C2-Lancaster Farming, Saturday, April 22, 1995
CHAOS, CERTAINTY
AND CHICKENS
Dr. William B. Roush
Associate Professor
Poultry Science
When the word “chaos” is men
tioned, usually the picture that
comes to mind is complete disor
der and utter confusion.
In the past 20 years, a new sci
ence has been developed by physi
cists and mathematicians which
gives a new meaning for chaos.
The mathematical definition of
chaos is random behavior in an
ordered system. That is, the results
of research in the area of physics
and mathematics show cases
where some responses which
appear to be random can be shown
to have order.
The phenomenon of chaos was
found by Edward Lorenz, a
meteorologist In 1960, Lorenz
was examining a computer model
of a simple weather system. He
found something strange. While
inadvertently putting numbers in
the program that differed only by
less than one part per thousand, he
discovered that the resulting
weather projections soon diverged
farther and farther until they bore
no discernible relation to each
other.
With this simple model and pri
mitive computer equipment, he
had discovered that systems which
are described by nonlinear equa
tions can be extremely sensitive to
small changes, often resulting in
“chaotic” behavior. Linear sys
tems, on the other hand, are more
robust in that small differences in
initial conditions lead only to small
differences in the final result.
Robert May, a theoretical biolo
gist, wrote a paper in 1976 that is
now a classic in chaos theory. He
gave an example of a simple equa
tion used to study population
growth. The equation is X’ = AX
(1-X), where X’ is the population
for one year, X is the population of
the preceding year, and A is a para
meter that varies between 0 and 4.
To examine chaos theory, here
is something you can try with your
calculator. Or, if you have a com
puter you might try the following
BASIC program:
10 INPUT A
20 X = .3
30 For I = 1 TO 150
40 X - A*X*(l-X)
50 PRINT X
60 NEXT I
70 END
To make things simple, suppose
X and X’ are numbers between 0
and 1 with the true population a
million times these values. Sup
pose the population (X) is .3 (ie -
300,000) and A = .2. Plugging in
the numbers 2(.3)(.7), you get .42.
To obtain the next year’s popula
tion plug .42 (this is the new X or
old X’) into the formula
2(.42)(.58) or approximately
.4872. Using the same procedure,
three years and thereafter, the
population stabilizes at .5. In fact,
whatever the original population
size, the population will stabilize
at .5. If A is increased to 2.6 the
population eventually stabilizes at,
approximately, .62.
Now increase the value for A to
3.2. The population no longer sta
bilizes to one number, but eventu
ally alters between two values,
approximately .5 and .8. As you
raise the value of A to 3.5 the num
bers alter between four numbers,
approximately .38, .83, .5, and .88.
Increasing the A value a little more
causes an alteration between eight
numbers. The doubling of the
number of values continues as the
value for A is increased.
Then suddenly, at approximate
ly A = 3.57, the number of values
grows to infinity the population
goes into “chaos.” The effect is
very apparent if it is graphed. All
this from the seemingly simple
equation AX(I-X).
An interesting attribute of this
equation is that there is a paradox
to this chaotic behavior. Although
the responses appear chaotic over a
certain time period, if the
responses are plotted as one time
period against another time period
(X versus X’), a form or structure
for the responses becomes
apparent
The discovery of the mathema
tics of chaos dispels an illusion that
mathematics is always certain. In
science there is always the goal to
define biological organisms and
their surroundings in a certain,
accurate, and precise manner.
While in certain controlled situa
tions mathematics can define out
comes precisely, attempts to mod
el nature and its factors are more
complex and variable.
The results of chaos and the
behavior of nonlinear systems
have implications in a number of
fields such as physiology, chemi
stry, and economics. For example,
it has been shown that physiologi
cal systems, including the hor
mone system, heart rhythms, and
breathing can exhibit chaotic beha
vior under certain circumstances.
So what does this have to do
with chickens? It so happens that
the equation illustrated by May in
1976 is very similar to an equation
used to describe growth in ani
mals. This observation has led to
studies at Penn State with broiler
chickens that have shown that day
to-day growth rate shows evidence
of chaotic responses (random but
ordered values). These mathemati
cal studies with poultry may, give
insight into the control of disease
conditions such as ascites which
are associated with increased
growth rate.
For more information on the
mathematics of chaos, the books
“Chaos: Making A New Science”
by James Gleick and “Does God
Play Dice: The Mathematics Of
Chaos” by lan Stewart are sug
gested. Both of these can be
obtained from or ordered at your
local bookstore.