Algebraic

You can call me Al.

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

$n_{i}(t+1)=n_{i}(t)R\!$

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

$n_{i}(t+\Delta t)\!$	$=n_{i}(t)R^{\Delta t}\!$
$n_{i}(t+\Delta t)-n_{i}(t)\!$	$=n_{i}(t)R^{\Delta t}-n_{i}(t)\!$
${\frac {n_{i}(t+\Delta t)-n_{i}(t)}{\Delta t}}$	$={\frac {n_{i}(t)R^{\Delta t}-n_{i}(t)}{\Delta t}}$
${\frac {n_{i}(t+\Delta t)-n_{i}(t)}{\Delta t}}$	$=n_{i}(t){\frac {R^{\Delta t}-1}{\Delta t}}$
$\lim _{\Delta t\to 0}\left[{n_{i}(t+\Delta t)-n_{i}(t) \over \Delta t}\right]$	$=\lim _{\Delta t\to 0}\left[n_{i}(t){\frac {R^{\Delta t}-1}{\Delta t}}\right]$
${\frac {dn_{i}(t)}{dt}}$	$=n_{i}(t)\lim _{\Delta t\to 0}\left[{\frac {R^{\Delta t}-1}{\Delta t}}\right]$
${\frac {dn_{i}(t)}{dt}}$	$=n_{i}(t)\ln R\!$

where the last simplification follows from L'Hôpital's rule. Explicitly, let $\epsilon =\Delta t$. Then

$\lim _{\Delta t\to 0}\left[{R^{\Delta t}-1 \over \Delta t}\right]$	$=\lim _{\epsilon \to 0}\left[{\frac {R^{\epsilon }-1}{\epsilon }}\right]$
	$=\lim _{\epsilon \to 0}\left[{\frac {{\frac {d}{d\epsilon }}\left(R^{\epsilon }-1\right)}{{\frac {d}{d\epsilon }}\epsilon }}\right]$
	$=\lim _{\epsilon \to 0}\left[{\frac {R^{\epsilon }\ln R}{1}}\right]$
	$=\ln R\lim _{\epsilon \to 0}\left[R^{\epsilon }\right]$
	$=\ln R\!$

The solution to the equation

${\frac {dn_{i}(t)}{dt}}=n_{i}(t)\ln R$

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.