Logistic growth model: dN/dt = rN(1 - N/K)
Carrying capacity (K) is the maximum population size an environment can sustain indefinitely given the available resources. The logistic growth equation dN/dt = rN(1 - N/K) describes how populations grow rapidly when small, slow as resources become scarce, and stabilize near K. Belgian mathematician Pierre Verhulst introduced this model in 1838. The population grows fastest at N = K/2, known as maximum sustainable yield — a critical concept in fisheries management. Real populations often overshoot K, leading to die-offs and oscillations. Factors that determine K include food availability, water, shelter, disease, predation, and space. Human activity frequently reduces K for wildlife through habitat destruction, pollution, and climate change.