EVOLUTION OF PERIODICITY IN PERIODICAL CICADASpeople.bath.ac.uk/pam28/...Mathematics,_University_of_Bath/.../cicadas.pdf · EVOLUTION OF PERIODICITY IN PERIODICAL CICADAS ... Periodical

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Ecology, 86(12), 2005, pp. 3200–3211q 2005 by the Ecological Society of America

EVOLUTION OF PERIODICITY IN PERIODICAL CICADAS

NICOLAS LEHMANN-ZIEBARTH,1,2 PAUL P. HEIDEMAN,1,2 REBECCA A. SHAPIRO,1,2 SONIA L. STODDART,1,2

CHIEN CHING LILIAN HSIAO,1,2 GORDON R. STEPHENSON,1,2 PAUL A. MILEWSKI,1 AND ANTHONY R. IVES2,3

1Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706 USA2Department of Zoology, University of Wisconsin-Madison, Madison, Wisconsin 53706 USA

Abstract. Periodical cicadas present numerous puzzles for biologists. First, their periodis fixed, with individuals emerging as adults precisely after either 13 or 17 years (dependingon species). Second, even when there are multiple species of either 13- or 17-year cicadasat the same location, only one or rarely two broods (cohorts) co-occur, so that periodicalcicada adults appear episodically. Third, the 13- or 17-year periods of cicadas suggest thereis something important about prime numbers. Finally, single broods can dominate largeareas, with geographical boundaries of broods remaining generally stable through time.

While previous mathematical models have been used to investigate some of these puzzlesindividually, here we investigate them all simultaneously. Unlike previous models, we takean explicitly evolutionary approach. Although not enough information is known aboutperiodical cicadas to draw firm conclusions, the theoretical arguments favor a combinationof predator satiation and nymph competition as being key to the evolution of strictly fixedperiods and occurrence of only one brood at most geographical locations. Despite ecologicalmechanisms that can select for strictly fixed periods, there seem to be no plausible ecologicalmechanisms that select for periods being prime numbers. This suggests that the explanationfor prime-numbered periods, rather than just fixed periods, may reside in physiological orgenetic mechanisms or constraints.

Key words: Allee effects; evolution of periodicity; Magicicada; rock–paper–scissors competition;spatial dynamics.

INTRODUCTION

Since the discovery of the periodical cicadas, Mag-icicada spp., in eastern North America some 300 yearsago (Oldenburg 1666, Walsh and Riley 1868), biolo-gists have been fascinated by their periodicity. Thereare seven species of periodical cicadas that divide intotwo categories: four species that live for 13 years, andthree species that live for 17 years (Williams and Simon1995, Marshall and Cooley 2000). Generally, the geo-graphical ranges of the 13- and 17-year cicadas arenonoverlapping, with 13-year cicadas occurring to thesouth and west of 17-year cicadas. Periodical cicadasemerge as a group in late spring, forming large andnoisy mating congregations for roughly a month, andthen the adults die. Because individuals emerge afterexactly 13 or 17 years (depending on the species), thepopulations are divided into discrete cohorts, or broods,that emerge together. In any one geographical location,there is typically only one or rarely two broods of agiven species, and when there are multiple species(which is often the case), their broods emerge in thesame years (Lloyd and Dybas 1966a, Dybas and Lloyd1974, Williams and Simon 1995). Therefore, periodical

Manuscript received 25 October 2004; accepted 15 December2004; final version received 11 February 2005. CorrespondingEditor: A. A. Agrawal. For reprints of this Special Feature, seefootnote 1, p. 3137.

3 Corresponding author. E-mail: [email protected]

cicadas only emerge at a given location once or rarelytwice every 13 or 17 years, causing a strikingly epi-sodic pattern of species that, when present, are strik-ingly noticeable.

Periodical cicadas present numerous biological puz-zles. What were the evolutionary forces that createdsynchrony in the emergence of broods? Not only isemergence timed to be exactly 13 or 17 years, butemergence in the spring occurs over a narrow window,with most individuals emerging over a few days (Wil-liams and Simon 1995), implying strong selection forthe emergence of large numbers of cicadas together.Why does only one brood typically dominate in a givengeographical location? This suggests that there are ad-vantages not only in emerging within a large brood,but also in emerging periodically so that cicadas arenot present every year. And why are the periods ofperiodical cicadas both prime numbers? Since periodlength is evolutionarily labile, this seems hardly co-incidental. The seven species actually consist of threesets of sister species that are morphologically, behav-iorally, and (in some cases [Martin and Simon 1988])genetically distinct, with each of the three sets con-taining one 17-year species, and one or two 13-yearspecies (Simon et al. 2000, Cooley et al. 2003). Thedifference among the species within these three sets isprimarily period length, suggesting that period lengthcan change evolutionarily between 13 and 17 relativelyeasily.

