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.
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 ?
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?
- DIt cannot be determined.
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
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.
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?
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?
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.
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. One 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.