Worksheet: The Logistic Model

In this worksheet, we will practice using the logistic differential equation to model situations where the growth of a quantity is limited by a carrying capacity.

Q1:

Suppose a population grows according to a logistic model with a carrying capacity of 7,500 and ๐‘˜=0.006. Write the logistic differential equation for this information.

  • A d d ๐‘ƒ ๐‘ก = 0 . 0 0 6 ๏€ผ 1 โˆ’ ๐‘ƒ 7 , 5 0 0 ๏ˆ
  • B d d ๐‘ƒ ๐‘ก = 0 . 0 0 6 ๐‘ƒ ๏€ผ 1 โˆ’ ๐‘ƒ 7 , 5 0 0 ๏ˆ
  • C d d ๐‘ƒ ๐‘ก = 0 . 0 0 6 ๐‘ƒ ๏€ผ 1 + ๐‘ƒ 7 , 5 0 0 ๏ˆ
  • D d d ๐‘ƒ ๐‘ก = ๐‘ƒ ๏€ผ 0 . 0 0 6 + ๐‘ƒ 7 , 5 0 0 ๏ˆ
  • E d d ๐‘ƒ ๐‘ก = ๐‘ƒ ๏€ผ 0 . 0 0 6 โˆ’ ๐‘ƒ 7 , 5 0 0 ๏ˆ

Q2:

Suppose a population grows according to a logistic model with a carrying capacity of 2,500 and ๐‘˜=0.004. Write the logistic differential equation for this information.

  • A d d ๐‘ƒ ๐‘ก = 0 . 0 0 4 ๏€ผ 1 โˆ’ ๐‘ƒ 2 , 5 0 0 ๏ˆ
  • B d d ๐‘ƒ ๐‘ก = 0 . 0 0 4 ๐‘ƒ ๏€ผ 1 โˆ’ ๐‘ƒ 2 , 5 0 0 ๏ˆ
  • C d d ๐‘ƒ ๐‘ก = 0 . 0 0 4 ๐‘ƒ ๏€ผ 1 + ๐‘ƒ 2 , 5 0 0 ๏ˆ
  • D d d ๐‘ƒ ๐‘ก = ๐‘ƒ ๏€ผ 0 . 0 0 4 + ๐‘ƒ 2 , 5 0 0 ๏ˆ
  • E d d ๐‘ƒ ๐‘ก = ๐‘ƒ ๏€ผ 0 . 0 0 4 โˆ’ ๐‘ƒ 2 , 5 0 0 ๏ˆ

Q3:

Suppose a population grows according to a logistic model with an initial population of 1,000 and a carrying capacity of 10,000. If the population grows to 2,500 after one year, what will the population be after another three years?

Q4:

The logistic population model assumes an upper limit ๐ฟ, beyond which growth cannot occur. The population ๐‘ฆ(๐‘ก) has a rate of change that satisfies dd๐‘ฆ๐‘ก=๐‘˜๐‘ฆ๏€ป1โˆ’๐‘ฆ๐ฟ๏‡ for some positive constant ๐‘˜. A suitable function ๐‘ฆ(๐‘ก) involves a second parameter ๐‘, determined by how rapid the initial growth is. Without integrating, which of the following could be ๐‘ฆ(๐‘ก)?

  • A ๐‘ 1 + ๐ฟ ๐‘’ ๏Šฑ ๏‡ ๏
  • B ๐‘ ๐ฟ ๐‘’ โˆ’ 1 ๏Šฑ ๏‡ ๏
  • C ๐ฟ 1 + ๐‘ ๐‘’ ๏Šฑ ๏‡ ๏
  • D ๐ฟ ๐‘ ๐‘’ โˆ’ 1 ๏‡ ๏
  • E ๐ฟ ๐‘ ๐‘’ โˆ’ 1 ๏Šฑ ๏‡ ๏

Q5:

Unlike exponential growth, where a population grows without bound, the logistic model assumes an upper limit ๐ฟ, beyond which growth cannot occur. The population ๐‘ฆ(๐‘ก) has a rate of change that satisfies dd๐‘ฆ๐‘ก=๐‘˜๐‘ฆ๏€ป1โˆ’๐‘ฆ๐ฟ๏‡ for some positive constant ๐‘˜. Given a population with ๐‘ฆ(0)=๐ฟ2, at what population is the growth zero?

