Lesson Worksheet: The Logistic Model Mathematics • Higher Education

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.

  • Add๐‘ƒ๐‘ก=0.006๏€ผ1โˆ’๐‘ƒ7,500๏ˆ
  • Bdd๐‘ƒ๐‘ก=0.006๐‘ƒ๏€ผ1โˆ’๐‘ƒ7,500๏ˆ
  • Cdd๐‘ƒ๐‘ก=0.006๐‘ƒ๏€ผ1+๐‘ƒ7,500๏ˆ
  • Ddd๐‘ƒ๐‘ก=๐‘ƒ๏€ผ0.006+๐‘ƒ7,500๏ˆ
  • Edd๐‘ƒ๐‘ก=๐‘ƒ๏€ผ0.006โˆ’๐‘ƒ7,500๏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.

  • Add๐‘ƒ๐‘ก=0.004๏€ผ1โˆ’๐‘ƒ2,500๏ˆ
  • Bdd๐‘ƒ๐‘ก=0.004๐‘ƒ๏€ผ1โˆ’๐‘ƒ2,500๏ˆ
  • Cdd๐‘ƒ๐‘ก=0.004๐‘ƒ๏€ผ1+๐‘ƒ2,500๏ˆ
  • Ddd๐‘ƒ๐‘ก=๐‘ƒ๏€ผ0.004+๐‘ƒ2,500๏ˆ
  • Edd๐‘ƒ๐‘ก=๐‘ƒ๏€ผ0.004โˆ’๐‘ƒ2,500๏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 1 year, what will the population be after another 3 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)?

  • A๐‘1+๐ฟ๐‘’๏Šฑ๏‡๏
  • B๐‘๐ฟ๐‘’โˆ’1๏Šฑ๏‡๏
  • C๐ฟ1+๐‘๐‘’๏Šฑ๏‡๏
  • D๐ฟ๐‘๐‘’โˆ’1๏‡๏
  • E๐ฟ๐‘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?

  • A๐ฟ2
  • B0
  • Cnever
  • DIt cannot be determined.
  • E๐ฟ

Q6:

Bacteria are growing at a rate of 15%โ€Ž 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%โ€Ž per month to be fully grown. If the initial number of fully grown trees in the garden is 20, what will the number of fully grown trees be after 9 months?

Q8:

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

  • A๐‘ƒ(๐‘ก)=4๐‘’1,200+๐‘’๏Šฆ๏Ž–๏Šฆ๏Šฎ๏๏Šฆ๏Ž–๏Šฆ๏Šฎ๏
  • B๐‘ƒ(๐‘ก)=1,200๐‘’4+๐‘’๏Šฆ๏Ž–๏Šฆ๏Šฎ๏๏Šฆ๏Ž–๏Šฆ๏Šฎ๏
  • C๐‘ƒ(๐‘ก)=2๐‘’1,200โˆ’๐‘’๏Šฆ๏Ž–๏Šฆ๏Šฎ๏๏Šฆ๏Ž–๏Šฆ๏Šฎ๏
  • D๐‘ƒ(๐‘ก)=1,200๐‘’2+๐‘’๏Šฆ๏Ž–๏Šฆ๏Šฎ๏๏Šฆ๏Ž–๏Šฆ๏Šฎ๏
  • E๐‘ƒ(๐‘ก)=1,200๐‘’2โˆ’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).

  • A๐‘ƒ(๐‘ก)=90019โˆ’๐‘’๏Šฑ๏Šฆ๏Ž–๏Šฆ๏Šญ๏
  • B๐‘ƒ(๐‘ก)=90019โˆ’๐‘’๏Šฆ๏Ž–๏Šฆ๏Šญ๏
  • C๐‘ƒ(๐‘ก)=90017+๐‘’๏Šฑ๏Šฆ๏Ž–๏Šฆ๏Šญ๏
  • D๐‘ƒ(๐‘ก)=9001+17๐‘’๏Šฑ๏Šฆ๏Ž–๏Šฆ๏Šญ๏
  • E๐‘ƒ(๐‘ก)=9001+17๐‘’๏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?

  • A๐‘ƒ(๐‘ก)=6๐‘’600+๐‘’๏Šฆ๏Ž–๏Šฆ๏Šช๏๏Šฆ๏Ž–๏Šฆ๏Šช๏
  • B๐‘ƒ(๐‘ก)=600๐‘’6+๐‘’๏Šฆ๏Ž–๏Šฆ๏Šช๏๏Šฆ๏Ž–๏Šฆ๏Šช๏
  • C๐‘ƒ(๐‘ก)=600๐‘’4โˆ’๐‘’๏Šฆ๏Ž–๏Šฆ๏Šช๏๏Šฆ๏Ž–๏Šฆ๏Šช๏
  • D๐‘ƒ(๐‘ก)=4๐‘’600โˆ’๐‘’๏Šฆ๏Ž–๏Šฆ๏Šช๏๏Šฆ๏Ž–๏Šฆ๏Šช๏
  • E๐‘ƒ(๐‘ก)=600๐‘’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. 1 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.

Q15:

A population is growing following the logistic growth model with an initial population of 7,700 and then a population of 8,500 at ๐‘ก=2years. Assuming a ๐‘…-value of 0.05, after how many years will the population reach 100,000? Round your answer to one decimal place.

Q16:

It is believed that some foxes have started colonizing a small town. Observations suggest that there are currently 14 foxes. Using a logistic growth model with ๐‘˜=0.15 (using a year as a unit of time) and a carrying capacity of 68 for this town, estimate the number of foxes that will be in this town in 5 years. Give your answer to the nearest whole number.

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