COMPUTING SCIENCE
Bugs That Count
Brian Hayes
Along the East Coast of the United States, a cohort of periodical
cicadas known as Brood X occupies the prime turf from New York City
down to Washington, D.C. Brood Xers are the hip, urban cicadas, the
inside–the–Beltway cicadas, the media–savvy
celebrity cicadas. When they emerge from their underground existence
every 17 years, they face predatory flocks of science writers and
television crews, hungry for a story.
This year was a Brood X year. Back in May and June, all along the
Metroliner corridor, the air was abuzz with cicada calls, echoed and
amplified by the attentive journalists. Then, in just a few weeks,
it was all over. The cicadas paired off, fell silent, laid eggs and
died. The press moved on to the next sensation. Perhaps a few
straggler cicadas showed up days or weeks late, but no one was there
to notice, and their prospects cannot have been bright. I worry that
the same fate may befall an article about cicadas appearing weeks
after the great emergence, at the very moment when most of us want
to hear not another word about red–eyed sap–sucking
insects for at least 17 years. I beg my readers' indulgence for one
long, last, lonely stridulation as the summer comes to a close.
Periodical cicadas are remarkable in many ways, but I want to focus
on one of the simplest aspects of their life cycle: the mere fact
that these insects can count as high as 17. Some of them count to 13
instead—and it has not escaped notice that both of these
numbers are primes. Of course no one believes that a cicada forms a
mental representation of the number 17 or 13, much less that it
understands the concept of a prime; but evidently it has some
reliable mechanism for marking the passage of the years and keeping
an accurate tally. That's wonder enough.
The physiological details of how cicadas count will have to be
worked out by biologists in the lab and the field, but in the
meantime computer simulations may help to determine how precise the
timekeeping mechanism needs to be. Computer models also offer some
hints about which factors in the cicada's ecological circumstances
are most important in maintaining its synchronized way of life. On
the other hand, the models do nothing to dispel the sense of mystery
about these organisms; bugs that count are deeply odd.
A Clockwork Insect
Cicadas spend almost all of their lives underground, as
"nymphs" feeding on xylem sucked out of tree roots; they
come to the surface only to mate. Most species have a life cycle
lasting a few years, but individuals are not synchronized; all age
groups are present at all times, and each year a fraction of the
population emerges to breed. True periodicity, where an entire
population moves through the various stages of life in synchrony, is
extremely rare. Of 1,500 cicada species worldwide, only a handful in
the genus Magicicada are known to be periodical; all of
them live in North America east of the Great Plains.
The taxonomy of the Magicicada group is somewhat
controversial and more than a little confusing. If you sort a
collection of specimens by appearance or mating call or molecular
markers, they fall into three sets; but it turns out that each of
these sets includes both 13–year and 17–year forms. Are
there six species, or only three? Complicating matters further, John
R. Cooley, David C. Marshall and Chris Simon of the University of
Connecticut have recently identified a seventh variety that has a
13–year period but shows genetic affinities to a 17–year group.
Then there is the division into broods. A brood is a synchronized
population, in which all individuals are the same age. Generation
after generation, they go through life in lockstep. One might expect
that a brood would consist of a single species, but that's generally
not the case. Brood X, for example, includes the 17–year forms
of all three species. Geographically, adjacent broods tend to have
sharp boundaries, with little overlap. Where two broods do share the
same real estate, they are chronologically isolated, typically with
four years between their emergences.
If you are a cicada trying to emerge in synchrony with all your
broodmates, there are two problems you need to solve: First you must
choose the right year, and then the right day (or night, rather)
within that year. The latter task is easier. Cicadas synchronize the
night of their emergence by waiting for an external cue: They crawl
out of their burrows when the soil warms to a certain temperature,
about 64 degrees Fahrenheit.
Keeping track of the years is more challenging. First you need an
oscillator of some kind—a device that goes
tick–tick–tick at a steady pace. The cicada oscillator
presumably ticks once per year. Second, you need to tally the
successive ticks, like a prisoner scratching marks on the wall of a
cell. Finally you have to recognize when the tick count has reached
the target value of 13 or 17.
