What is logistic growth?

Logistic growth is a model where a population initially grows exponentially but slows as it approaches the carrying capacity K due to limited resources. The result is an S-shaped (sigmoid) curve.

What does the carrying capacity K represent?

K is the maximum population size that the environment can sustain indefinitely. Once the population reaches K, net growth stops because births equal deaths.

What is the intrinsic growth rate r?

r is the per-capita growth rate when population density is very low (far from K). A value of 0.3 means the population grows at ~30% per time unit under ideal conditions.

What happens if P0 is greater than K?

If the initial population exceeds the carrying capacity, the logistic model predicts the population will decline back toward K, reflecting overshoot and correction.

What time units does this calculator use?

The calculator is dimensionless — time units match whatever you use for r. If r is per year, then t is in years. If r is per day, then t is in days.

How is this different from exponential growth?

Exponential growth assumes unlimited resources and grows without bound. Logistic growth adds a ceiling (K) so growth slows near capacity, which is more realistic for most real populations.

Logistic Growth Calculator

Compute population size at any time using the logistic growth equation with carrying capacity, initial population, and growth rate.

Logistic Growth Parameters

P(t) = K / (1 + ((K − P₀) / P₀) × e^−rt)

Carrying Capacity (K)

Maximum sustainable population size.

Initial Population (P₀)

Population at time zero. Must be > 0.

Growth Rate (r)

Intrinsic per-capita growth rate per time unit.

Time (t)

Time point to evaluate. Use same units as r.

Quick Examples

Enter parameters on the left and click Calculate to see the logistic growth results.

Population at time t

— individuals

—

P₀

—

% of carrying capacity —

Growth Table

10 equal time steps up to t

Time (t)	Population P(t)	% of K	Net Change

Summary

The logistic growth model describes how a population grows quickly at first and then slows as it approaches its carrying capacity K. The formula P(t) = K / (1 + ((K - P0) / P0) * e^(-rt)) captures this S-shaped (sigmoid) curve, where P0 is the initial population, r is the intrinsic growth rate, and t is time. This calculator lets you compute the population at a specific time and view a growth table over multiple intervals.

How it works

Enter the carrying capacity K — the maximum sustainable population.
Enter the initial population P0 at time zero.
Enter the intrinsic growth rate r (e.g. 0.3 for 30% per time unit).
Enter the time t at which you want to know the population.
Click Calculate to see P(t) and a growth table across 10 equal time steps.

Use cases

Ecology: model animal or plant population dynamics in a limited habitat.
Epidemiology: estimate disease spread under resource or immunity constraints.
Biology courses: demonstrate the sigmoid growth curve for homework or lab reports.
Business: model adoption curves or market saturation using logistic principles.
Conservation: predict whether an endangered species can recover over time.
Agriculture: estimate crop or livestock population growth within farm limits.

Frequently Asked Questions

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