Populate or Perish: Logo and the Logistic EquationAuthor(s): Paul ScottSource: Mathematics in School, Vol. 29, No. 3 (May, 2000), pp. 14-16Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212339 .

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POPULATE OR PERISH LOGO AND THE LOGISTIC EQUATION

by Paul Scott

The Incredibly Boring and Absolutely Fascinating Quadratic After exploring the properties of the general linear function y = ax + b (= 0), the student moves on to study the general quadratic function y = ax2 + bx + c (=0). It does not take long to recognize that this function has very limited possibilities: after all, you either get a parabola which looks like the entrance to a cave or the cross-section of the bed of a lake. What could be more boring than that? [In fact, this is an oversimplification. A brief look at, for example, Lockwood (1967) will reveal that the parabola has some quite surprising and unexpected properties.]

How, then, is it possible that a well-known scientist can write (May, 1976): 'I would therefore urge that people be introduced to equation (3) [a quadratic equation called the logistic equation] early in their mathematical education. This equation can be studied ... by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the students' intuition about non-linear systems.'

To help understand the significance of this statement, we shall describe the logistic equation, and give a simple program which allows students to investigate its behaviour.

Hare-splitting and Mathematical Modelling Much of mathematics is concerned with modelling real life situations. This means that we try to find equations which describe the real life happening, and then use the equations to forecast what will happen in the future. If the forecast is a good one, we will obtain an accurate prediction of the future.

One area of concern is that of population - whether it be rabbits or people. Let us try to find an equation which models the way a population grows.

A first attempt

Suppose at a certain time we have a population po where the subscript O here stands for old or original. Suppose the population grows at 10% per year. Then after a year, the new population PN will be given by pN = Po + 10/100 Po = po (1 + 10/100) = 1.1 po. In general it is clear that we will have PN = a po for some non-negative constant a.

Now, how good is this model? For a small population, it is probably very good. But we notice that over a period of years

! oAo. G6UESS WI7HA / LrARNT TO MULTIPLY AT

6 49LO

{; o /I TIV

O(

the population becomes increasingly large. It is clear that a population cannot increase indefinitely, for it will run out of space or food or both. Therefore we need to modify our equation.

A second attempt

Suppose we have a population of rabbits living on an island, and suppose that L is the largest population which the island can support. This suggests that we might like to include a factor (L - po) in the equation - a factor which will get small as Po approaches L, and thus decrease the prediction for the new population PN. So let us try the new equation

PN = a po(L - po)

It is customary to 'normalize' this equation by taking L = 1. For example, if the maximal population is 1000000, then we can agree to measure the population in millions. Also, instead of using O and N for the old and new populations, we write p, for the population in year n; this allows us to explore what happens in year 0, year 1, year 2, .... We now have the so- called logistic equation

Pn+1 = a pn(1 - Pn), n = 0, 1, 2, 3, .... (1)

Notice that we are assuming that our population lies between 0 and 1. The term Pn(1 - pn) is easily seen to lie between 0 and 1/4, this largest value being assumed when p, = 1/2. Hence to ensure that 0 pn+l < 1, we must take a to lie between 0 and 4, that is 0 . a . 4.

Thus, if we know the population Pn in year n, then the equation will predict the population Pn + in the following year.

14 Mathematics in School, May 2000

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We notice that this equation is precisely of the form

y = ax(1 - x) (2) a quadratic equation with a graph which is an inverted parabola. What adds interest to this equation is the fact that we are going to iterate it: starting with po we will determine

pl, this is then fed into the equation to obtain P2, and so on. We thus obtain a sequence of population values P0o, P, P2, P3, ... which predict the population in successive years. The interesting question now is, what happens to this sequence? Do the terms grow indefinitely as n gets larger? Or do they tend to zero, or to some other constant value?

As Robert May suggests, these questions can be approached with the help of a calculator, but a simple computer program makes the investigation much easier and much more instructive. And there are some amazing surprises in store for the student!

Clever Computers and Pretty Pictures

The material illustrated here was prepared using HyperStudio"M. You may well prefer to use the ideas and present them your own way.

