Exponential and Logistic Growth

Exponential growth assumes unlimited resources: no predators pressing in, no food running short, no space crunch. Each year the population grows by the same percentage of its current size, so the actual number added keeps climbing even though the rate stays fixed. That is why a pond weed covering half the pond on day 29 covers the whole pond on day 30 — doubling time stays constant, but the last doubling happens on top of a much bigger base. Plotted over time this traces a J: slow-looking at first, then nearly vertical.

Real environments push back. As a population approaches the carrying capacity (K) of its habitat, food, water, mates, and space start running out. Growth rate slows because births drop or deaths rise, and the curve bends from a J into an S: fast growth near the bottom, a slowing midsection, and a flat plateau at K. This logistic model is the exponential equation with a brake built in — the growth rate gets multiplied by a term that shrinks toward zero as population size nears K.

The two models sit side by side because real populations often start exponential, growing freely when resources are abundant, then shift logistic once limits bite. The extension case matters too: if K itself rises — more rain, a new food source, habitat restoration — the plateau lifts, and the population climbs again toward the new ceiling. This is how the growth curves connect forward to carrying capacity and resource limits as the controlling variables in the unit.