Biological processes described by the exponential function

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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 $a$.