Logistic Growth & Carrying Capacity Lab
Run the real logistic growth equation, add a harvesting rate, and find the maximum sustainable yield or push a population to collapse.
How to use: set the growth rate, carrying capacity, starting population and a harvesting rate, then press Run and watch the population follow the real logistic growth equation. Tip: tap or click anywhere on the graph to instantly restart at that population.
Challenge 1: maximum sustainable yield
Set the harvesting rate to about half of r (h = r/2) and run long enough to see the population settle near K/2 - the classic maximum-sustainable-yield equilibrium.
Challenge 2: overharvest to collapse
Set harvesting rate above r and run the population until it crashes toward zero.
About the Logistic Growth & Carrying Capacity Lab
Free logistic growth & carrying capacity lab. Run the real logistic growth equation, add a harvesting rate, and find the maximum sustainable yield or push a population to collapse. Drag, change the sliders and see the result live. No sign-up, works on phone and computer. Built for environmental science, the logistic growth & carrying capacity lab runs instantly in your browser: change a setting or drag an object and the result updates at once, so you learn by trying things out rather than only reading about them.
Run the real logistic growth equation, add a harvesting rate, and find the maximum sustainable yield or push a population to collapse. Use it to explore environmental science ideas at your own pace, then check what you found against the key ideas further down this page.
How to use the Logistic Growth & Carrying Capacity Lab
- Use the controls to change Intrinsic growth rate r (per year), Carrying capacity K, Starting population N₀, Harvesting rate h (per year). The simulation reacts instantly.
- Press "▶ Run", "Reset run", "Reset to defaults", "📄 Lab Report" to start, reset or change what is happening.
- Where you see a glowing handle, object, weight or atom, drag it with your mouse or finger. Everything responds in real time.
- Watch the readouts and graphs update as you experiment, and compare what you see with the key ideas below.
Things to try
- Run the model with no harvesting and watch the classic S-shaped logistic curve.
- Set h to exactly r/2 and watch the population settle at K/2.
- Push h just above r and watch a previously stable population collapse.
- Start at a tiny N₀ near zero and see how long it takes to reach carrying capacity.
Key ideas you can learn
- The logistic growth equation dN/dt = rN(1 − N/K) starts nearly exponential at low population, grows fastest at N = K/2, and levels off as N approaches the carrying capacity K.
- Adding a constant-rate harvest term, dN/dt = rN(1 − N/K) − hN, shifts the stable equilibrium down to N* = K(1 − h/r).
- The maximum sustainable yield is taken at N = K/2, where the population's own natural growth rate is at its peak (rK/4) - harvesting there removes the most individuals without shrinking the population over time.
- If the harvest rate h exceeds the intrinsic growth rate r, there is no positive equilibrium left and the population is driven to extinction.
Where this is used in the real world
Fisheries and wildlife managers set catch and hunting quotas using exactly this maximum-sustainable-yield logic from the logistic growth model to avoid collapsing the populations they depend on.
Who is this simulation for?
Students, teachers and curious learners of all ages.
For teachers: project it on the board, let students predict what will happen, then run it together. For students: change one thing at a time and write down what changes.
Frequently asked questions
Why is N = K/2, not N = K, the best population level to harvest at?
The logistic curve's own growth rate rN(1−N/K) is a downward parabola in N that peaks exactly at N = K/2; harvesting there lets you remove the largest possible amount each year while the remaining population still regrows to replace it, whereas near K the population is barely growing at all.
What happens if a fishery sets its harvest rate higher than the fish population's intrinsic growth rate?
Mathematically the only stable equilibrium becomes N = 0: the harvest term hN removes fish faster than reproduction rN(1−N/K) can ever replace them at any population size, so the stock is driven toward collapse regardless of the starting population.
Is the Logistic Growth & Carrying Capacity Lab free to use?
Yes. It is completely free, with no signup, no download and no ads inside the simulation. It runs in your web browser.
Does the Logistic Growth & Carrying Capacity Lab work on a phone or tablet?
Yes. It uses touch as well as the mouse, so you can drag objects with your finger. A larger screen makes the controls easier to see.