What is carrying capacity?

Carrying capacity (K) is the maximum population size that an environment can sustain indefinitely given the available resources such as food, water, and space.

What does the intrinsic growth rate r mean?

r is the per-capita rate of population increase under ideal conditions (births minus deaths per individual per time unit). A higher r means the population grows faster initially.

Why does growth slow near K?

As the population approaches K, resources become limiting. The term (1 - N/K) in the logistic equation acts as a brake, reducing the effective growth rate toward zero as N approaches K.

What is the half-K time?

Half-K time is when the population first reaches K/2, which is the inflection point of the logistic curve where growth rate is at its maximum. It equals ln((K - N0) / N0) / r.

Can I use any time units?

Yes. The calculator is unit-agnostic. If r is per year, then t is in years. If r is per hour, t is in hours. Just be consistent.

How do I find carrying capacity from a logistic equation?

In the logistic differential equation dN/dt = r·N·(1 − N/K), the carrying capacity K is the value of N where growth stops (dN/dt = 0). On a graph it is the horizontal asymptote the S-curve flattens toward. If you have the closed form N(t) = K / (1 + A·e^(−r·t)), K is simply the numerator — the population ceiling.

Carrying Capacity Calculator

Enter carrying capacity K, initial population N0, growth rate r, and time t to compute logistic population growth N(t) with a chart.

Logistic Growth Parameters

Carrying Capacity (K)

Max sustainable population

Initial Population (N₀)

Population at time t = 0

Growth Rate (r)

Per-capita rate per time unit

Time (t)

Time units elapsed

Population at Each Time Step

Time (t)	Population N(t)	Saturation (%)	dN/dt

Summary

The Carrying Capacity Calculator uses the logistic growth model to predict population size over time. Given a maximum sustainable population (K), starting population (N0), and intrinsic growth rate (r), it computes N(t) at any time step, shows when the population reaches half of K, and plots the full logistic S-curve.

How it works

Enter the carrying capacity K — the maximum population the environment can sustain.
Enter the initial population N0 and the intrinsic per-capita growth rate r.
Set the elapsed time t, using the same time units as r.
The calculator computes N(t) = K / (1 + ((K − N0)/N0)·e^(−r·t)) and plots the logistic S-curve.
Read off N(t), the saturation percentage N/K, and the half-K inflection time ln((K − N0)/N0)/r.

Use cases

Ecology: project deer or fish population growth toward a habitat limit.
Microbiology: model bacterial colony growth in a medium with finite nutrients.
Conservation biology: estimate recovery time for an endangered species.
Agriculture: predict livestock population growth within a fixed pasture area.
Epidemiology: model the spread of an infection in a finite host population.

Frequently Asked Questions

Last updated: 2026-09-30 · Reviewed by Nham Vu