Without markers. "Too many hopes and dreams won't see the light" (Allen Parsons Project) The Amoeba gets stuck in a local maximum under the constraint.

With markers. "Something in the air, turning me around and guiding me right". Notice the even the Amoeba gets the tune at the end.

This videos are from I.Hecht (without sound) and is from Science News. In both movies, the computerized amoeba has a chemical gradient to help navigating. To the right, there is an additional faint repelling marker which helps the amoeba to avoid trying a wrong direction several times.

The music added for this exhibit is from "Ammonia Avenue" by the Allen Parsons Project.

A rough mathematical model for an amoeba moving without gradient is the random walk or Brownian motion with obstacles. With the gradient, it is brownian motion with drift. Together with markers, it is a selfavoiding random walk in an obstacle with drift. Processes like this are studied in probability theory.

For us, in multivariable calculus, we note that that the strength of the chemical marker is a function f(x,y) of two variables. The gradient of f is a vector field, which is called the chemical gradient. In this case, it serves as a "force field" for the amoeba. The fact that the amoeba gets stuck in the gradient flow under constraint will be explained when we look at Lagrange multipliers.