Populations

Early on, a population with plenty of food and space grows exponentially: each individual reproduces at roughly the same rate, so the whole group doubles again and again, and the curve shoots upward faster and faster. That rabbit population exploding isn't breaking any rule, it's just multiplying by the same factor over and over on a short timescale. But no environment has infinite food, space, or tolerance for waste and disease. As numbers climb, each individual has less to work with. Births slow down, deaths pick up, and the two rates drift toward each other. That's the bend: the curve stops accelerating, slows, and flattens into an S shape, called logistic growth. The flat part sits near the carrying capacity, the population size an environment can sustain given its actual resources right now.

Carrying capacity isn't a ceiling painted on the ecosystem forever. It's just wherever births and deaths balance out given current conditions, so it moves whenever the resources move. A drought that shrinks forage lowers it; a wet year that grows more grass raises it. That's why a population can overshoot, crash, and resettle at a new, lower plateau, or recover toward a higher one once a limiting resource returns. This idea connects directly to the logistic curve itself: restoring a resource partway through a crash doesn't just stop the decline, it bends the curve back upward toward a new, higher flat region, since the ceiling itself just moved.