ANN ARBOR—Whether monitoring earthquakes and avalanches or tallying insect populations on plants, scientists studying natural phenomena often notice a consistent relationship between size and frequency: large is rare and small is common.

But while researchers frequently see such a pattern?known as a power function or power law?they don’t always know how to explain it.

In a paper in the Feb. 17 issue of the journal Science, University of Michigan researchers John Vandermeer and Ivette Perfecto offer a biological explanation for why the distribution of scale insects on coffee plants in an organic coffee farm follows a power function. They also show that subtle deviations from that function are due in part to a mutually beneficial relationship between the scale insects and ants. Their findings may be useful to pest managers trying to keep tabs on their targets.

“A large number of things in nature follow this power law, and in some cases the physical or biological rules that give rise to it are known, but in most cases it’s just an observation,” said Vandermeer, who is the Margaret Davis Collegiate Professor of Ecology and Evolutionary Biology and a professor of natural resources and environment.” What we present in this paper is a situation where a natural phenomenon follows the power law and clear biological reasons as to why it follows the power law. Furthermore, there are systematic deviations from that power law at high and low population densities, and we can explain those at high densities through this very key, mutualistic relationship between the ants and the scale insects.”

The green coffee scale (Coccus viridis) is a flat, featureless insect that lives on coffee bushes and is a known pest of the crop. On some bushes, tree-nesting ants (Azteca instabilis) protect the scale insects from predators and parasites and in return collect honeydew?a sweet, sticky liquid the scale secretes. To study the distribution of scale insects, Vandermeer and Perfecto set up a 45-hectare (111-acre) plot on a coffee farm in southwestern Mexico.

Because large concentrations of scale insects often are found on coffee bushes near trees where the ants live, the researchers located all the shade trees in the plot and checked the trees for ant colonies. Then they chose five sites that were near colonies and four that were far enough away to be outside the ants’ sphere of influence. In each site they counted scale insects on 50 to 100 coffee plants.

When Vandermeer and Perfecto plotted the data on a logarithmic scale, they saw the telltale, straight slope that is typical of a power function. Many bushes had small populations of scale insects, a few had large populations, and intermediate numbers of bushes had medium-sized populations. But at the very top and bottom of the graph?where population size was very low or very high?the points didn’t fit so well. There were too many bushes with very few or no insects and too many bushes with extremely large populations.

Considering everything they know about the system they’ve been studying for the past eight years, Vandermeer and Perfecto came up with an explanation for both the power function and the deviations from it. It goes like this: Scale insects looking for a home encounter coffee bushes at random, and the population on a particular bush grows as insects that have taken up residence there reproduce and others join them. At first, large numbers of bushes have few or no scale insects, just because none happened to find them yet. That accounts for the excess of bushes with few or no insects and explains the deviation at the low end of the graph.

Once all or almost all the bushes have at least one scale insect, all the populations increase exponentially, giving rise to the typical power function.

On bushes with no ants to protect the scale insects, predators and parasites impose limits on how large the scale populations can grow. But on bushes where scales enjoy protection, their numbers can reach higher levels. So when the sample included bushes that were ant infested, there was an excess of bushes with high populations of scales. When the researchers excluded the ant-infested bushes from the sample, the deviation from the power law disappeared. The protection given by the ants explains why so many coffee plants have extremely high populations of scale insects, said Perfecto, an associate professor in the U-M School of Natural Resources and Environment.

Finding biological explanations for the power law and deviations from it is interesting in itself, but the results also may have practical applications, Vandermeer said.

“Pest managers often want to know the rate of growth of a pest population,” he said.” But if you have an area which is extended in space, as most agricultural systems are, it’s hard to say what ‘the’ population is doing because you’ll have a small population of insects on this plant and another one on that plant. But if you measure a few of them and find that they follow a power law, that’s an indication that they’ve reached the critical point where the entire field is acting like one population. Then you can project the population into the future to determine when control activities may be necessary.”

The research was supported by the National Science Foundation.

John VandermeerIvette PerfectoScience magazine