Limited Resources

Calculus 2 · Axiom Academy

A few bacteria in a dish double, and double again — until the food starts running out. Watch growth bend toward a ceiling and trace the S-curve. Drop a few bacteria into a dish of nutrients and the early days look explosive: each one splits into two, two into four, four into eight. Pure exponential growth would keep that up forever. But a real dish holds a fixed amount of food — so somewhere, growth has to bend, slow, and stop. The shape that bend produces is one of the most common curves in all of nature. Watch the population climb. Early on it races upward like an exponential, but as the count rises the curve eases off and flattens against the dashed line at the top — the carrying capacity K , the largest population the dish can sustain. The S it traces is logistic growth . The slow start, the steep middle, the leveling off — that three-part shape is the signature of growth under limited resources. The ceiling sets the whole story What decides how high the population can go? The environment. Drag the carrying capacity K — a bigger dish, more nutrients — and watch the entire S-curve redraw. No matter where you set it, the curve always bends toward that ceiling and flattens against it. K is the horizontal asymptote the population can approach but never pass. Growth is fastest right in the middle, as the curve crosses K⁄2 — then it eases off for the long approach to the ceiling.

This is the written version of the interactive lesson above. See the full Calculus 2 course.