# 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 . Write the logistic differential equation for this information.

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Q2:

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

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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 , beyond which growth cannot occur. The population has a rate of change that satisfies for some positive constant . A suitable function involves a second parameter , determined by how rapid the initial growth is. Without integrating, which of the following could be ?

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Q5:

Unlike exponential growth, where a population grows without bound, the logistic model assumes an upper limit , beyond which growth cannot occur. The population has a rate of change that satisfies for some positive constant . Given a population with , at what population is the growth zero?

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• B0
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• DIt cannot be determined.
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Q6:

Bacteria are growing at a rate of ‎ 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 ‎ 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 . If the initial population of the fish is 400, what is the population of the fish at any given time?

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Q9:

Suppose a population’s growth is governed by the logistic equation , where . Write the formula for .

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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 . If the initial population is 120 citizens, what is the population of the citizens in the village at any time?

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Q12:

The population growth of wolves in a national park follows a logistic growth model with an initial population of 15 wolves, a -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 , using day as a unit of time. At days, a Cerastium plant’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 , 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 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 . Assuming a -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 (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.