The Logistic Equation and Saturating Growth · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Growth with a ceiling

Mathematics · Differential Equations · ages 19-20
Name ______________________   Date ____________

  1. Same lake, two forecasts. What differs between exponential and logistic predictions over a long horizon?

    • Exponential grows without bound while logistic levels at K
    • Both level off at the same K
    • Exponential levels off while logistic keeps growing
    • Both grow without bound at the same rate

  2. For dP/dt equals 0.4 P times (1 minus P over 100), what are the equilibrium solutions?

    • P equals 0 only
    • P equals 0 and P equals 100
    • P equals 50 and P equals 100
    • P equals 40 only

  3. A logistic solution is P of t equals 100 divided by (1 plus e to the minus 0.4 t). What is P of 0?

    Answer: ______________

  4. For the same equation, which equilibrium is stable?

    • P = 0
    • Both are stable
    • P = 100

  5. A logistic solution is P = 100 over (1 + e to the minus 0.4 t). What is P(0)?

    Answer: ______________

  6. For dP/dt = 0.4 P times (1 minus P over 100), the equilibrium P = 0 is stable.

    Circle one:   True   False

  7. Why does logistic growth slow down as P nears K?

    • The bracket (1 minus P over K) shrinks toward zero
    • The rate r keeps increasing
    • The population stops breeding entirely

  8. Same lake, two forecasts. What differs between exponential and logistic predictions over a long horizon?

    • Both level off at the same K
    • Exponential grows without bound while logistic levels at K
    • Exponential levels off while logistic keeps growing

  9. A herd of 120 lives in a range with K = 100. What happens next under the logistic model?

    • It keeps rising past 120
    • It stays exactly at 120
    • It declines toward 100

  10. For dP/dt = 0.4 P times (1 minus P over 100), the rate at P = 1 is 0.4 times 1 times 0.99. What is it?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Growth with a ceiling W1-mt_7Y_48Ga33L-s1

  1. Exponential grows without bound while logistic levels at K · Only the logistic model has the braking term, so only it saturates.
  2. P equals 0 and P equals 100 · Setting the rate to zero gives P equals 0 or 1 minus P over 100 equals 0, so P is 0 or 100.
  3. 50 · At t equals 0 the exponential is 1, so P is 100 over 2, which is 50.
  4. P = 100 · Small populations grow away from 0 and toward 100.
  5. 50 · At t = 0 the exponential is 1, so P is 100 over 2.
  6. False · A tiny introduced population grows, moving away from 0.
  7. The bracket (1 minus P over K) shrinks toward zero · The braking factor vanishes exactly at the ceiling.
  8. Exponential grows without bound while logistic levels at K · Only the logistic model has the braking term, so only it saturates.
  9. It declines toward 100 · Above K the bracket turns negative, so the rate pushes down.
  10. 0.396 · 0.4 times 0.99 equals 0.396, positive, so P grows.
Worksheet · LightMySky