Differential equation (math)

MAT 275 Online Activity 2

Population growth is usually limited by the carrying capacity of the environment.

The growth is initially approximately exponential, but then it slows down as the number of

bacteria, animals or people approaches the maximum number that can be sustainably

supported by the environment.

This kind of limited population growth obeys the logistic equation:

𝑑𝑦

𝑑𝑡 = 𝑟 (1 −

𝑦

𝐾 )𝑦

In this model, 𝑦(𝑡) is the number of individuals in the population at time t, and 𝑟 and 𝐾 are

positive constants.

Questions:

1. The constant 𝐾 is the carrying capacity of the environment. Confirm this by showing,

directly based on the logistics equation above, that

a. 𝑦(𝑡) = 𝐾 is the only positive equilibrium solution of the logistic equation,

b. 𝑦′(𝑡) > 0 if 𝑦(𝑡) < 𝐾 for a positive solution 𝑦(𝑡) of the logistic equation,

c. 𝑦′(𝑡) < 0 if 𝑦(𝑡) > 𝐾 for a positive solution 𝑦(𝑡) of the logistic equation.

d. Explain in your own words what a. - c. are saying about the solutions 𝑦(𝑡). Is it

impossible for the population to exceed 𝐾?

2. The constant 𝑟 is called the intrinsic growth rate, which is the growth rate as if there

were no environmental limits to growth. Give a mathematical explanation of that, based

directly on the logistic equation given above.

On all these questions, you must find your own solutions. Do not just repeat information from

the textbook or found from other sources.