The Logistic Equation

Differential Equations · Axiom Academy

A fundamental model of population growth with limited resources, showing how a carrying capacity bends exponential growth into an S-curve. 1. The Logistic Differential Equation Start from exponential growth dy/dt = ry and multiply in a braking factor (1 - y/K) , where y(t) is the population, r is the intrinsic growth rate, and K is the carrying capacity — the maximum population the environment can sustain. 2. Equilibria and the Fastest-Growth Point Equilibria occur where dy/dt = 0 — the population isn't changing. Setting the right side to zero factors cleanly into two solutions. A population that starts just above zero keeps growing away from it. The equilibrium repels nearby solutions. A population that starts near K , above or below, is drawn back to it. The equilibrium attracts nearby solutions. 3. The S-Curve: Solution Trajectories Solving the logistic equation exactly (separation of variables + partial fractions) gives a closed form. Every solution — no matter the starting population y_0 — converges to the carrying capacity K . You've seen how a single braking factor turns unlimited exponential growth into a self-limiting S-curve that always settles at the carrying capacity. Scroll up to revisit any step.

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