carrying capacity
An environment can support only so much life. A pasture feeds a certain number of cows; a pond holds a certain number of fish; an island sustains a certain population of birds. Push past that number and there is not enough food, water, or space to go around, so the population stops growing — or shrinks back. That ceiling is the carrying capacity, written K.
In the logistic equation dP/dt = rP(1 − P/K), the symbol K is the carrying capacity, and it plays a precise mathematical role: it is the stable equilibrium of the model. When P = K the bracket (1 − P/K) is zero, so dP/dt is zero and the population holds steady. More than that, K is an attractor — start below K and the population grows up toward it; start above K (an overstocked pasture) and the population falls back down toward it. So whatever you begin with, the population is drawn to K over time. The other equilibrium, P = 0, is unstable: any small population grows away from extinction.
Carrying capacity is one of the most important ideas in ecology and resource management, telling you the long-run size a habitat will support. But it is a modeling idealization: real carrying capacity is not a fixed number — it shifts with seasons, weather, disease, and human impact. Treating K as a permanent constant is convenient, but a model that lets K vary is often closer to the truth.
If a lake's carrying capacity for trout is K = 5000, a stocked population of 8000 will decline toward 5000, while a depleted population of 500 will recover up toward 5000 — both approaching the same stable level.
K is the stable equilibrium of logistic growth: populations are drawn to it.
Carrying capacity is rarely truly constant; droughts, disease, and habitat change all move K, so a population can appear to overshoot or crash when really the ceiling itself shifted.