Jane Jackson's note prompts me to share with you a problem I pose in my
course for biology students. In the same course we do RC circuits and
radioactive decay, so all these problems are first order differential
equations to which the students must learn solutions. Malthus wasn't a
physicist, but I think he would laugh heartily today if someone were to
fire that popular oxymoron "sustainable growth" at him. He did understand
exponentials.
Here is the problem as I gave it in the spring semester.
The Big Crunch
The following problem is given to you as a challenge. It illustrates the
dominance of the exponential function over a power law. We will encounter
exponential growth (or decay) again later in the course.
Imagine an infinite three dimensional sea of nutrient. In this sea we
place one bacterium of volume a^3. This bacterium metabolizes, doubles
its volume, and divides into two new bacteria identical to the original
in time T. During the intervening time it can swim with a maximum speed v.
If this colony of bacteria does not expand by swimming the bacteria will
soon run out of nutrient. Worse still, they will fill the volume of space
available to them. If the bacteria swim sufficiently cleverly they can
occupy a volume of the sea which increases in time. Eventually, however,
their exponential growth will catch up with their expansionist strategy
and they will run out of nutrient and fill all available space completely.
It is clear that the optimal strategy will involve filling a sphere of
maximum size attainable under the circumstances given.
1. Given a =3D 50 =B5m, T =3D 1200 s, and v =3D 2.0 mm/s, calculate how=
long they
can survive.
2. Given the same bacteria, but supposing they can swim with v =3D 3x10^7 m=
/s
(one tenth the speed of light), what is the radius of the colony on doomsda=
y?
I hope that you (and perhaps some of your students) will enjoy this.
(And yes, Jane, I'm well over 40.)
Leigh
Leigh Hunt Palmer
Department of Physics
Simon Fraser University
Burnaby, BC V5A 1S6
Canada