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?
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 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. 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.
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.
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.