Carrying Capacity Estimate Calculator
The carrying capacity K is the population size an environment can support over the long term. In the logistic model, the instantaneous growth rate is r times N times (1 minus N over K), so if you have measured the current population, the intrinsic growth rate, and the observed instantaneous growth rate, you can solve for K. This calculator performs that rearrangement and also reports the per-capita growth rate. The result is meaningful only when density dependence is slowing growth, that is when the observed per-capita rate is below r.
Carrying capacity formula
logistic rate: dN/dt = r * N * (1 - N / K)
per-capita rate = (dN/dt) / N
solve for K: K = N / (1 - (dN/dt) / (r * N))
valid when (dN/dt) / N is less than r
With N = 600, r = 0.5, and dN/dt = 120: per-capita rate is 0.2, so K = 600 / (1 - 0.2/0.5) = 600 / 0.6 = 1,000.
Carrying capacity facts
- K is the population an environment can sustain indefinitely.
- Per-capita growth falls linearly from r toward zero as N nears K.
- The estimate requires the observed per-capita rate to be below r.
- Growth in absolute terms peaks when the population is half of K.
- r and the observed rate are empirical, population-specific measurements.
Carrying capacity: frequently asked questions
What is carrying capacity?
Carrying capacity, written K, is the maximum population size an environment can sustain indefinitely given its resources. In the logistic model, population growth slows as the population approaches K and stops when it reaches it, because the per-capita growth rate falls to zero there.
How is K estimated from a growth observation?
The logistic rate equation dN/dt = r times N times (1 minus N over K) can be rearranged for K. Given the current population N, the intrinsic growth rate r, and the observed instantaneous growth rate dN/dt, the carrying capacity is K = N divided by (1 minus dN/dt over (r times N)).
Why must the observed growth be below the maximum?
The rearrangement is only meaningful when the observed per-capita growth rate, dN/dt over N, is less than the intrinsic rate r. That is the regime where density dependence is reducing growth. If the observed rate equals or exceeds r times N, the formula does not give a finite positive K.
What units should I use?
Use consistent units. Population N and the resulting K share the same count unit. The intrinsic rate r is per unit time, and the observed growth rate dN/dt is in population per the same unit time, so the ratios are dimensionless.
Where do r and the growth observation come from?
Both the intrinsic growth rate r and the observed instantaneous growth rate are empirical, measured from field or experimental data for the specific population. You supply them; the calculator performs only the deterministic logistic rearrangement.
Official sources
- USGS: USGS population dynamics research.
- National Park Service: NPS wildlife and population science.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.