The Logistic Model

Calculus 2 · Axiom Academy

Exponential growth meets a hard ceiling — and the result is the S-shaped curve that real populations actually follow. Pure exponential growth says — the bigger the population, the faster it grows, with no limit. The logistic model multiplies that by a braking factor that shrinks toward zero as P nears the carrying capacity L . P(t) — the population at time t k — the intrinsic growth rate (k > 0) L — the carrying capacity (maximum sustainable population) 2. Population Behavior Over Time Solving the logistic equation gives an S-shaped (sigmoid) curve. Watch the population dot: it inches up at first, then races through the middle, then flattens as it presses against the carrying-capacity line L — three distinct phases from one equation. P is small, so even a high per-capita rate moves few individuals. Absolute growth is modest. Near P = L/2 the product is largest, so the curve climbs fastest. As the brake dominates and growth fades, approaching L but never crossing it. At P = L/2 the curve switches from concave up to concave down — the moment growth is maximal. Unconstrained, would bend upward forever (a J). The braking factor turns the late stage downward, splicing exponential lift-off onto a flat plateau — the classic sigmoid. Plot the growth rate against the population P itself and you get a downward parabola. It is zero at the empty population and again at carrying capacity, and it peaks exactly halfway between — the population grows fastest when it is at half of L .

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