These models can fulfill many different purposes in the middle and high school curriculum. They are simple, yet rich, and can be used to help students learn some very important mathematics. In particular, they lend themselves to numerical, graphical, and symbolic study. Graphs involving linear models are very easy to draw by hand and give students a solid foundation for working with graphs later that are best drawn using a computer or graphing calculator. In addition, linear models are used to study many important real-world phenomena. We look at three examples.

• Population models involving immigration.
• Supply and demand models for changing prices.
• The natural cleaning of a polluted lake.

Population Models Involving Immigration

There is a thriving colony of birds on the nearby lush mainland and each year 1,000 birds from the lush mainland migrate to the barren island. Thus, the population change on the island can be described by the model

Notice that this is a linear model in the sense that we are using the word "linear" in this module and from now on.

• Compute and graph p₂, p₃, ... p₁₀ for this model with the initial condition
  p₁ = 50.

• Compute and graph p₂, p₃, ... p₁₀ for this model with the initial condition
  p₁ = 500.

• Compute and graph p₂, p₃, ... p₁₀ for this model with the initial condition
  p₁ = 10,000.

• Describe the long term behavior of the bird colony on this island.

We can write this model as

or

where

This is a particularly good way of looking at this model because it focuses our attention on the function

that describes how each year's population is determined by the population during the preceding year. Notice that this function is linear. This is the reason that this kind of model is called a linear model.

The graph below shows the function f(p) in red and the function g(p) = p in black. The function g(p) = p is the function we would use if the population did not change from one year to the next.

Notice that the two graphs cross at the point p = 5,000. This point is called an equilibrium point because next year's population is the same as the current population.

The graph above shows how we can determine this equilibrium point graphically by finding the point at which the function f(p) crosses the diagonal line g(p). This determination can be very accurate since the graph of f(p) is a straight line and thus is easy to draw.

We can verify that p = 5,000 is an equilibrium point numerically by computing

and we can find this equilibrium point algebraically by solving the equation

For each of the following linear models find the equilibrium point graphically and algebraically and verify that your answer is correct numerically.

• p_n = 0.90 p_n-1 + 1,000

• p_n = 0.90 p_n-1 + 500

• p_n = 0.75 p_n-1 + 1,000

• p_n = 0.75 p_n-1 + 500

• p_n = 1.2 p_n-1 + 1,000

• p_n = m p_n-1 + b, where m and b are constants.

• In the exercise above if b is positive, for what values of m is the equilibrium point positive and for what values of m is the equilibrium point negative?

• In the first chapter we looked at supply and demand models. Suppose the supply function and the demand function for a particular product are given by
  S(p) = 1000 p - 400

  D(p) = 5000 - 500 p

  where p denotes price. Suppose that the change in prices is described by the discrete dynamical system

  p_n = p_n-1 + k(D(p_n-1) - S(p_n-1))

  where k is a positive constant. Show this is a linear model and find its equilibrium point. Show that the equilibrium point is the price at which supply and demand are equal.

The Natural Cleaning of a Polluted Lake

Next we turn our attention to a lake that has been polluted. We suppose that the lake was polluted by someone who has been caught and that no additional pollutant will be added in the future. Suppose the present level of pollution is

The abbreviation ppm stands for parts per million.

Suppose that our lake is part of a chain of lakes connected by rivers and that each year 10% of the water in the lake flows downstream and is replaced by clean water from the lakes upstream and from rainfall. This leads to the model

• Suppose that the people who polluted the lake were sued by the state and found guilty. The court has imposed a fine of $50,000 for each year that the lake is unsafe for swimming starting with the current year with the pollution level 20 ppm. An expert from the state university testifies that the lake is unsafe for swimming as long as the level of pollution is above 5 ppm. What fine will the company have to pay if the court accepts the expert's testimony?

• The company calls its own expert who testfies that it is perfectly safe to swim in the lake when the level of pollution is below 15 ppm. What fine will the company have to pay if the court accepts the company's expert's testimony?

• An environmental group called Friends of the Lake files a friend of the court brief in which they argue that the company's expert is not impartial. She has earned a fair amount of money traveling around the country testifying on behalf of convicted polluters and being paid for her testimony. Friends of the Lake also argues that even the university expert's testimony is suspect because his university has accepted several research grants from the polluter's parent company. His testimony did not mention that pregnant women are particularly sensitive to this pollutant. Friends of the Lake files a report by a third expert who argues that the safe level of pollution is 1 ppm. What fine will the company have to pay if the court accepts this argument?

In practice, the water flowing into a lake like the lake above is rarely completely free of the pollutant. If the level of pollution in the water flowing into the lake is b ppm then the model above would be replaced by

Explore the model above.

• Begin by looking at the model given by
  p_n = 0.90 p_n-1 + 0.10 b

  with b = 0.025 and p₁ = 20. Find the equilibrium point for this model. Find the long term behavior of this model.

• Next look at the model given by
  p_n = 0.90 p_n-1 + 0.10 b

  with b = 0.075 and p₁ = 20. Find the equilibrium point for this model. Find the long term behavior of this model.

• Next look at the model given by
  p_n = 0.90 p_n-1 + 0.10 b

  with b = 0.15 and p₁ = 20. Find the equilibrium point for this model. Find the long term behavior of this model.

• Formulate and prove a general theorem. What is the equilibrium point for the model
  p_n = 0.90 p_n-1 + 0.10 b?