This HyperStudio stack consists of just three cards, of which the first is a simple title card (Figure 1). On the second card

POPULATI ON

GROWTH

Fig. 1

the user is able to enter a value for the constant a which is then inserted in equation (1). The repeated iteration takes place, producing a sequence of population values. These population values are then plotted against time, and the points joined to give a continuous graph (for example, Figure 2 shows the

1 Population p

11.2

Time it:

UalTuefa :j :

m. 'm Fig. 2

graph when a = 3). To gain a better understanding of what is happening, the user can move to the third card which shows the axes, the parabola with equation (2) (having the same value of a), and the diagonal liney = x. See Figure 3 for the

1.

68, ,3eraions: Blue.after 380

itatuewaf a is 3I

Fig. 3

case a = 3. The initial population value is assumed to be x =

Po = 0.2, although there is nothing particularly significant about this number. The vertical line from this point meets the parabola at (0.2, y) = (po, Pi). There is then a horizontal segment across to the diagonal line y = x, giving rise to the point (p1,P1). This defines the new x-valuepl, and the process begins again with pi in place of po. In fact the program runs 400 iterations with the first 200 in green and the final 200 in blue. This means that if the iterations 'settle down' by the 200th iteration, a final pattern is easily observed. When the mouse button is clicked, card 2 appears again, and the process can be repeated with a new value of a.

To see the possibilities of the program, let us consider the output for some values of the variable a. We have already looked at the case a = 3. Figures 4, 5 and 6 give the outputs when a = 3.2, 3.5 and 4.

S Population p

Time t A

3.2.li,i,-) Ualue ofaiil :i

mn

Fig. 4a

Mathematics in School, May 2000 15

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..

58. iterations: Blue after 3.88

.2 alue of ais3.2 "il

Fig. 4b

SPopulation d p

SaTimete t

malue ofa LIZ]O

14

5005 iterations Blue after: 38_

. ....... ..... .

. .. .:... . . .

----alu e Q"is ',,5

Fig. 5

Popu aton p

Time

ialue of aI

Fig. 6a

590 :iterations Blue -after. 380

S Ualue of a i-:4

Fig. 6b

These outputs suggest that * (in fact) for a = 2, the population approaches a single

limit point; * for a = 3 and 3.2, the population approaches a steady

oscillation between two fixed values; * for a = 3.5, the population finally cycles between four

fixed values; * for a = 4, there is no discernible pattern appearing, even

after many iterations.

Possible Problems and Credible Conjectures This is a marvellous situation for students to explore and make conjectures. It has very practical implications, and is also relatively recent in discovery (Feigenbaum was working on this in 1975!). Here are some questions which students could investigate using the program. * At what point a = b, does the population ultimately

move from the single limit to an oscillation? * Is there any value for a which the population cycles

between three values? * At what point a = b2 does the population ultimately

change from a 2-cycle to a 4-cycle? * At what point a = b3 does the population ultimately

change from a 4-cycle to an 8-cycle? * Look at the sequence of points b1, b2, b3, b4, .... Do you

expect there to be a limiting value? What might it be? * There is something special about the point a = 2. (Hint:

where does y = x cut the parabola?) * The point a = 3.5699 ... is the Feigenbaum point - the

entry into chaos. Investigate this experimentally. A theoretical discussion of these issues can be found in the

book by Peitgen, Jiirgens and Saupe (1991). A nice coverage of the population problem can be found in Tuck and de Mestre (1991). M

References Lockwood, E. H. 1967 A Book of Curves, Cambridge University Press. May, R. M, 1976 'Simple Mathematical Models with very Complicated

Dynamics', Nature, 261, p. 459. Peitgen, H.-O., Jtirgens, J. and Saupe, D. 1991 Fractals for the Classroom

Vol. 2, Springer. Tuck, E. O. and de Mestre, N. J. 1991 Computer Ecology and Chaos, Longman

Cheshire.

Keywords: Logistic Equation; Quadratic; Iteration; Modelling; Chaos.

Author Prof. Paul Scott, Department of Pure Mathematics, University of Adelaide, S. Australia 5005. e-mail: pscott@maths.adelaide.edu.au

16 Mathematics in School, May 2000

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