Solving the Logistic Equation

Calculus 2 · Axiom Academy

LESSON Solving the Logistic Equation Separate variables, split with partial fractions, and watch the famous S-curve fall out of the algebra. 1. The Equation With a Built-In Brake The logistic model multiplies the exponential rate kP by a braking factor that shrinks as the population P rises toward the carrying capacity L : P = population · t = time · k = growth rate · L = carrying capacity Plot the growth rate against the population P and a story appears. Watch how fast the population is growing as P climbs from 0 to L : Separate the variables — all the P 's on one side, t on the other — then integrate. The left side looks hard until you realize the messy fraction splits cleanly into two simple pieces : Watch the single fraction break apart by partial fractions , then each half integrate to a natural log: — both numerators come out to exactly 1 . Because , that hidden factor of L is exactly what turns the usual coefficients into clean 1 's — the algebra is tailor-made for this equation. Exponentiate to clear the logs, then solve for P . The constant A is fixed by the starting population P_0 = P(0) . The result is the logistic function — and every logistic solution has this same S-shape: Watch it draw itself from the starting value, bend through the inflection at , and flatten against the carrying capacity: Concave up — growth accelerates, almost exponential while the brake is still weak. Steepest point — growth rate is maximal, then begins to slow.

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