exponential growth and decay

What do a savings account earning compound interest, a colony of bacteria, and a lump of radioactive rock have in common? In each, the amount present right now controls how fast it changes. Twice as much money earns twice the interest; twice as many bacteria produce twice as many offspring per hour; twice as many radioactive atoms decay twice as fast. This single idea — rate of change proportional to current amount — is the most fundamental model in all of differential equations.

In symbols it reads dy/dt = ky, where y(t) is the quantity and k is the proportionality constant. If k is positive the quantity grows; if k is negative it decays. The solution is y(t) = y0 e^(kt), where y0 is the starting amount: the exponential function is precisely the function that equals its own rate of change (up to the factor k). Growth means the curve sweeps upward ever faster; decay means it falls steeply at first and then flattens, approaching zero but never quite reaching it.

Exponential models are exact only while the proportionality holds. Real populations cannot grow exponentially forever — food and space run out — so the model is a good description only early on, before crowding bites; the logistic equation repairs this. But for radioactive decay the law is essentially perfect, because each atom decays independently of the others, with no crowding to interfere.