Logistic Growth Calculator
Model population growth that slows as it approaches a carrying capacity, using the logistic equation to find the size at any time and how close it is to the ceiling.
How it works
N(t) = K ÷ (1 + ((K − N₀) ÷ N₀) · e^(−r·t))
Logistic growth adds a ceiling — the carrying capacity K — to exponential growth, producing an S-shaped curve. Growth is nearly exponential while the population is small, then slows as crowding, limited food and competition bite, and finally levels off as the size approaches K. The parameter r sets how fast the curve rises through its steep middle phase, where growth is fastest at K/2.
Worked example
With N₀ = 100, K = 1,000, r = 0.3 and t = 10: e^(−0.3×10) = e^−3 ≈ 0.0498, and (K − N₀)/N₀ = 900/100 = 9. So N = 1,000 ÷ (1 + 9 × 0.0498) ≈ 1,000 ÷ 1.448 ≈ 691, about 69% of the carrying capacity.
Sources: Logistic function — Wikipedia
Frequently asked questions
What is carrying capacity?
Carrying capacity K is the maximum population an environment can sustain given its food, space and other resources. In the logistic model, growth stops when the population reaches K, because births and deaths come into balance.
How does logistic growth differ from exponential?
Exponential growth accelerates without limit, while logistic growth is checked by the carrying capacity. The two curves are almost identical when the population is far below K; they diverge sharply as the population approaches the ceiling.
When is growth fastest?
The absolute growth rate peaks when the population is at half the carrying capacity, K/2. Below that the population is too small to add many individuals; above it, crowding slows the gain — so the steepest part of the S-curve is in the middle.
Related calculators
Educational estimate, not medical advice — verify formulas, units and assumptions for your specific case.