Carrying capacity ($K$)

A carrying capacity is an abundance scale at which a specified environment's net population growth vanishes. In the logistic growth equation, positive unharvested populations approach $K$. It depends on the species, resources and environment rather than being an immutable universal population limit.

The useful distinction is between a growth rate and a sustainable abundance. In a logistic differential equation, $r$ controls how quickly a population responds, while the carrying capacity $K$ sets the positive equilibrium point. Increasing $r$ with $K$ fixed accelerates approach without raising that equilibrium. Increasing $K$ changes the destination. The logistic differential equation diagrams show populations approaching the same carrying capacity from below and above.

In ecological use, the carrying capacity describes what a particular environment can support under specified conditions. Nutrient supply, available space, and accumulated waste can change those conditions; it is not a permanent property of a species alone. A population temporarily above a carrying capacity is possible. In the simplest constant-$K$ logistic growth equation, its net growth becomes negative; real population dynamics can include delays, seasonal variation, and changing resources that the basic mathematical model omits. ^[ref].