About the Logistic Growth Calculator
This logistic growth calculator models a population that grows almost exponentially at first and then slows as it approaches the environment’s carrying capacity K. Enter the starting population, the carrying capacity, the intrinsic growth rate r and the elapsed time to get the population at that moment, the instantaneous growth rate, and the time when growth is fastest.
It suits biology and ecology students studying the Verhulst model, microbiologists estimating when a culture will reach stationary phase, and anyone modelling adoption or spread that saturates. The S-shaped chart makes it easy to see the lag, rapid-growth and plateau phases, and the table lists values across the whole curve.
The model assumes a constant growth rate and carrying capacity, no time delays, and a continuously reproducing population. Real populations can overshoot, oscillate or be affected by seasons and harvesting, so treat the curve as an idealised baseline. Use the same time unit for r and t — if r is per day, t must be in days.
With the default inputs, the population at time t is 450.85. Change any value above to recalculate instantly.
How to use the logistic growth calculator
- 1Enter the starting population P₀ and the carrying capacity K.
- 2Enter the intrinsic growth rate r as a percent per time unit.
- 3Enter the elapsed time in the same unit and pick that unit for labels.
- 4Read the population, then check the inflection time and S-curve chart.
Formula and method
The logistic (Verhulst) model modifies exponential growth with a crowding term: each individual still reproduces at rate r, but the factor (1 − P/K) shrinks growth to zero as the population P nears the carrying capacity K. Solving that differential equation gives the closed-form S-curve above, which this calculator evaluates directly.
Growth is fastest at P = K/2, the inflection point, reached at t = ln((K − P₀)/P₀) ÷ r, where the rate is rK/4. More generally the time to reach a fraction f of K is ln(((K − P₀)/P₀) × f/(1 − f)) ÷ r. The time unit is whatever unit r is expressed in.
- P(t)
- Population at time t
- P₀
- Initial population
- K
- Carrying capacity (maximum sustainable population)
- r
- Intrinsic (per-capita) growth rate per time unit
- t
- Elapsed time, in the same unit as r
Worked examples
100 animals, K = 1,000, r = 20%/year, after 10 years
P(10) = 1000 ÷ (1 + 9e^(−2)) = 450.9. The population is just short of half capacity, and growth peaks at 50 per year when it reaches 500 at t = ln 9 ÷ 0.2 ≈ 10.99 years.
Fish stocked in a pond: 50 fish, K = 500, r = 50%/year, after 5 years
P(5) = 500 ÷ (1 + 9e^(−2.5)) ≈ 287.6 fish. The inflection point is at ln 9 ÷ 0.5 = 4.39 years and the pond reaches 90% of capacity after about 8.8 years.
Bacterial culture: 2 cells, K = 10,000, r = 69.3%/hour, after 12 hours
A rate of 0.693 per hour is a doubling time of about one hour while the culture is small. After 12 hours P = 10000 ÷ (1 + 4999e^(−8.316)) ≈ 4,499 cells, close to the inflection at 12.29 hours.
Frequently asked questions
What is the logistic growth formula?+
P(t) = K / (1 + ((K − P₀)/P₀)·e^(−rt)), where K is the carrying capacity, P₀ the initial population, r the intrinsic growth rate and t time. It comes from the differential equation dP/dt = rP(1 − P/K).
What is carrying capacity?+
Carrying capacity K is the largest population an environment can sustain indefinitely given its food, space and other resources. In the logistic model growth slows to zero as the population approaches K.
What is the difference between logistic and exponential growth?+
Exponential growth keeps accelerating without limit (J-curve). Logistic growth starts out nearly exponential but slows as resources become limited, producing an S-shaped curve that levels off at the carrying capacity.
When is logistic growth fastest?+
At the inflection point, when the population equals half the carrying capacity (K/2). The maximum rate there is rK/4 individuals per time unit, which is the basis of maximum sustainable yield in fisheries.
How do I find r from a doubling time?+
In the early, nearly exponential phase r ≈ ln 2 ÷ doubling time. A population that doubles every 3 years has r ≈ 0.231, or 23.1% per year.