Experiment 1
Keep K = 100 and r = 0.4. Change initial abundance from 10 to 150.
Check the prediction
The population now declines toward 100. K is an attracting level in this model, not a wall that prevents initial values above it.
Population dynamics
Crowding changes the growth rate.
Separate total and per-capita growth, explain logistic carrying capacity, and recognize missing ecological interactions.
Start with: Rates of change. Per-capita means per individual; abundance is the population size.
01 · Commit to a prediction
What do you expect?
No. There are few individuals to reproduce near zero. In the continuous logistic model, total growth peaks at half the carrying capacity.
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02 · Change an assumption
Predict. Change. Explain.
Before changing a control, say what should move and why. Start with the experiments below. Reset restores the starting model; it preserves your written responses.
Experiment 2
Reset. Set initial abundance to zero.
Check the prediction
Zero stays zero. Without immigration or another source, density dependence cannot create organisms from nothing.
0 to 300 · step 5
10 to 200 · step 10
0 to 1 · step 0.05
Calculated model output. The table gives the same values. Displayed values are rounded; calculations keep full precision.
Read the plotted data as a table
03 · Connect the mechanism
From the picture to the quantities.
The logistic model multiplies abundance by a per-capita growth rate that declines linearly with abundance. It describes a balance of births and deaths without tracking each separately. Negative feedback stabilizes positive populations near K. Smooth fractional abundance is a deterministic approximation, not a count of a simulated animal.
dN/dt = rN(1−N/K)
Per-capita growth = r(1−N/K), for N > 0
N(t) = K / [1 + (K/N₀−1)e−rt], for N₀ > 0
Per-capita growth = r(1−N/K), for N > 0
N(t) = K / [1 + (K/N₀−1)e−rt], for N₀ > 0
N and K are abundances in individuals, K > 0. N₀ is initial abundance, 0 to 300 here. r is 0 to 1 day⁻¹. t is days, 0 to 20. dN/dt is individuals per day. Per-capita growth is undefined at N = 0 as an observed ratio; its limiting model coefficient is r. At N₀ = 0 use N(t) = 0 directly.
Worked example
At K = 100 and r = 0.4/day, total growth at N = 10 is 0.4 × 10 × 0.9 = 3.6 individuals/day. At N = 50 it is 10/day, and at N = 90 it is 3.6/day again. Completing the square gives rN(1−N/K) = rK/4 − (r/K)(N−K/2)², proving the maximum occurs at K/2.
Assumptions and limits
One species, constant resources, immediate density dependence, no age structure, immigration, harvest, stochasticity or delays. The exact continuous solution is plotted, not an Euler approximation or logistic map. K is an environmental model parameter, not a universal species constant. Competition and coexistence need multiple populations and interaction assumptions; this lesson does not infer them from one S-shaped curve.
The annotated sources distinguish established results from this lesson’s original examples.
04 · Follow the structure
Where else does this apply?
A microbial culture
Resource limitation can slow growth as density rises.
Boundary: Changing nutrients, waste, death phases and multiple strains can violate constant K and r.
Adoption of a practice
An S-shaped aggregate can resemble logistic growth.
Boundary: Social imitation and market limits are different mechanisms. A similar curve does not establish biological density dependence.
05 · Retrieve without hints
Close the explanation. Try a new case.
Write an answer before opening its feedback. Later, return directly here without rereading above. Recognition, explanation and transfer are separate outcomes. No page action or answer reveal measures mastery.
Recognition and explanation
Reveal reasoning and rubric
Total growth multiplies the per-capita rate by abundance. A small group can have fast individual growth but a small total increase.
Self-check: State both quantities and their units.
Calculation
Reveal reasoning and rubric
N = 100. The maximum is rK/4 = 0.2 × 200/4 = 10 individuals/day.
Self-check: Give the abundance and rate with different units.
Novel transfer
Reveal reasoning and rubric
No. The closed-population model excludes immigration. Add an inflow term or specify a later nonzero initial condition. The model also cannot give an extinction probability for a small population.
Self-check: Identify the omitted source and distinguish deterministic abundance from stochastic extinction.
On a later day, try again and record actual evidence in the curriculum. A later unaided explanation and a fresh transfer problem give stronger evidence than immediate familiarity. No reminder is scheduled.
Sources & scope
What supports the lesson?
- OpenStax, Concepts of Biology §19.2. Logistic growth, limiting resources and carrying capacity.
Original teaching examples. Reference links need a connection; the lesson itself does not. Built 2026-10-11. Learner understanding is not assessed.