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.