Logistic Carrying Capacity Calculator
The logistic model is the realistic counterpart to exponential growth: a population grows fast while small but slows as it approaches its carrying capacity K, the limit the environment can support, tracing an S-shaped curve that levels off at K. This calculator uses the closed-form logistic solution to project the population at a chosen time from the starting size, growth rate and carrying capacity, and reports the instantaneous growth rate at that point. Enter the rate as a percentage and keep the starting population at or below K.
Logistic growth formula
r = rate% / 100
A = (K - N0) / N0
N(t) = K / (1 + A * e^(-r * t))
dN/dt = r * N(t) * (1 - N(t) / K)
Population approaches K as t grows large
The term (1 minus N over K) slows growth near the carrying capacity, giving the characteristic S-shaped curve.
Logistic model context
- Carrying capacity K is the sustainable maximum population.
- Growth is fastest at N equal to half of K, then slows toward K.
- The model needs K greater than zero and a positive starting population.
- Use consistent time units for r and t.
- Real ecosystems can overshoot or oscillate around K; logistic is an idealised baseline.
Logistic carrying capacity: frequently asked questions
What is the logistic growth model?
Logistic growth describes a population that grows quickly when small but slows as it nears its carrying capacity K, the maximum the environment can support. The closed-form solution is N(t) = K divided by (1 plus A times e to the minus r t), where A depends on the starting population.
What is carrying capacity?
Carrying capacity, K, is the largest population an environment can sustain indefinitely given its resources. As the population approaches K, the per-individual growth rate falls toward zero, and the population levels off at K.
How does logistic growth differ from exponential growth?
Exponential growth has no upper limit and accelerates forever. Logistic growth includes the term (1 minus N over K), which throttles growth as N approaches K, producing an S-shaped curve that plateaus. Logistic is the more realistic model when resources are finite.
What is the instantaneous growth rate?
It is dN/dt = r times N times (1 minus N over K), the rate of change of the population at the current size. It is largest at N equal to half of K and approaches zero as N nears K or zero.
What inputs do I need?
You need the starting population N0, the carrying capacity K, the intrinsic growth rate r as a percentage, and elapsed time t. The calculator requires K greater than zero and N0 between zero and K for a standard logistic curve.
Official sources
- NCBI Bookshelf: National Center for Biotechnology Information.
- U.S. Geological Survey: USGS home.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.