Definition

Carrying capacity is the biggest number of one kind of living thing that a place can feed and shelter over the long term, so a small pond with enough insects and hiding spots for about $50$ frogs will lose most of $200$ hatching tadpoles to starvation or predation as the population drops back toward that limit; it is not a wall a population hits and stops at, since populations often overshoot, crash, and wobble around the limit instead. More formally, carrying capacity, written $K$, is the maximum population size an environment can sustain indefinitely given available food, water, shelter, and other resources, and as a population nears $K$, limiting factors such as competition, predation, and disease slow its growth into an S-shaped logistic curve; $K$ itself can shift with the environment, rising after a wet year or falling after drought or habitat loss. At the mathematical level, $K$ is the equilibrium population size in the logistic model $\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)$, where per-capita birth and death rates balance exactly, and because it is a property of the population-environment pair rather than the species alone, real populations chase a moving target, with time lags between resource depletion and population response producing overshoot or sustained oscillation; in fisheries and wildlife management the maximum sustainable yield occurs at $N = K/2$, where logistic growth peaks at $rK/4$.