Lesson Worksheet: The Logistic Model Mathematics • Higher Education

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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:

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?

Q2:

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?

Q3:

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.

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 ?

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?

A
B0
Cnever
DIt cannot be determined.
E

Q6:

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?

Q7:

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.

Q8:

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 , 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.

Q9:

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?

Q10:

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.

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