Population Growth: A Model Introduction

Just as we need tools such as the microscope to help us extend our powers of observation, so we need mental "tools" to help us extend our thinking. One such mental tool is called a model. The model we are discussing here is not an object; it is a mental image. This kind of model simplifies a complex real situation so that we can more easily understand it. Because the model is a simplification, it differs in some respects from the real situation. The simplifications we make are called assumptions. To simplify, we assume certain things that may be only approximately true. We must keep these assumptions in mind whenever we use the model to try to understand a real situation.

Should the model give results similar to those given in the real situation, we can have confidence that the real situation "works" in the same way the model does. Of course, the two will never match exactly, but the degree of matching will determine the extent of our confidence in the model.

In this activity you will use a model to investigate the way in which a population might grow.

Materials

Ordinary (arithmetic) graph paper

Semilogarithrnic graph paper

Procedure

1. Set up the model. Let us begin with a real organism - the house sparrow. Now imagine an island, and on that island, in the spring of 1997, an imaginary population of 10 house sparrows - 5 male-female pairs.

Assumption 1: Each breeding season (spring), each pair of sparrows produces 10 offspring, always 5 males and 5 females.

Assumption 2: Each year all the breeding (parent) birds die before the next spring.

Assumption 3: Each year all offspring live through the next breeding season. (In most real situations some parents would live and some offspring would die. But taken together, Assumptions 2 and 3 tend to balance each other, thus reducing the difference between our model and a real situation.)

Assumption 4: During the study the population is closed.

2. Growth of the population. Now we want to see how this hypothetical population will grow. To do this, we must calculate the size of the population at the beginning of each breeding season. According to Assumption 1, in 1997 we have 5 pairs, each producing 10 offspring, a total of 50 offspring. According to Assumption 2, the 10 breeding birds of 1997 die before the next spring. According to Assumption 3, all of the 50 offspring live to the spring of 1998. Thus, at the start of the 1998 breeding season, there will be 50 house sparrows on the island. According to Assumption 1, these will be 25 males and 25 females (25 pairs), each of which will produce 10 offspring. Continue with this kind of reasoning to calculate what the island's sparrow population will he at the beginning of the breeding season in 1999, 2000, and 2001.

You now have a series of numbers; you can get a clearer idea of the way the population grows by plotting the numbers on a line graph. Construct the graph so that the years are shown along the horizontal axis and the number of birds along the vertical axis. You should make the vertical scale large enough to show the small 1997 population. Plot as many generations as you can.

No doubt you had difficulty plotting all the data on ordinary graph paper. This difficulty can he overcome with another tool - semi-logarithmic (usually called "semi-log") graph paper. Construct your semi-log graph with the same data you used before.

Questions

1. What advantage(s) does the semi-log graph have over the ordinary graph for plot- ting data on population growth?

2. Put the two graphs in front of you. How does the slope of the line connecting the plotted points change as you read from left to right (from year to year) across the ordinary graph?

3. What does this mean in terms of rate of population growth?

4. What kind of line shows the same thing on the semi-log graph?

5. If you were to continue to use the same set of assumptions to calculate populations for an indefinite number of years and plot them on a graph, what would happen to the slope of the line on the ordinary graph? On the semi-log graph?

6. Now you need to relate your results to the purpose of the investigation. In one or two sentences describe the growth of a hypothetical population that is limited by the assumptions stated in the model.

7. Do you think any real population might grow in this way? Why or why not?

For further investigation

You can examine the effects that changes in assumptions have on the growth of the model population. This will give you a better understanding of factors involved in population changes.

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 unchanged. Calculate the population size of each generation. Compare these results with the results from the original assumptions by drawing a graph on the grid used for the original data.

2. Change Assumption 3 as follows: Each year 2/5 of the offspring (equally males and females) die before the next breeding season. All other assumptions remain unchanged. As before, calculate the populations and draw a comparative graph.

3. Change Assumption 4 as follows: Each year 50 new house sparrows (equally males and females) arrive on the island from elsewhere. None leave. All other assumptions remain unchanged. Calculate the populations and draw a comparative graph.

4. Devise other problems for yourself by changing the assumptions in other ways.