon}\right] }[/math] [math]\displaystyle{ =\lim_{\epsilon \to 0} \left[\frac{\frac{d}{d\epsilon}\left(R^{\epsilon} - 1\right)}{\frac{d}{d\epsilon}\epsilon}\right] }[/math] [math]\displaystyle{ =\lim_{\epsilon \to 0} \left[\frac{R^{\epsilon}\ln R}{1}\right] }[/math] [math]\displaystyle{ =\ln R \lim_{\epsilon \to 0} \left[R^{\epsilon}\right] }[/math] [math]\displaystyle{ =\ln R\! }[/math]
The solution to the equation
- [math]\displaystyle{ \frac{d n_i(t)}{dt} = n_i(t) \ln R }[/math]
- [math]\displaystyle{ n_i(t) = n_i(0) e^{t\ln R} = n_i(0) R^{t}.\! }[/math]
Continuous rate of change
Let [math]\displaystyle{ r_1 }[/math] and [math]\displaystyle{ r_2 }[/math] be the instantaneous rates of increase of type 1 and type 2, respectively. Then
- [math]\displaystyle{ {dn_i(t) \over dt} = r_i n_i(t). }[/math]
With the total population size
- [math]\displaystyle{ n(t) = n_1(t) + n_2(t) }[/math]
we have the proportion of type 1
- [math]\displaystyle{ p(t) = {n_1(t) \over n(t)} }[/math]
Define the fitness advantage
- [math]\displaystyle{ s \equiv s_{12} = r_1 - r_2\! }[/math]
Given our interest in understanding the change in gene frequencies, our goal is to compute the rate of change of [math]\displaystyle{ p(t) }[/math].
[math]\displaystyle{ {\partial p(t) \over \partial t} }[/math] [math]\displaystyle{ = {\partial \over \partial t}\left({n_1(t) \over n(t)}\right) }[/math] [math]\displaystyle{ = {\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} }[/math] [math]\displaystyle{ = {\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) }[/math] [math]\displaystyle{ = {\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) }[/math]