Lightning Growth

We created the animation below to give the students some kind of “hands-on” experience with the fundamental process of exponential growth.

In this animation, we have a population of particles moving randomly inside a circle. Assume that we have a “lightning storm” with one strike every second at a random place inside the circle. Any particle that is within a fixed distance δ of the strike (keep in mind that a particle is a point) produces an “offspring” that immediately become a new particle and goes off in a random direction.  That’s how the population grows.

We ask the students: is the growth exponential, and if so, what is its equation? They almost always reply: “Well the more particles that are in the circle, the greater the chance of an offspring, so the faster the population grows. That means it’s exponential.” Just like commentators on the radio, they have missed the defining property of exponential growth, that the growth rate of the population is proportional to its size.