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COMPUTING SCIENCE

Bugs That Count

Brian Hayes

No Thank You, Not Another Bite

If periodicity confers some current selective advantage, a widely favored candidate for that benefit is "predator satiation." Cicadas are large, noisy, clumsy and tasty (or so I'm told). All in all, they are a bonanza for birds and other predators. But the emerging adults appear in such enormous numbers—often hundreds per square meter—that the feast is more than the diners can finish. If the same mass of cicadas were spread out in yearly cohorts one–seventeenth as large, predators could gobble them up, year by year.

Figure 2. Predator satiation helps...Click to Enlarge Image

Predator satiation and timekeeping errors have the potential to interact in interesting ways. By eliminating off–year stragglers, predation tends to sharpen the peak of the emergence, but of course it also diminishes the height of the peak itself. If predation is too light, then error suppression will be ineffective and synchronization will fail. If predation is too severe, the whole population risks extinction. This analysis suggests the possibility of a system stable only within a narrow range of parameters—always a cause of skepticism, since the parameters are unlikely to be so well–behaved in nature. To my surprise, however, a simple model of predator satiation proved to be highly robust. I defined the satiation threshold—the maximum number of cicadas eaten by predators—as a fraction of the total carrying capacity, or in other words the maximum possible cicada population. I found that synchronization was maintained even with a satiation threshold as low as 0.1; at the other end of the scale, predation did not lead to immediate extinction until the threshold was raised above 0.7.

Other aspects of the model are not so confidence–inspiring. Having added a death rate via predation, it is necessary to include a birth rate as well, or else the population would inevitably dwindle away. The numerical value assigned to the birth rate is somewhat arbitrary. The only guidance comes from field studies showing that successful female cicadas lay a few hundred eggs. Fortunately, the behavior of the model is not overly sensitive to choices of birth rate within the plausible range.

A closely related issue is how to impose a limit on cicada numbers when excess births cause the population to exceed the carrying capacity of the environment. In formulating the computer model, I chose to limit total population by reducing the newborn generation as necessary. Hatchlings could survive only if there were vacancies available for them; newborns could never displace older cohorts. This decision was based on an observation by Karban that infant mortality dominates cicada demography; if a cicada survives its first two years, it will likely last to maturity.

Figure 3. Intergenerational competition appears...Click to Enlarge Image

Further experiments revealed that the capacity limit and the strategy chosen for enforcing it can have significant effects on the outcome of a simulation. For example, steady attrition in the nymph population, even at a low background rate of mortality, leaves the brood much more vulnerable to extinction. In contrast, allowing intergenerational competition—in which newborns compete with their elders on an equal basis for the available resources—helps stabilize the synchronized, periodical mode of reproduction. As a matter of fact, it is only in models with this form of competition that I have seen synchronization arise spontaneously from an initially random state. The other models described here can stabilize existing periodicity but have a hard time generating it in the first place.

The crucial role of a finite carrying capacity in cicada population models was pointed out almost 30 years ago by Frank C. Hoppensteadt and Joseph B. Keller, then of the Courant Institute of Mathematical Sciences at New York University. I had read an account of their computer simulations before attempting my own, but only after some direct experience did I understand the emphasis they put on carrying capacity. The importance of intergenerational competition was stressed by M. G. Bulmer of the University of Oxford at about the same time.

A static population near the carrying capacity, low mortality except at the extremes of the age range, reproduction postponed until the last possible moment—these are characteristics of the Magicicada way of life. I can't help noting that the same traits will soon describe the human population. Intergenerational conflict over resources is also conspicuous in human affairs. Perhaps the cycles of baby booms in recent decades are signs of incipient synchronization in human reproductive practices.





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