Carrying Capacity

What sets K

Imagine a pond stocked with goldfish. Add ten fish; they thrive. Add a hundred; some still thrive. Add a thousand; oxygen drops, ammonia builds, and most starve or suffocate. Somewhere between ten and a thousand is the population the pond's resources can keep alive indefinitely. That is K.

K is not chosen by the species; it is imposed by the habitat. Population ecologists track four broad classes of factors that set or shift K:

1. Energy and material inputs. Net primary production for herbivores. Prey biomass for carnivores. Nutrient supply rate for plants. The amount of carbon, nitrogen, phosphorus, and water flowing through the ecosystem per year sets a hard ceiling.
2. Habitat structure. Number of nest sites for cavity-nesting birds, number of refugia for prey, vertical strata for foraging. A grassland with no trees has zero K for woodpeckers regardless of insect supply.
3. Density-dependent biotic pressures. Competition, predation, parasitism, and disease all intensify as density rises. They lower the effective K below the pure resource limit.
4. Behaviour and physiology. Territoriality caps spacing for many birds and mammals. Many species set their own K below the resource ceiling because aggressive defence excludes additional individuals.

Liebig's law of the minimum

The German chemist Justus von Liebig framed the principle for crops in 1840: yield is determined by the resource in shortest supply, not by total resource abundance. A field with abundant nitrogen, potassium, and water but only trace phosphorus will be limited by phosphorus. Adding more nitrogen does nothing.

The principle generalizes to any population. The textbook image is a wooden barrel made of staves of different heights — water rises only to the lowest stave. K is the lowest stave, and you must identify which one it is before predicting how the population will respond to change.

	Abiotic factors	Biotic factors
Type	Non-living conditions	Living interactions
Examples	Water, temperature, light, nutrients, salinity, oxygen	Predation, competition, disease, parasitism, mutualism
Density-dependence	Usually independent	Strongly dependent
Time scale	Days to seasons	Generations
Sets ceiling on K?	Yes — physical hard limit	Yes — softer, dynamic limit
Adjusts to N?	No (resource pool fixed)	Yes (predators numerically respond)
Example dominance	Desert plants → water	Pelagic fish → predation

Most real systems involve both. Boreal voles experience density-dependent predation by owls (biotic) on top of harsh winter mortality (abiotic). Disentangling them requires manipulative experiments, not curve fitting.

The logistic equation

Carrying capacity enters formal ecology through the logistic growth model:

dN/dt = rN(1 − N/K)

Three regimes follow from one equation:

• When N is much less than K, the term (1 − N/K) is near 1. Growth approaches exponential at rate r.
• When N approaches K, (1 − N/K) approaches zero. Growth slows.
• When N exceeds K, (1 − N/K) goes negative. The population shrinks back toward K.

K is therefore a stable equilibrium: small departures in either direction return to K. The shape of N(t) is a sigmoid curve — slow start, steep middle, asymptote at K.

Worked example: African elephant

An average tropical savanna in Tsavo East has net primary productivity of about 600 g dry mass per m² per year. An adult elephant eats about 150 kg of forage per day, or 55 tons per year. Allow 10 percent of NPP to be available to elephants without crashing other herbivores; that is 60 g per m² per year, or 60 tons per km² per year.

60 tons divided by 55 tons per elephant gives just over 1 elephant per km² from forage alone — but elephants knock down trees and open canopy, increasing their effective K through ecosystem engineering. Long-term census data from Tsavo show 0.5 to 2 elephants per km² in productive years, occasionally peaking near 5/km² in mosaic landscapes. Continent-wide K estimates for closed-canopy forest are about 50 elephants per km² because forage is denser per unit area.

Overshoot and crash

If a population grows faster than its resources can adjust — common after a good breeding season or with introduced species — it overshoots K. The classic case study is the St. Matthew Island reindeer.

• 1944: 29 reindeer introduced to a 332 km² Alaskan island as wartime emergency food.
• 1957: 1350 reindeer.
• 1963: 6000 reindeer — five times the island's lichen-based K.
• 1966 winter: 42 reindeer left, all female, all old, all of which subsequently died.

