Population Changes - A Model

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Bob Height


Just as we need physical tools such as microscopes and telescopes to extend our powers of observation, we also need mental tools to extend our thinking. One such mental tool is called a model. Here the word does not mean an object. Instead, it means something we construct in our minds. A mental model simplifies a complex real situation, so that we understand it more easily. Because a model is a simplification, it differs in some respects from a real situation. To simplify a situation, we make assumptions. That is, we assume certain things that may be only approximately true. We must keep these assumptions in mind whenever we use a model to look at a real situation.

If a specific model gives results similar to the observations we make in some real situation, we conclude that the real situation works in the same way the modal does. Of course, this conclusion may be wrong: a diff erent modal might also give results that fit the real situation.


  1. Let us begin with real organisms house sparrows. Imagine an island. For our model we will start with a hypothetical (imaginary) population of house sparrows. In the spring of 1990 there were 10 sparrows: 5 male-female pairs. Our assumptions are:

    Assumption #1: Every breeding season (spring), each pair of sparrow's produces 10 offspring, always 5 males and 5 females.
    Assumption #2: Each year, before the next spring, all the breeding parents die.
    Assumption #3: Each year all offspring live through the next breeding season to breed, then they die. (In most real situations some parents would live and some offspring would die. But taken together, Assumptions 2 and 3 balance each other to reduce the difference between the model and a real situation.)
    Assumption #4: During the study, no other sparrows arrive on the island, and none leave.

  2. Now let us use this model. Calculate the size of the hypothetical population at the beginning of each breeding season. According to Assumption #1, in the spring of 1990 there are 10 sparrows (5 male and 5 female or 5 pair). Each of the 5 pairs of sparrows produced 10 offspring for a total of 50. The original parents die, leaving the population to be 50 sparrows in the spring of 1991. The 50 breeding birds in 1991 represent 25 pairs of sparrows with each pair having 10 offspring. The breeding parents die leaving 250 birds to breed in the spring of 1992. Continue using these assumptions and calculate the sparrow population for the years 1993, 1994, and 1995.
  3. Place your data into a data table that is a visual method scientists use to organize their statistics before analysis. Make the table neat by using a straight edge. Your teacher will show you some suggestions for constructing and labeling the table.
  4. You know have a series of numbers (statistics) that are arranged within a data table. To get a clearer idea of the population change, plot the numbers on a line graph (arithmetic graph). Show the years along the horizontal axis (time should always be found on the X-axis) and the number of sparrows along the vertical axis. Be sure to make the scale large enough to show the 1995 population with out going off the graph. Remember that the value you set for scale cannot vary along an axis. Plot the six generations.
  5. Now plot your data using another tool semi-logarithmic graph paper. (Usually called semi-log paper) You do not need to understand fully the mathematics of logarithms to use this tool. Your teacher will explain what you need to know to plot the data. Construct your semi-log graph with the same data you used before.


Look at the two graphs that you constructed. Answer the following questions using complete sentences.

  1. What advantage(s) does the semi-log graph have over the arithmetic graph for plotting data on the population growth?
  2. Look at the arithmetic graph. How does the slope of the line connecting the plotted points change as you read from left to right (year to year)?
  3. What does this mean in terms of population change?
  4. Now compare the graphs. What kind of line shows the same thing on the semi-log graph?
  5. If you continued using the same assumptions to calculate populations for an indefinite number of years and plotted them on a graph, what would happen to the slope of the line of the arithmetic graph?
  6. What would happen to the slope of the semi-log graph?
  7. In one or two sentences, summarize the change in a population that is supported by the assumptions stated in the model.
  8. Do you think any real population might change this way? Why or why not?
  9. If this were a real situation, what are some consequences of high density in a population and what are some solutions?

Further Investigations:

Sometimes a model gives results that are very different from observed situation. To make a model more useful, you can change one or two assumptions and compare the new results with reality. The closer the results of a model are to the observed situation, the more useful the model. Some suggestions for changing assumptions follow. In each case calculate the population, construct an appropriate table, and plot the data on your arithmetic and semi-log graph paper. Use different colors for your lines and include a legend. Both graphs should show the 6 different variations.

Each of the following changes pertains to the original model. You are changing only one assumption at a time. (Remember to change only the original model each time)

Change #1. Change Assumption #2 as follows: Each year 2/5 of the breeding birds (equally males and females) live to breed again a second year and then die. All other assumptions remain the same.

Change #2. Change Assumption #3 as follows: Each year 2/5 of the offspring (equally males and females) die before the beginning of the next breeding season. All other assumptions remain the same.

Change #3. Change Assumption #4 as follows: Each year 20 new house sparrows (equally males and females) immigrate to the island. None leave. All other assumptions remain the same.

Change #4. Change Assumption #4 as follows: Each year 40 house sparrows (equally males and females) emigrate from the island. All other assumptions remain the same.

Change #5. You can devise a more complex problem by changing two or more assumptions simultaneously.

Answer Key - Population Modeling:

Base Problem:

1990 10
1991 50
1992 250
1993 1250
1994 6250
1995 31250

Change #1:

1990 10
1991 54
1992 290
1993 1558
1994 8370
1995 44966

Change #2:

1990 10
1991 30
1992 90
1993 270
1994 810
1995 2430

Change #3:

1990 10
1991 70
1992 370
1993 1870
1994 9370
1995 46870

Change #4:

1990 10
1991 10
1992 10
1993 10
1994 10
1995 10

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