# 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.

**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 ?

- A
- B
- C
- D
- E

**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

**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.

**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* plantโs 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.

**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.