The two graphs below show the percentage of the population in the age groups 0-4, 5-9, 10-14, ..., and 95 and up for the United States and for Bangladesh. Not surprizingly, the graphs are quite different. The United States is often classified as a "more developed country" and it exhibits an age distribution typical of such countries. Because health care and the food supply are well-developed, people tend to live longer. Can you explain the population bulge around the age of forty?

Bangladesh is often classified as a "less developed country" and it exhibits an age distribution that is typical of such countries. Notice how rapidly the population percentage decreases for older age groups. The differences between the two countries would be even more striking if the age distribution was broken down into smaller groups. One of the biggest problems in less developed countries is the high level of infant mortality -- the percentage of babies that die very young.

A wealth of population data, by country and age group, including death rates and birth rates by age group, is available over the World Wide Web. The links below lead to three examples, all at the United States Census Bureau. Using this data and the ideas developed in this module you can build very detailed population models.

• United States National Population Projections from the United States Census Bureau.

• World Population Profile: 1996

• United States Census Bureau International Data Base

In this module we use discrete dynamical systems to model population growth for populations, like human populations, for which age is important. In some ways continuous dynamical systems might be better because human population is changing continuously. In practice, however, our data is discrete and, so, discrete models are often used. Moreover, human populations do exhibit some of the kinds of behavior that we have already seen for discrete dynamical systems and that we do not see with the simplest continuous models. The reason for this is that the effects that humans have on our environment can be very persistant. One example of this phenonmenon is global warming, caused in part by increased burning of fossil fuels. If our actions result in global warming then it will take some time for the earth to return to its former condition. Another example is the depletion of rain forests. The links below lead to further information about these two problems.

• The Environmental Protection Agency's Global Warming Site

• The Problem of Forest Loss -- World Resources Institute

• One School-Based Project on Rain Forests

We will look at a fairly simple model of population growth that considers age. You can build more complex and realistic models using data from the United States Census Bureau links above but we will be able to learn a great deal about these kinds of models with our simpler models.

We will work with a mythical species whose life span is up to five years and we will use five sequences

Each year a certain percentage of the population in each age group survives and moves into the next age group. This leads to four change equations of the form

represents the births each year. The numbers S₁, S₂, S₃, S₄, S₅ represent the birth rate for each age group. These equations can be expressed more compactly using vectors and matrices.

One example of these change equations is given below.

In this example 80% of the population in their first year of age survives to their second year; 95% of the population in their second year of age survives to their third year; 85% of the population in their third year of age survives to their fourth year; and 65% of the population in their fourth year of age survives to their fifth year. Note that none of the population in their fifth year of age survives to a sixth year.

Note also that the birth rate for the part of the population in their first year is 0.50; for the part of the population in their second year it is 1.50; for the part of the population in their third year it is 1.50; for the part of the population in their fourth year it is 1.00; and for the part of the population in their fifth year it is 0.50.

• Use your CAS window to study the first few terms of the model above using the initial conditions

  A₁ = 150

  B₁ = 145

  C₁ = 140

  D₁ = 125

  E₁ = 100

• Look at the same model and the same initial conditions as above and describe as completely as you can what happens over the short term and the long term to this population.

• Consider a particular species for which the change equations are
  A_n = 0.25 A_n-1 + 1.00 B_n-1 + 0.75 C_n-1 + 0.50 D_n-1 + 0.25 E_n-1.

  B_n = 0.50 A_n-1

  C_n = 0.75 B_n-1

  D_n = 0.75 C_n-1

  E_n = 0.25 D_n-1

  Suppose that 100 individuals in their third year move into a previously unihabited area. Describe what happens.

• The change equations above might represent a species with a relatively high birth rate and a relatively low survival rate. In human populations this is common in "less developed" countries. Now suppose that after twenty years this species becomes "more developed" and the birth rate drops and the survival rate increases so that the change equations become
  A_n = 0.25 A_n-1 + 0.45 B_n-1 + 0.40 C_n-1 + 0.25 D_n-1 + 0.15 E_n-1.

  B_n = 0.85 A_n-1

  C_n = 0.95 B_n-1

  D_n = 0.90 C_n-1

  E_n = 0.50 D_n-1

  Describe what happens.