The population crashed because lichens grow about 1 cm per decade and were stripped to bare rock. Recovery would take centuries. Overshoot-and-crash dynamics are common in introduced ungulates, irruptive insects, and some marine invertebrates after upwelling pulses.

Logistic K vs alternative models

	Exponential	Logistic (constant K)	Allee-effect
Equation	dN/dt = rN	dN/dt = rN(1 − N/K)	dN/dt = rN(N/A − 1)(1 − N/K)
Growth at low N	Fast (exponential)	Fast (exponential)	Negative below threshold A
Carrying capacity	None	Yes — fixed K	Yes — fixed K
Equilibria	None (or 0)	0 (unstable), K (stable)	0 (stable), A (unstable), K (stable)
Predicts crash from low N?	No	No	Yes
Use case	Bacteria first hours, invasion	Density-regulated populations	Endangered species, social mating
Empirical fit	Short windows only	Many lab populations	Cooperative breeders, schooling fish

Real populations rarely match any of these exactly. Time-lagged logistic models with delayed density dependence (May, 1976) generate the limit cycles seen in lemmings and forest insects. Chaos becomes possible when r and the lag length are large.

Empirical patterns

• Yeast in flask culture (Gause, 1932). The original logistic confirmation. Yeast counts climb sigmoidally to a flat K determined by glucose supply; double the glucose and K doubles.
• Wildebeest in the Serengeti. A population grew from 250,000 to 1.4 million between 1961 and 1977 after rinderpest eradication, then stabilized near K ≈ 1.3 million for four decades, with year-to-year variation tracking rainfall.
• Phantom-midge (Chaoborus) in salmon hatcheries. Stocking density above the K imposed by oxygen and zooplankton supply causes mass mortality within weeks. Hatchery operators back-calculate K and set stocking 60 percent below it as a safety margin.
• Kaibab deer (Arizona, 1906–1939). A predator removal program let deer climb from 4000 to 100,000 by 1924 — well above K — then crash to 10,000 by 1939 after browsing destroyed forage. Cited as a textbook example for decades, though re-analysis shows the data are noisier than the simple narrative implies.

Diagram sketch

• Panel A. Sigmoid logistic curve: N rises from N₀ through an inflection at K/2, asymptotes at the dashed K line.
• Panel B. Overshoot-and-crash: N rises above K, then decays back through damped or undamped oscillation.
• Panel C. Liebig's barrel — staves labelled water, nitrogen, phosphorus, light, predator pressure; the lowest stave caps yield.

Pitfalls

• K is not constant. Treating it as a fixed number bakes in stability that real ecosystems do not have. Drought halves grassland K; a fire resets succession and K with it.
• K is not the same as ecosystem health. An invasive species can have a high K while degrading the system around it. Population at K does not imply ecosystem at equilibrium.
• Logistic growth ignores age structure. A population at K with all old adults is on the brink of crashing; one with many juveniles is stable. Leslie matrix models are needed when age matters.
• Logistic growth ignores spatial structure. Local patches can be at K while regional metapopulation has room. Conservation units must match the dispersal scale.
• Anthropogenic shifts. Climate change, fertilizer drift, and habitat alteration move K faster than evolution can re-tune r. Many populations are now chasing a moving K.
• Confusing biotic potential with K. r and K are independent parameters. r-selected species (high r, low K) and K-selected species (low r, high K) are a textbook simplification, not a strict dichotomy.

Variants

• Theta-logistic. dN/dt = rN(1 − (N/K)^θ) generalizes the curvature near K. θ > 1 fits ungulate populations that crash hard at the limit; θ < 1 fits insects that brake gently.
• Time-lagged logistic. dN/dt = rN(1 − N(t − τ)/K). Density dependence acts on past population size, generating cycles when rτ exceeds 1.5.
• Stochastic K. K(t) is drawn from a distribution each year; long-run population variance scales with the K variance.
• Density-dependent K (technology-driven). Used in human demography: K(N) rises with N as larger populations support agriculture, trade, and energy infrastructure. Predicts saturating-but-not-fixed equilibria.