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CalculusBy Educator Today

C31 · Unit C7 Differential equations

Exponential and logistic models

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 7.9 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

Exponential growth y minutes equals ky gives y equals y sub 0 e^(kt); its fractional growth rate k is constant. Logistic growth P minutes equals kP(1 minus P over K), with k,K>0, has equilibria 0 and K. For 0<P<K, population rises toward K and grows fastest at P equals K over 2. The logistic component of this lesson belongs to BC; AB students study the exponential model and its interpretations.

Worked example

For P minutes equals 0.2P(1 minus P over 1000), carrying capacity is 1000 and maximum rate occurs at P equals 500, where P minutes equals 50 people per year. Below K over 2 the solution is concave up; between K over 2 and K it is concave down.

Explore the mathematics

Exponential and logistic models

Compare growth above and below a stable carrying capacity.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict how growth changes as the population approaches K.xy1234y′=y(1−y/5), y(0)=1

Predict before continuing

Predict how growth changes as the population approaches K.

Read the complete explanation
  1. Predict how growth changes as the population approaches K.
  2. For 0<y<K, y′ is positive; at y=K it is zero.
  3. For y>K the rate is negative, pointing back toward the carrying capacity.
  4. The equilibrium K=5 is stable for positive starting values. The model also has equilibrium 0.
See the idea: Exponential and logistic models
See the idea: Exponential and logistic modelsHorizontal axis: population P. Vertical axis: growth rate P′. The logistic rate is zero at P=0 and P=1000 and reaches 50 at P=500. This is a rate-versus-population graph, not a population-versus-time graph.xy00250155003075045100060maximum rate 50

Horizontal axis: population P. Vertical axis: growth rate P′. The logistic rate is zero at P=0 and P=1000 and reaches 50 at P=500. This is a rate-versus-population graph, not a population-versus-time graph.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Carrying capacity is an equilibrium level, not the maximum derivative. Distinguish k in kP(1 minus P over K) from a coefficient in kP(K minus P).

Investigate and explain

Compare exponential and logistic predictions for the same small initial population. Explain why short-term similarity does not validate long-term exponential growth.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.