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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\!
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
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) = n_1(t) + n_2(t)\!

we have the proportion of type 1

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

Define the fitness advantage

s \equiv s_{12} = r_1 - r_2\!

Given our interest in understanding the change in gene frequencies, our goal is to compute the rate of change of p(t).

{\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)
= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n_1(t) + r_2 n_2(t)\right)
= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n_1(t) + (r_1-s)(n(t)-n_1(t))\right)
= {r_1 n_1(t) \over n(t)} - {n_1(t) \over n(t)^2}\left(r_1 n(t) -s n(t) + s n_1(t))\right)
= {n_1(t) \over n(t)^2}\left(s n(t) - s n_1(t))\right)
= s{n_1(t) \over n(t)}\left(1  - {n_1(t) \over n(t)}\right)
= s p(t)(1-p(t))\!

The logit function \mathrm{logit} (p) = \ln {p \over 1-p}, which takes p \in [0,1] \to \mathbb{R}, induces a more natural space for considering changes in frequencies. In logit terms,

Diffusion approximation

Insert math here.
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