Biological processes described by the exponential function

What are biological or biochemical processes that are described (stochastically or deterministically) by exponential function $f(x) = e^{ax}$?

Edit: What about examples that are not related to the population growth? I'm especially interested in biochemical examples.

• The canonical example in college algebra texts is the growth of bacteria populations (or populations in general, really). – Austin Mohr Nov 23 '11 at 21:53
• See kinetics section of wiki page of chemical reactions. You would also find some of the demonstrations on chemical kinetics from Wolfram Demonstrations project useful – Sasha Nov 23 '11 at 22:31

whenever a process evolves at a speed proportionnel to its own advancement, it satisfies the differential equation $y'=a y$ which leads to exponential solutions.

this is a very basic model for all kind of population growth (as you'd say that the bigger the population is, the more baby they make per time unit).

it is universal in the sense that if $y$ satisfies a more complicated evolution equation $y'=f(y)$ near an equilibrium point (for example, if you have a very small initial population, or a starting process) then linearizing this equation you obtain something like $y' \approx f'(y_0) y$ where $y_0$ is the equilibrium point.

Addendum : radioactive decay follows a decreasing exponential law because of the time-independance nature of the probability for a particle to split.

Basic equations in kinetic chemistry are exponential laws (though they get more complicated for complex reactions), I suspect mainly because of the universal property I mentionned : it is "the most simple evolution to an equilibrium state" in a sense.

When a drug is eliminated from the body, typically that is described by giving a "half life" ... which means it is an exponential decay $Ce^{at}$ with $a < 0$. Then $T$ is the half-life if $e^{aT}=1/2$, so it is related to the constant $a$.

Even when more than one species is involved, the simplest models of population evolution (e.g., the simplest predator-prey models) lead to exponentials (or linear combinations thereof).

Also, people can work out how many hours ago you died by applying Newton's Law of Cooling, which amounts to an exponential function.

• I'm still alive! – Leo Nov 23 '11 at 22:12
• I'm glad to hear that. But, you know, it pays to plan ahead. – Gerry Myerson Nov 23 '11 at 22:49

PCR (polymerase chain reaction) is the best example of an exponential function in biochemistry IMO. It's also population growth but it's a lot cleaner than bacterial growth. And people actually do math on it - quantitative PCR attempts to calculate the starting population from later, detectable amounts of DNA. It is a sensitive enough technique to quantify a couple of handfuls starting molecules.

This answer is probably 5 years too late though.

• Quantitative PCR is a typical example where exponential growth is mitigated by growth limitations which imply a gradual transition to linear growth. – Did Sep 1 '16 at 8:20
• What's an example of exponential growth without growth limitations? – grofte Sep 1 '16 at 8:24

an example of a BIOLOGICAL exponential decay process is the concentration of a drug in the bloodstream, after it has benn intravenously injected into the body :)

• This was in GEdgar's answer in 2011. – Gerry Myerson May 31 '17 at 0:08