  • A ๐ฟ 2
  • B0
  • Cnever
  • DIt cannot be determined.
  • E ๐ฟ

Q6:

Bacteria is growing at a rate of 15% per minute in a closed container. If the initial number of bacteria is 2 and the carrying capacity of the container is 2 million cells, how long will it take the bacteria to reach 1 million cells? Give your answer to the nearest minute.

Q7:

A garden has a carrying capacity of 200 trees and needs a rate of 3% per month to be full grown. If the initial number of full-grown trees in the garden is 20 trees, what will the number of full-grown trees be after 9 months?

  • A21 trees
  • B27 trees
  • C34 trees
  • D45 trees
  • E25 trees

Q8:

An aquarium contains fish with a carrying capacity of 1,200 and a growth rate of 8%. If the initial population of the fish is 400, what is the population of the fish at any given time?

  • A ๐‘ƒ ( ๐‘ก ) = 4 ๐‘’ 1 , 2 0 0 + ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏
  • B ๐‘ƒ ( ๐‘ก ) = 1 , 2 0 0 ๐‘’ 4 + ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏
  • C ๐‘ƒ ( ๐‘ก ) = 2 ๐‘’ 1 , 2 0 0 โˆ’ ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏
  • D ๐‘ƒ ( ๐‘ก ) = 1 , 2 0 0 ๐‘’ 2 + ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏
  • E ๐‘ƒ ( ๐‘ก ) = 1 , 2 0 0 ๐‘’ 2 โˆ’ ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šฎ ๏

Q9:

Suppose a populationโ€™s growth is governed by the logistic equation dd๐‘ƒ๐‘ก=0.07๐‘ƒ๏€ผ1โˆ’๐‘ƒ900๏ˆ, where ๐‘ƒ(0)=50. Write the formula for ๐‘ƒ(๐‘ก).

  • A ๐‘ƒ ( ๐‘ก ) = 9 0 0 1 9 โˆ’ ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šญ ๏
  • B ๐‘ƒ ( ๐‘ก ) = 9 0 0 1 9 โˆ’ ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šญ ๏
  • C ๐‘ƒ ( ๐‘ก ) = 9 0 0 1 7 + ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šญ ๏
  • D ๐‘ƒ ( ๐‘ก ) = 9 0 0 1 + 1 7 ๐‘’ ๏Šฑ ๏Šฆ ๏Ž– ๏Šฆ ๏Šญ ๏
  • E ๐‘ƒ ( ๐‘ก ) = 9 0 0 1 + 1 7 ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šญ ๏

Q10:

A cage can hold up to 1,000 birds, and 200 birds are initially moved into the cage. Suppose that the population of birds grows according to the logistic model. If after 2 months there are 400 birds in the cage, after how many months will the population reach 800 birds? Give your answer to the nearest month.

Q11:

The population of the citizens inside a village has a carrying capacity of 600 and a growth rate of 4%. If the initial population is 120 citizens, what is the population of the citizens in the village at any time?

  • A ๐‘ƒ ( ๐‘ก ) = 6 ๐‘’ 6 0 0 + ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏
  • B ๐‘ƒ ( ๐‘ก ) = 6 0 0 ๐‘’ 6 + ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏
  • C ๐‘ƒ ( ๐‘ก ) = 6 0 0 ๐‘’ 4 โˆ’ ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏
  • D ๐‘ƒ ( ๐‘ก ) = 4 ๐‘’ 6 0 0 โˆ’ ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏
  • E ๐‘ƒ ( ๐‘ก ) = 6 0 0 ๐‘’ 4 + ๐‘’ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏ ๏Šฆ ๏Ž– ๏Šฆ ๏Šช ๏

Q12:

The population growth of wolves in a national park follows a logistic growth model with an initial population of 15 wolves, a ๐‘˜-value of 0.05 (using a year as the unit of time), and a carrying capacity of 80. In roughly how many years does the model predict a wolf population of 60?

Q13:

The biomass of Cerastium is assumed to follow a logistic growth model with an initial biomass of 0.1 g and a proportionality factor ๐‘˜=0.055, using day as a unit of time. At ๐‘ก=75 days, a Cerastium plantโ€™s biomass was 3.0 g. Find the final biomass of this Cerastium plant when it is fully grown. Give your answer to one decimal place.

Q14:

It is assumed that the snail population in a kitchen garden can be modeled with a logistic growth model. At ๐‘ก=0, there were 6 snails in the patch. One month later, there were 12 snails. Given that the carrying capacity of the kitchen garden is 36 snails, at what time ๐‘ก in months does the model predict a snail population of 32? Round your answer to the nearest whole number of months.

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