December 2005 3201EMPIRICALLY MOTIVATED ECOLOGICAL THEORY

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These questions have a mathematical component,and it was for this reason that periodical cicadas wereselected as a topic for an undergraduate summer re-search program involving all of the authors. The goalof this program was to give undergraduate studentsexperience in research at the interface of biology andmathematics. Because the question of periodical cicadaperiodicity is inherently both biological and mathe-matical, it gave a compelling empirical problem todemonstrate the value of mathematics in the biologicalsciences. Developing ecological theory to address spe-cific empirical problems–the theme of this Special Fea-ture–may be useful not just in research, but also ineducation.

Below, we first review the biology of periodical ci-cadas. We then give an overview of previous theoreticalmodels addressing periodical cicadas, showing that ourunderstanding of the evolution of cicada’s periodicityis far from complete. Using a mathematical model thatincorporates and generalizes features from many pre-vious models, we investigate different mechanisms thatcould be responsible for the evolution of strict peri-odicity. We then investigate the more specific problemof explaining prime-numbered periods. Finally, we de-velop a spatial model of periodical cicada dynamics,which gives a strong argument in favor of one of theseveral mechanisms that could lead to cicada period-icity.

STUDY SYSTEM

The life cycle and key ecological features of peri-odical cicadas are reviewed in depth by Lloyd and Dy-bas (1966a, b) and Williams and Simon (1995); herewe present an abbreviated overview to explain the con-struction of models. The adult stage starts as a givenbrood of nymphs digs its way from the ground at nightto emerge over a 7–10 d period (Heath 1968, Williamset al. 1993). After mating in large congregations, fe-males lay up to 600 eggs. Nymphs then hatch, drop tothe forest floor, and burrow to underground roots. Thetotal adult life span is roughly 2–6 weeks. Adult mor-tality from bird predation can be high (Karban 1984),although birds become satiated when emerging broodsare large (Karban 1982, Williams et al. 1993). Ander-son (1977) and Nolan and Thompson (1975) both re-corded an increase in fledging success of birds duringyears of cicada emergence, showing that cicadas rep-resent an important food source for some species ofbirds. In a recent study, Koenig and Liebhold (2005)analyzed 37 years of North American Breeding BirdCount data for 24 species that potentially eat cicadas.Of these, 15 showed some population abundance re-sponse to cicada emergences, with one species (theRed-headed Woodpecker) showing an 18% increase inabundance in the year following emergence. Increasesin abundance, however, tended to be short-lived, lasting1–3 years. A few species had lower than average abun-dance in the years preceding cicada emergence, which

might suggest a very long-term effect of the previousemergence, although this pattern was far less strikingthan the short-lived increase in abundance of other spe-cies following emergence.

Nymphs feed on xylem from the roots of a varietyof deciduous tree species (White and Strehl 1978). Ina study investigating underground survival, Karban(1997) found high nymph mortality in the first twoyears following egg laying, but subsequent low mor-tality until emergence. Furthermore, early mortalitywas apparently density dependent; from five studysites, the two with much higher density than the othersalso had higher mortality. This could be explained bycompetition among nymphs for food (Williams and Si-mon 1995). Nymphs grow through five instars, andonce they have reached the last nymph instar, growthis arrested even if this occurs several years before theirtimed emergence at 13 or 17 years (White and Lloyd1975). Arrested growth suggests strong evolutionaryforces underlying the strict periodicity of emergence.Occasionally, however, ‘‘mistakes’’ are made, in whichcase a 17-year cicada generally emerges at 13 years,although rarer mistakes are made in which emergenceis off by only one year (Williams and Simon 1995).Several authors argue that the phenotypic differentia-tion between 13- and 17-year periods may be governedby a single, diallelic locus (Lloyd et al. 1983, Cox andCarlton 1988), suggesting that there is a genetic switchmechanism between prime-numbered periods.

THEORETICAL APPROACH

Previous models

Numerous theoretical models have been used to in-vestigate mechanisms that could create cicada peri-odicity. Here we review some of the key models thatexplore different facets of periodicity.

