What it shows
A population with plenty of food and space grows exponentially: the more individuals there are, the more babies they have, so the curve bends up into a J and never stops. Real places run short of food and space, so growth slows down and stops at the carrying capacity K, the most the place can support. That is logistic growth, an S-shaped curve.
This page draws both curves from three sliders: the start size N₀, the intrinsic rate of natural increase r, and K. A second graph plots the growth rate dN/dt against N: a straight line for exponential growth, an upside-down U for logistic growth, highest at N = K/2. A third card shows the three age pyramid shapes: expanding, stable and declining.
It fits Class 12 Biology: organisms and populations (population attributes, age pyramids, exponential and logistic growth). For age-structured models with a Leslie matrix see population growth models; what happens when one population eats another is in predator and prey, and you can draw a real pyramid from your own numbers in population pyramid.
How to use it
- Pick a story (bacteria in a flask, deer on an island, fish in a pond) or slide your own numbers.
- Slide time t to move the dots along the curves. The facts show both values of N.
- Use Show to see just one curve.
- Look at the dN/dt graph: the dot shows how fast the logistic population is growing right now.
- Tap Expanding, Stable or Declining to compare age pyramids.
Worked example
Bacteria story: N₀ = 10, r = 0.5 per hour, K = 1000. After t = 10 hours:
- exponential: N = 10 × e^(0.5 × 10) = 10 × e⁵ ≈ 1484.1;
- logistic: N = 1000 / (1 + 99 e^(−5)) ≈ 599.9, about 60 % of K;
- the doubling time of exponential growth is ln 2 / 0.5 ≈ 1.39 hours;
- logistic growth is fastest at N = K/2 = 500, where dN/dt = rK/4 = 125 per hour, reached at t ≈ 9.19 hours.
What the options mean
- r: births minus deaths per individual per unit time, when nothing is short.
- K: the carrying capacity. Above K, deaths beat births and N falls back.
- Age pyramids: bars show the share of people in each 10-year age group. A wide base means many young people (expanding); straight sides mean steady (stable); a narrow base means shrinking (declining). The shapes here are made up.
Check yourself
Five fresh questions: exponential growth after a time, logistic dN/dt, where growth is fastest, a doubling time, and naming a pyramid shape. Nothing is saved.
FAQ
Why do real populations not grow exponentially for long? Food, space, water and nesting places run out, and disease and predators increase. These limits are what K stands for.
Is the logistic curve exact for real animals? No; it is a simple model. Real populations wobble around K, overshoot, or crash, but the S shape is a useful first picture.
Is this from my textbook? It covers the standard school topic with its own examples. It is an independent study aid.
Where the numbers come from
- Standard population ecology: exponential growth N = N₀e^(rt), logistic (Verhulst-Pearl) growth dN/dt = rN(1 − N/K), doubling time ln 2 / r, and age pyramids. Written for fibub; no textbook text, figures or exercises are used.
- Story numbers and the three pyramid shapes are made up for teaching.
- Checked against scipy.integrate.solve_ivp (rtol 1e-12) and scipy.optimize.brentq for the curves and the time to K/2.
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Last updated 2026-11-09.