The cicada oscillator is probably an annual variation in some
property of the xylem the insects consume, reflecting a deciduous
tree's yearly cycle of growing and shedding leaves. Support for this
hypothesis comes from an ingenious experiment conducted by Richard
Karban, Carrie A. Black and Steven A. Weinbaum of the University of
California, Davis. They reared cicadas on orchard trees that can be
forced to go through two foliage cycles in a single year. Most of
the cicadas matured after 17 of the artificially induced cycles,
regardless of calendar time.
The cicada's tally mechanism remains unknown. One example of a
biological counting device is the telomere, a distinctive segment of
DNA found near the tips of chromosomes in eukaryotic cells. Each
time a cell divides, a bit of the telomere is snipped off; when
there's none left, the cell ceases to replicate. Thus the telomere
counts generations and brings the cell line to an end after a
predetermined number of divisions. Perhaps the cicada employs some
conceptually similar countdown mechanism, although the biochemical
details are surely different.
There is no reason to suppose that cicadas count strictly by ones.
Indeed, the coexistence of 13–year and 17–year periods
suggests other possibilities. For example, the two life cycles might
be broken down as (3x4)+1=13 and (4x4)+1=17. In other words, there
might be a four–year subcycle, which could be repeated either
three or four times, followed by a single additional year. An
appealing idea is to identify such subcycles with the stages, or
instars, in the development of the juvenile cicada. And it's notable
that what distinguishes 17–year from 13–year forms is a
four–year prolongation of the second instar. Unfortunately for
the hypothesis, the rest of the nymphal stages are not uniform,
four–year subcycles. Nymphs pass through them at different
paces. Only at the end of the cycle do the members of a brood
get back in synch.
Missing a Beat
The synchronization of cicada emergence is impressive, but not
perfect. There are always at least a few clueless unfortunates who
turn up a year early or a year late. Four–year accelerations
and retardations are also common. Evidently, the year–counting
mechanism can go awry. How much error can the system tolerate before
synchronization is lost entirely?
Several authors have proposed that Magicicada periodicity
evolved during the Pleistocene epoch, as a response to the
unfavorable and uncertain climate of glacial intervals. Conditions
have changed dramatically since the glaciers retreated, and so it
seems unlikely that the same selective pressures are still working
to maintain synchronization. What does maintain it? Before
considering more complicated hypotheses, it seems worthwhile to ask
whether periodicity could have survived as a mere vestigial
carryover, without any particular adaptive value in the current
environment. If the timekeeping device is never reset, how
accurately would it have to work to maintain synchronization over
the 10,000 years or so since the end of the Pleistocene?
The answer depends in part on what kinds of errors can disrupt the
counting. The simplest model allows individual cicadas to make
independent errors. Each year, each cicada has some small likelihood
of either failing to note the passage of the year or interpolating a
spurious extra year. Under this model, the error rate needs to be
kept below 1 in 10,000.
The weakness of this model is the assumption that cicadas would make
independent errors. If all the cicadas are trying to read the same
chemical signal in the tree sap, errors could be strongly
correlated. In a bad year with a short growing season, the signal
might never reach the threshold of detection for many individuals. A
double oscillation is also a possibility, for example if the trees
are defoliated by predators and then put out a second growth of leaves.
An error model that allows for such correlations works like this: A
cicada's probability of correctly recording the passage of a year
depends on the strength of the xylem signal, which varies randomly
from year to year but is the same for all the cicadas. If the signal
is very strong, almost everyone detects it correctly. If the signal
is extremely feeble, nearly all miss it. Although this latter
event must be counted as a timekeeping error, it does not break
synchronization; instead it retards the entire population by a year.
What spoils synchronization is an ambiguous signal, one in the gray
area where half the cicadas detect it and the other half don't. This
splits the population into two groups, which will mature and emerge
a year apart. Four or five such splittings over 10,000 years would
be enough to wipe out synchronization.
A drawback of this error model is that it depends on two variables,
which are hard to disentangle: the frequency of ambiguous signals in
the xylem and the cicada's acuity in reading those signals. If the
signal is usually near the extremes of its range, then even with a
crude detector, the population will almost always reach a consensus.
