Carrying Capacity Calculator
The carrying capacity K is the maximum population an environment can sustain. Choose what you want to solve for: the instantaneous growth rate dN/dt, the population at a future time N(t), the carrying capacity K, the intrinsic rate r, or the time to reach a target population. The calculator also supports the simpler exponential model so you can compare the two.
Formula
Worked example
Logistic N(t): with N0 = 50, K = 1000, r = 0.5, t = 10 time units: A = (1000-50)/50 = 19; e^(-0.5 x 10) = 0.00674; denominator = 1 + 19 x 0.00674 = 1.128; N(10) = 1000/1.128 = 886 individuals (88.6% of K). To solve for K given N = 200, dN/dt = 80, r = 0.5: K = 200/(1 - 80/(0.5 x 200)) = 200/(1-0.8) = 1000.
How the logistic growth model works
The logistic equation dN/dt = rN(1 - N/K) describes population growth that is limited by resources. The first part, rN, is the unconstrained exponential term: every individual contributes r new individuals per time unit. The second part, the factor (1 - N/K), is the density-dependent brake. When N is tiny compared with the carrying capacity K, this factor is close to 1 and growth is nearly exponential. As N rises toward K, the brake shrinks toward zero, slowing growth. At N = K the factor is exactly zero, so dN/dt = 0 and the population stops changing. This produces the familiar S-shaped (sigmoid) curve that rises steeply and then levels off. Integrating the logistic equation gives the closed-form solution N(t) = K / (1 + ((K - N0)/N0) * e^(-r*t)), which this calculator uses for the N(t) and t modes.
Carrying capacity and the inflection point
The carrying capacity K is the maximum population an environment can support indefinitely given its food, water, space, and other limiting factors. A key property of the logistic model is that growth is maximised at the inflection point N = K/2, where the rate equals rK/4. Below K/2 growth is limited by small numbers; above K/2 it is limited by crowding. If N ever exceeds K, because of immigration or a sudden resource loss, the factor (1 - N/K) turns negative and the population is predicted to decline back toward K. This overshoot and correction is a hallmark of real regulated populations. The logistic model is an idealisation: actual populations also experience time lags, age structure, stochasticity, and environmental variation, so treat the output as an estimate of the underlying trend.
Choosing a solve-for mode
This calculator offers six modes. The default dN/dt mode is the instantaneous rate of change: useful for ecology textbook problems. The N(t) logistic mode projects the full S-curve to any future time point, given initial conditions. The K mode reverses the logistic equation to estimate carrying capacity from a measured growth rate, useful when K is unknown but r and current growth data are available. The r mode estimates the intrinsic growth rate from two census observations spaced in time, with K known. The t mode inverts the logistic formula to find the time at which the population will reach a chosen target below K. The exponential mode drops the K ceiling entirely and shows the simpler N0 * e^(r*t) trajectory for comparison, useful for early-phase populations far below carrying capacity.
Exponential vs. logistic growth
In exponential growth N(t) = N0 * e^(r*t), the population grows at a constant per-capita rate forever. This is realistic only at low densities far from any resource ceiling. In logistic growth the per-capita rate declines linearly from r (at N = 0) to 0 (at N = K). At low densities the two curves overlap closely; the divergence becomes obvious once N is above roughly 10-20 % of K. Real populations sit somewhere between the two models: their growth is slower than pure exponential because of density-dependent competition, but often faster than the basic logistic because carrying capacity itself changes with technology, migration, and ecosystem change.
Logistic growth rate and N(t) at different population sizes
| N (or time) | N as % of K | Brake (1 - N/K) | Growth rate dN/dt | N(t) from N0=50, t= |
|---|---|---|---|---|
| 100 | 10% | 0.90 | 45 | t=5: 179 |
| 250 | 25% | 0.75 | 93.75 | t=10: 500 |
| 500 | 50% | 0.50 | 125 | t=15: 821 |
| 750 | 75% | 0.25 | 93.75 | t=20: 953 |
| 1000 | 100% (K) | 0 | 0 | t=25: 987 |
Using r = 0.5 and K = 1000. Growth rate dN/dt peaks at N = K/2 = 500. N(t) values start from N0 = 50.
Frequently asked questions
What is the logistic growth rate formula?
The logistic growth rate is dN/dt = rN(1 - N/K), where r is the intrinsic per-capita growth rate, N is the current population, and K is the carrying capacity. The factor (1 - N/K) slows growth as the population approaches the environmental limit K.
What is the logistic population formula N(t)?
The closed-form solution is N(t) = K / (1 + ((K - N0)/N0) * e^(-r*t)). It tells you the population at any future time t given the initial population N0, carrying capacity K, and growth rate r. The curve starts nearly exponential at low densities and flattens as N approaches K.
How do I solve for carrying capacity K?
Rearranging the logistic equation gives K = N / (1 - dN/dt / (r*N)). You need the current population N, the observed instantaneous growth rate dN/dt, and the intrinsic rate r. Enter these in the "Solve for K" mode and the calculator does the algebra for you.
When is population growth fastest in the logistic model?
Growth is fastest at the inflection point N = K/2, exactly halfway to the carrying capacity. At that point the absolute growth rate equals rK/4. Below K/2 the population is too small to grow quickly; above K/2 crowding slows things down.
How do I estimate intrinsic rate r from census data?
If you have two population counts N1 at time t1 and N2 at time t2, and you know K, use r = ln((K/N1 - 1) / (K/N2 - 1)) / (t2 - t1). This is the "Estimate r from two time points" mode. Both N1 and N2 must be below K for the formula to work.
What happens when the population exceeds the carrying capacity?
If N > K, the factor (1 - N/K) becomes negative, so dN/dt is negative and the population declines. The logistic model predicts the population will shrink back toward K. In reality, overshot populations sometimes crash below K before recovering, a pattern not captured by the basic logistic equation.
What is the difference between exponential and logistic growth?
Exponential growth N(t) = N0 * e^(r*t) assumes unlimited resources and grows without bound. Logistic growth introduces the carrying capacity K as a ceiling: the per-capita rate declines as N rises toward K and hits zero at N = K. At low densities both models look similar; the difference becomes large once the population is above about 10-20% of K.