What this calculator does
Find out how big a population will be at any time, or how long it takes to reach a size you choose. No population can keep multiplying forever. Food, space, water and other resources run short, and growth slows as numbers rise.
The carrying capacity, K, is the largest population an environment can support over the long term. The logistic model describes a population that grows quickly while it's small. Then it levels off as it gets close to K.
This calculator works the logistic equation in both directions. Give it a time, and it returns the population. Give it a target population, and it returns how long the population takes to get there. It also marks the inflection point at K/2 (half of K), where growth is fastest. And it shows the curve in a table.
Where it's used
Why this math matters outside the classroom:
- Wildlife management. Wildlife managers use a habitat's carrying capacity to judge how many deer or elk a reserve can support.
- Fisheries. Fishery scientists note logistic growth is fastest at half the carrying capacity, so a stock with K = 10,000 grows quickest near 5,000.
- Yeast experiments. Biology students fit a logistic curve to yeast counts that rise quickly and then level off as nutrients run out.
- Comparing models. Ecology classes compare a population starting at 100 with K = 1,000 against unchecked exponential growth to see where the curves part.
How to use it
- Choose whether to find the population at a time or the time to reach a population.
- Enter the carrying capacity K, the starting population N₀ and the intrinsic growth rate r (how fast the population grows while it's still small).
- Enter either the time that has passed or the target population. Time uses whatever unit r is measured in: years, days, generations or hours.
- Read the result, the inflection point, and the table of population and growth rate over time.
What the inputs mean
- Carrying capacity (K)
- The long-run ceiling the population settles toward. It depends on the environment, not just the species, and can shift if conditions change.
- Initial population (N₀)
- The population at time zero. It can be above K. If it is, the model shows a decline toward K.
- Intrinsic growth rate (r)
- The growth rate per individual when the population is tiny and resources are unlimited. Use the same time unit as t. For example, 0.4 per year means the population would grow about 40% a year if crowding weren't a factor. Because growth is continuous, that works out to slightly more than 40% compounded.
- Time (t)
- In the same time unit as r.
- Target population
- Must lie between the starting population and K. The model gets closer and closer to K forever, so K itself is never reached.
Formula and method
N(t) = K ÷ (1 + A × e^(−r × t)), where A = (K − N₀) ÷ N₀ t = ln[ N × (K − N₀) ÷ (N₀ × (K − N)) ] ÷ r dN/dt = r × N × (1 − N ÷ K) inflection: N = K ÷ 2 at t = ln(A) ÷ r; maximum growth rate = r × K ÷ 4
N is the population and N(t) is the population at time t. N₀ is the starting population, K is the carrying capacity and r is the growth rate. A is a constant set by the starting point. e is the number 2.718…, used for continuous growth, and ln is its matching logarithm (the natural log). dN/dt means how fast the population is changing at that moment.
The logistic equation comes from the growth rule dN/dt = rN(1 − N/K). When N is small, the bracket is close to 1, so growth is nearly exponential (multiplying at a steady rate). As N gets close to K, the bracket shrinks toward zero and growth stalls.
Solving N(t) for t gives the second formula. That's how the calculator finds the time to reach a target. The growth rate rN(1 − N/K) is largest when N is exactly half of K. That's the inflection point, where the S-shaped curve switches from speeding up to slowing down.
Worked example
A population of 50 lives in a habitat with a carrying capacity of 1,000 and an intrinsic growth rate of 0.4 per year. What is the population after 10 years, and when does it reach 800?
- A = (1,000 − 50) ÷ 50 = 19.
- e^(−0.4 × 10) = e^(−4) = 0.01831564.
- N(10) = 1,000 ÷ (1 + 19 × 0.01831564) = 1,000 ÷ 1.347997 = 741.84.
- Time to 800: ln[800 × 950 ÷ (50 × 200)] ÷ 0.4 = ln(76) ÷ 0.4 = 10.827 years.
- Inflection: ln(19) ÷ 0.4 = 7.361 years, when the population is 500 and growing at its fastest, 0.4 × 1,000 ÷ 4 = 100 per year.
Result: After 10 years the population is about 742, or 74.2% of capacity. It reaches 800 after about 10.83 years, and its growth peaked at 100 a year around year 7.36.
Understanding the result
Before the inflection point, the population grows faster each year. After it, each year's increase is smaller than the last, even though the population keeps rising.
The time to reach a target rises steeply as the target gets close to K. Getting from 90% to 99% of capacity takes a little longer than getting from 10% to 50%.
If the starting population is above K, the model predicts a smooth decline toward K. The decline slows as the gap closes.
Assumptions and limitations
- K and r stay constant over the whole period.
- Growth responds instantly to crowding, with no time lags.
- The population is large enough to treat as a smooth number. The model ignores random births and deaths, migration and the mix of ages.
- Real populations often overshoot K, swing up and down, or crash. That's especially likely when there's a delay between crowding and its effect on births.
- Predators, disease, harvesting and seasonal changes aren't modeled. For predator and prey interactions, Lotka-Volterra models (which track both groups together) are a better fit.
- Estimating r and K from field data is hard. Small changes in either can shift the timing a lot.
Frequently asked questions
What is the difference between logistic and exponential growth?
Exponential growth assumes unlimited resources, so the population multiplies by the same factor every period forever. Logistic growth starts out nearly exponential. But it slows as the population fills the space available. That makes an S-shaped curve that levels off at K.
Why can the population never reach K exactly?
In the equation, the term A × e^(−rt) shrinks toward zero but never equals zero. So N gets ever closer to K without arriving. In practice, a population within a fraction of a percent of K is treated as having reached it.
What does the inflection point mean?
It's the moment when the population equals half the carrying capacity. At that point it's adding individuals faster than at any other time. Fisheries and wildlife managers use this idea, since a population held near K/2 replaces itself most quickly.
What time unit should I use?
Any unit, as long as r and t use the same one. If r is 0.4 per year, t is in years. If you know a per-day rate, enter time in days.
How this calculator is checked
Method: Verhulst logistic equation, closed-form solution of dN/dt = rN(1 − N/K); inflection at N = K/2. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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