Maximum Growth Rate

Calculus 2 · Axiom Academy

A logistic population grows fastest at exactly half its carrying capacity — and that's the secret to harvesting it forever. 1. Growth Is Steepest in the Middle In nature a population can't grow forever — resources run out at the carrying capacity L . The logistic equation builds that ceiling right in: the factor is the brake, shrinking toward zero as P climbs toward L . the growth rate of the population P(t) P(t) — the population at time t r — the intrinsic (unchecked) growth rate L — the carrying capacity, the largest population the environment can sustain Trace a point along the S-shaped curve below. Watch the tangent line glued to it: nearly flat at the bottom, steepest as it crosses the dashed P = L/2 line , then flattening again near the top. That tilt is the growth rate dP/dt — and it peaks right in the middle. Forget time for a moment and look at the growth rate purely as a function of how many individuals there are. Expanding the logistic right-hand side, — a downward parabola in P , zero at both P = 0 and P = L . A parabola peaks at its vertex, which we find the calculus way — differentiate g with respect to P and set it to zero: Watch the height bar below: as the population sweeps from 0 up to L , the growth rate rises, crests, and falls — and the crest lands squarely on P = L/2 . The curve is concave up — growth is still accelerating. Few individuals, room to spare. The vertex. Growth rate is at its absolute maximum — the sweet spot.

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