Population Changes - A Model
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Bob Height
Introduction:
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.
Procedure:
- 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.
- 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.
- 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.
- 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.
- 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.
Discussion:
Look at the two graphs that you constructed. Answer the following
questions using complete sentences.
- What advantage(s) does the semi-log graph have over the
arithmetic graph for plotting data on the population growth?
- 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)?
- What does this mean in terms of population change?
- Now compare the graphs. What kind of line shows the same
thing on the semi-log graph?
- 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?
- What would happen to the slope of the semi-log graph?
- In one or two sentences, summarize the change in a population
that is supported by the assumptions stated in the model.
- Do you think any real population might change this way?
Why or why not?
- 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 |