What is the difference between exponential and logistic growth?

Exponential growth assumes unlimited resources: the population grows at a constant rate r indefinitely. Logistic growth adds a carrying capacity K — the maximum sustainable population — so growth slows as the population nears K and follows an S-shaped (sigmoid) curve.

What units does the growth rate r use?

r is expressed per time step. If one time step equals one year and r = 0.1, the population grows by roughly 10% per year. The calculator works with any consistent unit (hours, days, years) as long as r and the time steps match.

What happens when the population exceeds carrying capacity K?

In the logistic model, a population above K will actually decline back toward K. The calculator handles this correctly — if N₀ > K, you will see the population decrease in the table.

How accurate are these models?

These are idealized mathematical models that assume a constant rate and no stochastic effects, migration, or age structure. They are excellent for educational purposes and first-order estimates but real populations require more complex modeling.

Can I use decimals for the initial population?

The calculator accepts decimal inputs (useful for representing thousands or millions as scaled units), but population counts are rounded to whole numbers in the output table for readability.

Population Growth Calculator

Compute exponential or logistic population growth from initial size, growth rate, and time — with a full step-by-step table.

Growth Parameters

Growth Model

Initial Population (N₀)

Growth Rate (r) per time step

Time Steps (1 – 100)

Initial Population

—

Final Population

—

Growth Rate (r)

—

Net Change

—

Step-by-Step Growth Table

Time Step	Population	Change	% Change

Set your parameters and click Calculate
to see the growth projection.

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Summary

This calculator models two classic population growth equations: exponential growth (N(t) = N₀ × e^(rt)) and logistic growth (N(t) = K / (1 + ((K − N₀)/N₀) × e^(−rt))). Enter the starting population, intrinsic growth rate, and number of time steps to instantly see future population size. Switch to logistic mode and add a carrying capacity K to observe how populations level off as they approach their environmental limit.

How it works

Select either Exponential or Logistic growth model.
Enter the initial population size (N₀).
Enter the intrinsic growth rate r (e.g. 0.1 = 10% per time step).
For logistic growth, enter the carrying capacity K.
Set the number of time steps to project.
Click Calculate to see the final population and a full growth table.

Use cases

Biology coursework on population dynamics and ecology.
Modeling bacterial colony growth in a petri dish.
Wildlife management projections for endangered species.
Estimating human population under limited resource constraints.
Comparing unchecked exponential growth vs. resource-limited logistic growth.
Teaching the concept of carrying capacity and sigmoid growth curves.
Public health modeling of disease spread in a susceptible population.
Fisheries management and sustainable harvest planning.

Frequently Asked Questions

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