logistic equation
Pure exponential growth — dy/dt = r*y — is a fantasy: it says a population, a bank balance, or a rumor doubles forever with no ceiling. Reality has limits. The logistic equation is the simplest honest fix, the model that lets growth start exponentially when resources are plentiful but gently brake as the population approaches the maximum its environment can support. It is the canonical first nonlinear model and the showpiece of qualitative theory.
Its form is dy/dt = r*y*(1 - y/K), where r is the intrinsic growth rate and K is the carrying capacity. Read the two factors: when y is small, the bracket is near 1 and growth is nearly exponential, dy/dt approximately r*y; as y climbs toward K, the bracket shrinks toward 0 and growth stalls; if y ever exceeds K, the bracket goes negative and y declines back. The equation is separable (and is also a Bernoulli equation with n = 2), and its solution is the famous S-shaped sigmoid curve y(t) = K/(1 + A*e^(-r*t)) — slow start, steep middle, leveling plateau at K. On the phase line it has two equilibria: y = 0 (unstable) and y = K (stable), so every positive starting population is funneled toward the carrying capacity.
The logistic equation is one of the most reused models in science: population biology, the spread of epidemics and innovations, autocatalytic chemical reactions, tumor growth, and the loading curves of resources all wear its S-shape. It is the perfect worked example tying this whole field together — separable, Bernoulli-solvable, with an instructive phase line and a clean stability verdict. The honest caveat is that it is a deliberate simplification: it assumes a fixed carrying capacity and a single species with no delays, age structure, or noise, and real populations often deviate.
A fish population with r = 0.5/year and K = 1000 follows dy/dt = 0.5*y*(1 - y/1000). Starting at 100, it rises slowly, accelerates through the steep middle, and levels off near 1000 — the S-curve, with 1000 the stable carrying capacity it never overshoots for long.
Exponential at first, then braking to the carrying capacity K — the classic sigmoid.
The logistic curve is not the only S-shaped growth law, and a smooth approach to K is a modeling assumption, not a law of nature. Adding a time delay or making r large enough turns the discrete version chaotic — the famous logistic map — a reminder that this clean continuous model is a simplification.