Let it Grow, Let it Grow, Let it Grow…
Version 3.141
Background
How does one come up with mathematical formulas for different situations? There are
three basic approaches one may use:
i.
Modeling from first principles: This is where one makes a few basic, yet
reasonable, assumptions and sees what mathematical formulas emerge.
ii.
Modeling from scientific principles: Similar to the above approach, except
here one begins with some basic scientific assumptions to derive the
mathematics. For example, one may assume that the only force a falling
object undergoes is the constant acceleration due to gravity. Given just this
physical assumption, one can derive formulas for various falling objects and
thrown projectiles.
iii.
Modeling from real data: If there is enough data regarding the specific
situation you are interested in, you can use these data to find the “best fit”
curve or formula. This formula can then be used to interpolate or extrapolate
additional information.
In the following activity we will follow (i) and see how just a few simple assumptions
will allow us to derive some fairly complicated mathematical formulas.
Biological Populations: For this task we will concern ourselves with modeling some
biological population. This could be the world human population, the deer population in
the Yakima Valley, or the wild salmon population in the ocean which return to our area
rivers and streams to spawn. Our main task will be to derive a reasonable mathematical
formula (or model) to describe how these populations change over time. Our driving
assumption about these populations stems from the reproductive nature of the species we
are studying. In all of the above examples, one can easily argue that the number of
offspring at any given time is directly proportional to the number of female members of
the population. Furthermore, since the number of female members of a population is also
directly proportional to the total population (for humans the number of females is almost
exactly ½ the population), we have that the number of offspring is simply a constant
multiple of the current population. We have,
Basic Assumption 1: The number of births is proportional to the current population. Call
this constant birth rate B.
Notice that this does not say that the number of births is constant, just that the number of
births is a constant multiple of the current population (if the population is larger, the
number of births will be larger).
Sadly enough, every population has a percentage of its members which die during any
given time period. This gives us
Basic Assumption 2: The number of deaths is proportional to the current population. Call
this constant death rate D.
The Basic Model Let us assume that we are modeling a population of the endangered
Gork. A small colony of 50 Gorks has just been introduced to the small pacific atoll
Halfway. Let us further assume that the given yearly birth and death rates for Gorks are
0.2 and .05 respectively.
1. With the initial population of 50 Gorks, what do our birth and death rates tell us
will happen during the first year on Halfway? How many Gorks die? How many
new Gorks are born? How many Gorks are on Halfway at the end of the first
year?
2. How many Gorks are on Halfway at the end of the second year?
3. The third year?
4. For the above computations you may have first calculated the number of births,
then the number of deaths, and used those two figures to determine the final
population of Gorks. Is there an easier way?
Brief Algebraic Interlude: If P0 represents the number of initial Gorks, and P1 is
the number of Gorks at the end of the first year, then
P1  P0  BP0  DP0
Factor out the common term to find a simpler way to calculate the population of
Gorks in each successive year.
5. Calculate the yearly population of Gorks for the first 10 years.
6. Create a graph of the population of Gorks over time.
Exploring the Model
7. Over time would you say that the population of Gorks increases or decreases?
Why is this the case? When would the reverse situation occur?
8. How long did it take the Gork population to double from its initial value of 50?
How long did take for it to double a second time? Without calculating explicitly,
when do you think the population of Gorks will reach approximately 400?
9. Would you say that the population of Gorks changes linearly? Give at least two
specific reasons for your answer.
10. How does the basic model change if we change:
a. The initial population of Gorks?
b. The assumed birth rate?
c. The assume death rate?
By “basic model”, we mean the overall features or characteristics of the model
such as its tendency to increase or decrease, how quickly or slowly it may
increase or decrease (doubling time or halving time), and the basic shape of the
graph.
Finalizing the Model
At this time our model can easily be used to calculate the population of Gorks at the end
of a year assuming we know the number of Gorks at the beginning of that year.
11. If there were 818 Gorks in year 20, how many Gorks would our model predict for
year 21?
If you hadn’t known how many Gorks there were in year 20, you may have found it hard
(or tedious at best) to compute the number of Gorks in year 21. Thus, our model would
be much more powerful if we had a formula to compute the population of Gorks in any
given year without first having to calculate the number for each intervening year. In
order to do this, some simple algebra comes to the rescue!
12. Back in problem (4) you had a formula for the number of Gorks at year 1:
P1  (1  r ) P0 .
What was r?
13. By similar reasoning, it should be true that
P2  (1  r) P1 .
Using the above two equations, find a formula for P2 which contains only r and
P0 .
14. Find a formula for P3 which contains only r and P0 .
15. If we let Pn represent the population of Gorks after n years, find an easy to use
formula for Pn .
