# 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 . 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 three 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 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?

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?

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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 the nearest whole number of years for durations over 10 years and to one decimal place for durations under 10 years.

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.