INTRODUCTION
Nearly 50 years ago, Cohen (1976) published an example of how stochastic
intrusions affect predictability. Start with an urn that contains one
red and one green ball. Choose one of the balls at random, and after
looking at the color, put it back in the urn and add another ball of the
same color as the one chosen. Repeat, adding a new ball to the urn each
time. What happens to the proportion of red and green balls in the urn
as the number of balls becomes very large?
The solution (Cohen points out that hardly anybody guesses correctly) is
that the proportion of red (or green) balls converges to some valuep . The unexpected part comes when the experiment is repeated. The
proportion of red (or green) balls again converges, but to adifferent value of p . When repeated enough times, pforms a uniform distribution between 0 and 1. p is known as a
“nondegenerate random variable,” referring to the fact that it is not
restricted to a particular value.
The exercise is known as “Polya’s urn scheme,” having been introduced
by Eggenberger and Polya (1923) to model the spread of infection in a
population. Cohen suggests its relevance to several ecological examples,
including two that hit close to home for those of us studying social
behavior: the possibility that differences in the sizes of lion
(Panther leo ) prides and troops of Japanese macaques
(Macaca fuscata ) reflect inherently variable outcomes of
identical underlying stochastic forces rather than deterministic
ecological differences. More generally, it raises a disturbing question:
might a long-term project, potentially encompassing one’s entire career,
yield entirely different results if one were to start over and repeat
the study?
Our project focuses on social behavior of the acorn woodpecker
(Melanerpes formicivorus ), a cooperatively breeding bird common
in oak woodlands of western North America. Inadvertently, the question
of whether the same results would emerge by starting the study over
again was tested, since at the same time we started our work in
California, Peter Stacey, then a graduate student at the University of
Colorado, began a 10-year study of acorn woodpeckers in Water Canyon,
New Mexico. Although we both addressed the same question—why do acorn
woodpeckers live in groups and breed cooperatively?—our answers were
very different, with ours focusing on ecological constraints “forcing”
birds to delay dispersal (Koenig and Pitelka 1981, Koenig et al. 1992)
while Stacey’s emphasized the potential benefits birds gained by
remaining in their natal group (Stacey and Ligon 1987, 1991). How much
of these different perspectives were a consequence of “deterministic
ecological differences” and how much were “inherently variable outcome
of identical underlying stochastic forces”?
Certainly there were basic ecological differences between the
populations (Koenig and Stacey 1990). But it would be hubris to claim
that this explains all the contrasting conclusions we drew from our
work. A better test would be to repeat both studies, with Stacey working
in California and us at Stacey’s site in New Mexico. Although none of us
is in a position to conduct this experiment, we can look at our own
long-term study and attempt to determine the extent to which our results
are a consequence of deterministic vs. stochastic forces. Here we first
briefly analyze some basic ecological data from our long-term study of
acorn woodpeckers with this goal in mind. We then offer thoughts
regarding the benefits and drawbacks of long-term studies.