Logistic Population Growth Calculator
Real populations rarely grow without limit; instead they slow as resources become scarce and approach a carrying capacity. The logistic model captures this with an S-shaped curve set by the initial size, the intrinsic growth rate, and the carrying capacity. This calculator returns the projected population at a chosen time and the instantaneous growth rate at that point. Enter your own measured growth rate and carrying capacity, which are population-specific parameters; the logistic equation is then applied deterministically.
Logistic growth formula
N(t) = K / (1 + ((K - N0) / N0) * e^(-r * t))
dN/dt = r * N * (1 - N / K)
N approaches K as t grows large
growth is fastest when N = K / 2
With N0 = 100, K = 1,000, r = 0.5, at t = 5: N(t) is about 591, and the instantaneous rate r times N times (1 minus N/K) is about 121 per unit time.
Population ecology facts
- The logistic curve is S-shaped, rising then levelling off at K.
- Growth is fastest at half the carrying capacity.
- At low population relative to K, growth is nearly exponential.
- The per-capita growth rate falls linearly as N rises toward K.
- r and K are empirical and specific to each population and environment.
Logistic growth: frequently asked questions
What is the logistic growth model?
The logistic model describes population growth that slows as it approaches a carrying capacity K. Its solution is N(t) = K divided by (1 plus ((K minus N0) over N0) times e to the power of minus r times t), giving an S-shaped curve that starts near exponential and levels off at K.
How does logistic growth differ from exponential growth?
Exponential growth assumes unlimited resources and rises without bound. Logistic growth adds a carrying capacity, so the per-capita growth rate falls as the population nears K and the curve flattens. At low population relative to K, the two models are nearly identical.
What is the instantaneous growth rate?
The logistic differential equation gives the instantaneous rate dN/dt = r times N times (1 minus N over K). It is largest when N is half of K and approaches zero as N nears the carrying capacity, which produces the characteristic S-shape.
What units should I use?
Use any consistent units. The intrinsic growth rate r is per unit time (for example per year), and time t is in the same unit. Population size N0 and carrying capacity K share the same count unit. The output population carries that count unit.
Where does the growth rate value come from?
The intrinsic growth rate r and the carrying capacity K are empirical parameters specific to a population and environment. Because they are measured, not universal, you enter them yourself; the calculator applies only the deterministic logistic equation.
Official sources
- USGS: USGS population dynamics research.
- U.S. Census Bureau: U.S. Census Bureau population topics.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.