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.
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.
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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