Beyond Exponential

Differential Equations · Axiom Academy

Watch a colony hit a wall the exponential model never saw coming — and discover the equation that actually predicts it. A model that forgets the dish has walls A single bacterium divides every 20 minutes. Early on, the differential equation nails it — population doubling like clockwork. But a petri dish only holds so much food and space. Watch what the exponential model predicts once the colony starts to crowd its dish, next to what the colony actually does. Press play: the blue curve is the exponential model's prediction, climbing forever. The green curve is the colony's real population. Watch them split apart as the dish fills up. The exponential model has no idea the dish is finite — it just keeps multiplying. Reality does have a ceiling. Drag the dish's capacity — the ceiling follows Slide K , the carrying capacity, up and down. The real-growth curve always bends over and settles exactly at K — a bigger dish just means a higher ceiling, not "no ceiling." The exponential curve never adjusts to K at all; it doesn't know K exists. Every dish size gives a different ceiling — but there's always a ceiling. That's the piece pure exponential growth is missing. Find the term that switches growth off Drag the point along the curve to change the current population P . Watch the growth-rate readout: it starts near its full exponential rate r when P is small, then shrinks toward zero as P closes in on K . That shrinking factor is — the one piece the exponential model never had.

This is the written version of the interactive lesson above. See the full Differential Equations course.