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Key concepts

  • When studying population functions, different assumptions—such as exponential growth, logistic growth, or threshold population—lead to different rates of growth.
  • The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.
  • The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.

Key equations

  • Logistic differential equation and initial-value problem
    d P d t = r P ( 1 P K ) , P ( 0 ) = P 0
  • Solution to the logistic differential equation/initial-value problem
    P ( t ) = P 0 K e r t ( K P 0 ) + P 0 e r t
  • Threshold population model
    d P d t = r P ( 1 P K ) ( 1 P T )

For the following problems, consider the logistic equation in the form P = C P P 2 . Draw the directional field and find the stability of the equilibria.

Solve the logistic equation for C = 10 and an initial condition of P ( 0 ) = 2 .

P = 10 e 10 x e 10 x + 4

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Solve the logistic equation for C = −10 and an initial condition of P ( 0 ) = 2 .

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A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2 % . If the initial population is 50 deer, what is the population of deer at any given time?

P ( t ) = 10000 e 0.02 t 150 + 50 e 0.02 t

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A population of frogs in a pond has a growth rate of 5 % . If the initial population is 1000 frogs and the carrying capacity is 6000 , what is the population of frogs at any given time?

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[T] Bacteria grow at a rate of 20 % per hour in a petri dish. If there is initially one bacterium and a carrying capacity of 1 million cells, how long does it take to reach 500,000 cells?

69 hours 5 minutes

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[T] Rabbits in a park have an initial population of 10 and grow at a rate of 4 % per year. If the carrying capacity is 500 , at what time does the population reach 100 rabbits?

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[T] Two monkeys are placed on an island. After 5 years, there are 8 monkeys, and the estimated carrying capacity is 25 monkeys. When does the population of monkeys reach 16 monkeys?

7 years 2 months

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[T] A butterfly sanctuary is built that can hold 2000 butterflies, and 400 butterflies are initially moved in. If after 2 months there are now 800 butterflies, when does the population get to 1500 butterflies?

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The following problems consider the logistic equation with an added term for depletion, either through death or emigration.

[T] The population of trout in a pond is given by P = 0.4 P ( 1 P 10000 ) 400 , where 400 trout are caught per year. Use your calculator or computer software to draw a directional field and draw a few sample solutions. What do you expect for the behavior?


A direction field with arrows down for P < 1,000, pointing up for 1,000 < P < 8,500, and pointing down for P > 8,500. Right above P = 8,500, the arrows point down and to the right.

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In the preceding problem, what are the stabilities of the equilibria 0 < P 1 < P 2 ?

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[T] For the preceding problem, use software to generate a directional field for the value f = 400 . What are the stabilities of the equilibria?


A direction field with arrows pointing down and to the right. Around y = 4,000, the arrows are more horizontal. The further the arrows are from this line, the more vertical the arrows become.
P 1 semi-stable

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[T] For the preceding problems, use software to generate a directional field for the value f = 600 . What are the stabilities of the equilibria?

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[T] For the preceding problems, consider the case where a certain number of fish are added to the pond, or f = −200 . What are the nonnegative equilibria and their stabilities?


A direction field with arrows pointing up for P < 10,000 and arrows pointing down for P > 10,000.
P 2 > 0 stable

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Questions & Answers

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Practice Key Terms 6

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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