If ambiguity is common, then the insect's decision mechanism needs
to be finely tuned. I have experimented with tree–ring data as
a proxy for the distribution of xylem–signal amplitudes, but
the results were not much different from those with a random distribution.
The cicadas' response to the signals is defined by an S–shaped
curve. If the curve is infinitely steep—a step
function—then the probability of registering a tick of the
clock is exactly 0 up to some threshold and exactly 1 above the
threshold. As the curve softens, the transitional region where
probabilities are close to ½ gets broader.
Running the simulation, it turns out that synchronization survives
only if the response curve is very steep indeed, with a vanishingly
narrow region of ambiguity. For ease of analysis, suppose we are
merely trying to synchronize the clocks of two cicadas that each
live for 10,000 years. To a first approximation, they remain in
phase only if they agree on the interpretation of the signal every
year throughout the 10,000–year interval. For a
90–percent chance of such uninterrupted agreement, the
probability of agreement each year must be at least 0.99999.
Is such accuracy plausible in a biological mechanism? Could
periodicity really be a historical relic, without adaptive
significance today? Probably not, but the models are too simplistic
to support quantitative conclusions. Nevertheless, the idea of
timekeeping errors introduced by ambiguities in environmental
signals may well have a place in the biology of cicadas. Suppose
there is a north–south gradient in signal amplitude; then
somewhere along the gradient there must be a zone of ambiguity.
Forty years ago, Richard D. Alexander and Thomas E. Moore of the
University of Michigan, Ann Arbor, pointed out that broods tend to
be arranged like shingles from north to south, with each brood
emerging one year later than the one above. It's a pattern that
might have been generated by successive population–splitting
events like those in the model.
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.
Primes and Other Conundrums
One factor not addressed by the models I have described so far is
feedback from prey to predator species. In an emergence year, birds
that feast on cicadas should be able to raise more young than usual;
the resulting increase in predator numbers will make life even
harder on straggler cicadas the following year, thus sharpening the
population peak. Many authors have drawn attention to this linkage,
and have offered it as the key to understanding why the
Magicicada species chose prime numbers for their
life–cycle periods. Because a prime has no divisors other than
itself and 1, the cicadas avoid falling into resonance with
predators whose abundance fluctuates on some shorter cycle. For
example, a hypothetical 12–year cicada would be susceptible to
predators with cycle lengths of 2, 3, 4 or 6 years.
G. F. Webb of Vanderbilt University has constructed computer
simulations in which such interactions of prey and predator cycles
favor the 13–year and 17–year cicada periods. Eric Goles
of the University of Chile and Oliver Schulz and Mario Markus of the
Max–Planck Institut für molekulare Physiologie in
Dortmund, Germany, have also published on this subject, referring to
cicadas as "a biological generator of prime numbers."
Whether the cyclic predator species exist remains an open question.
And some quite different explanations have also been put forward.
For example, Randel Tom Cox of Arkansas State University and C. E.
Carlton of Louisiana State University argue that the heart of the
matter is not predation but hybridization. Interbreeding between
broods that differ in period could disrupt synchronization for both
groups. Thus 13–year and 17–year broods are favored
because they emerge together only once every 221 years.
Much else about the lives of cicadas remains mysterious. In the
1960s Monte Lloyd of the University of Chicago and Henry S. Dybas of
the Field Museum of Natural History in Chicago offered a curious
meta–theory of cicada evolution. Any theory that seems too
plausible, they argued, is automatically suspect. "If there
were a broad, easy evolutionary highway towards periodicity, then
why would not more species have taken it?"
If one day we run out of cicada mysteries to solve, there are still
harder problems waiting. Synchronized, periodic breeding is known in
plants as well as animals. Daniel H. Janzen of the University of
Michigan has written about the bamboo Phyllostachys
bambusoides, which apparently maintains synchronized flowering
even when seeds are planted continents apart. And the length of the
plant's period makes the cicadas seem as ephemeral as mayflies. The
bamboo knows how to count not just to 17 but to 120.
© Brian Hayes