16. Use your formula to calculate P100 . Now, that’s a lot of Gorks! Comment on the
“long-term behavior” of this model. That is, as time goes on what can you say
about the population? Would this long-term behavior change if r or the initial
population changed? Why or why not?
Some History
In 1798, Thomas Robert Malthus published an essay on human population entitled
“Essay on the Principle of Population”. In this essay he stated (among many other
things): “population increases in a geometric ratio.” A geometric ratio is one in which
each succeeding value is a constant multiple of the previous value. That is, Malthus
basically derived the exact same model for population growth we did. Sometimes this
model is referred to as the Malthusian Model. Malthus used his reasoning regarding the
exponential growth possibilities of the human race to make many social, political, and
religious claims. His writings influenced many of the great political, social, and natural
scientific thinkers of the time (including Charles Darwin). See, for example,
http://www.blupete.com/Literature/Biographies/Philosophy/Malthus.htm for more
information on the ideas of Malthus.
Reality Check
One aspect of mathematical modeling is the “reality check”. All of the formulas and
values we calculated above came from our two basic assumptions. The power of our
model depends tremendously on the validity of those two assumptions. To see if our
model is a relatively reasonable predictor of biological populations, we could look at
some actual data for a particular species. Doing a web search for “human population
growth” produced the following graph
(http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/P/Populations.html) of the world
population over the last couple thousand years (the dotted line refers to a prediction of
future populations):
17. Does this graph have the “basic features” of the exponential, or Malthusian¸
model we have been working with?
18. See if you can find population data on another species. Compare and contrast the
actual data with our model.
19. Can such growth, either in a human population or another species, continue
indefinitely? Explain.
A Stronger Model
The graph above showing the word human population over time indicates that we have a
very strong model indeed. However, one might argue that such growth can not continue
indefinitely due to overcrowding, lack of natural resources, etc. One can see evidence of
this in other biological populations. Below is the graph of the number of duckweed
plants in a small pond observed over time (the units on the time axis are days):
Duckweed Plant Population
1600
Number of plants
1400
1200
1000
Series1
800
600
400
200
0
0
5
10
15
20
25
30
35
Days
Initially one might argue that the plants grew exponentially (the lower left portion of the
graph), but over time the rate of growth appears to be slowing. This could be caused by
overcrowding (there is only so much surface area on the small pond that can be inhabited
by duckweed plants). Depending on the area where the species live, there may be some
preset limit or “carrying capacity” to the natural expansion of the species. If one tried to
overcome the natural carrying capacity by dumping in a pondful of duckweed plants,
most likely many of the added plants would quickly die. Thus, one might be inclined to
begin our new model with two basic assumptions:
Basic Assumptions:
1. The change in the population is proportional to the current population (this is our
original assumption) for small populations, but
2. The change in population is negative if the population exceeds the carrying
capacity of the environment.
While it may take on a little time to decide how to use these assumptions to derive a
mathematical formula, one of the simplest formulas that meet the above two criteria is
P
Change in population = rP (1  )
L
Where r is the “exponential growth factor” from before, P represents the current
population, and L represents the carrying capacity for the environment. As you look at
the right hand side of this equation think of it as being composed of two parts. The first
part, rP , should look familiar (and corresponds to our basic assumptions from before)
P
while the second part, (1  ) , is new. Let’s see what this all means.
L
If P is small (say near zero), then the number in parentheses is close to 1. Which means
our formula for the change in population is just the exponential model we had before
(therefore our first basic assumption is satisfied). Furthermore, if P is greater than L then
P
P
is greater than 1 making (1  ) a negative quantity. This would mean that the
L
L
change in population is negative which is what our second assumption stated.
Calculating with the new Model:
Let’s go back to the idyllic isle of Halfway and use r  0.15 , L = 1000, and a starting
population of 50 Gorks.
20. Using our new formula for the change in population, what would the change in
population of Gorks be that first year?
21. If this change is added to the starting population, you will get the number of
Gorks in year 1. Does this differ from the exponential model?
22. Calculate the number of Gorks for the next 40 years. This should not be too
difficult if you make use of some form of technology to help with the repeated
calculation. For example in the TI-83 you could
a. Type 50 [enter]
b. 2nd ANS + .15*2nd ANS * (1- 2nd ANS/1000)
c. 2nd ENTRY
d. 2nd ENTRY
e. Etc.
23. Create a graph of the Gork population over time.
24. How does this graph compare with the graph of the exponential model?
25. Does this graph have the features of the duckweed data?
Reality Check
It turns out that the above model is extremely accurate in many diverse situations.
For instance, if one plots the number of new AIDS cases in the United States over
time (http://www.nlreg.com/aids.htm) one sees: