Logistic Models with Differential Equations

Plain exponential growth, dP/dt = kP, assumes there's always room and food for more. Real populations run into limits: space, resources, predators. The logistic equation builds that limit directly into the rate. It still starts with kP, the exponential push, but multiplies it by (1 - P/K), where K is the carrying capacity, the population size the environment can sustain.

That factor (1 - P/K) is the brake. When P is small compared to K, the fraction P/K is near 0, so (1 - P/K) is near 1, and the equation behaves almost exactly like exponential growth — fast and unchecked. As P climbs toward K, P/K approaches 1, so (1 - P/K) shrinks toward 0, and dP/dt shrinks with it. The growth rate is throttled continuously, not suddenly: there's no moment where the curve switches from exponential to flat. It just keeps slowing, smoothly, until P essentially stops changing at P = K, where dP/dt = 0. Also worth noticing: dP/dt is largest not at the start but when P = K/2, which is why the logistic curve is S-shaped — concave up then concave down.

This topic sits right after unrestricted exponential growth and solving separable differential equations, and it uses both. The logistic equation is separable, and solving it gives the closed form P(t) = K / (1 + ((K - P0)/P0) e^−kt), built from the same separation-and-integration moves used on dP/dt = kP. Letting K get arbitrarily large in that formula recovers ordinary exponential growth — logistic growth is exponential growth with a limit folded in.