Hoppensteadt and Keller (1976) and Bulmer (1977)investigated mechanisms that could drive periodicity.Both investigations demonstrated that a combinationof nymph competition and predator satiation could leadto one or a few broods dominating other broods anddriving them to extinction. Nymph competition actsacross broods, with high enough densities of one broodleading to the suppression of other broods in followingyears. This process is exacerbated by predator satiation(Bulmer 1977). When there is predator satiation, largebroods are favored over small broods, since by satiatingtheir predators large broods have higher per capita sur-vival. Thus, once a brood is diminished to levels atwhich predator satiation is no longer strong, the broodwill be extinguished. Similar effects could be producedvia parasitism rather than predator satiation (May1979), and between-year effects of parasitism or pre-dation could serve like competition to cause high den-sities of one brood to reduce the density of broods infollowing years (Bulmer 1977, Behncke 2000). Thesemodels all begin with the assumption that cicadas have

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3202 NICOLAS LEHMANN-ZIEBARTH ET AL. Ecology, Vol. 86, No. 12

the same fixed age at emergence and hence have aperiodic life cycle. They only address how a singlebrood might become dominant, rather than how peri-odicity might evolve in the first place.

Recent theoretical work has addressed why cicadaperiods are prime numbers. Building on an idea ofGould (1977), Webb (2001) assumed that cicada pred-ators consist of species having cyclic or ‘‘quasi-cyclic’’dynamics with either two- or three-year periods. Thisleads to high predator abundances, and high predationrates, in years divisible by either two or three. Becauseprimes are the only numbers between 10 and 18 thatare not divisible by 2 or 3, broods of prime-periodcicadas frequently escape high predation levels andhence tend to dominate hypothetical cicadas with non-prime periods. This mechanism for generating primenumbers relies on either externally driven two- andthree-year cycles of predators, or predators that havestrict fecundity schedules creating dynamics that tendto show two- or three-year oscillations.

Markus et al. (2002) built a stylized model in whichcicadas experience a negative impact when they emergewith predators, and predators experience a positive im-pact when they reproduce at the same time as cicadaemergence. Unlike Webb (2001), cicadas were restrict-ed to have single broods with strict period (i.e., theentire population was synchronized), and predatorswere assumed to be themselves strictly periodic witha single cohort. Markus et al. (2002) then allowed ci-cadas and predators to ‘‘evolve’’ by introducing mutantcicadas and predators with period lengthened or di-minished by one year. This eventually leads to cicadaswith prime periods. Nonetheless, this mechanism gen-erating prime-period cicadas relies upon predators hav-ing strictly periodic generations and fitness tightly cou-pled to cicada emergence, both of which are biologi-cally unlikely.

A final group of models is largely verbal, explainingprime periods as the consequence of evolution to avoidhybridization (Cox and Carlton 1988, Yoshimura1997). Here it is assumed that strictly periodic life cy-cles were previously established by some factor, butnumerous different strict periods (e.g., 12-year, 13-year, 14-year, etc.) co-occurred. Prime-numbered pe-riods are favored, because this limits the chance ofhybridization with other phenotypes; hybridizationwould be disfavored either if hybrids experienced ge-netic breakdown in the mechanism determining periodlength, or if hybrids had the period of neither parentalphenotype and hence emerged at the ‘‘wrong’’ time insmall numbers with few mates (Cox and Carlton 1988).A difficulty of this explanation is that prime-periodphenotypes might in fact be more likely to hybridize;if, for example, 12- and 13-year phenotypes co-occur,they will emerge together at least within 156 years,while 12- and 14-year phenotypes will never emergetogether if they initially emerge 1 year apart.

Despite numerous models, answers about the mech-anisms that could underlie cicada periodicity are sur-prisingly incomplete. Few models explicitly addressevolutionary processes, and models that address factorsleading to the dominance of only a few broods (e.g.,Hoppenstaedt and Keller 1976, Bulmer 1977) do notaddress why periods are prime numbers. Finally, weknow of no mathematical models addressing the spatialpattern of cicada emergence, with sharp and apparentlystable boundaries separating broods.

Model structure and analysis

To investigate the evolution of cicada periodicity,we constructed a model that amalgamates many of thefeatures found in the collection of existing cicada mod-els. This amalgamated model makes it possible to ad-dress simultaneously multiple factors that might leadto the evolution of periodicity. After developing andanalyzing the ‘‘base’’ model, we then modify it to ex-plore a possible explanation of periods being prime.Finally, we present a spatial version of the model toinvestigate whether the spatial pattern of brood dom-ination suggests one mechanism over others as the mainforce driving the evolution of periodicity. We do notaddress ideas about hybridization and historical gen-eration of periodicity during the last Ice Age (Cox andCarlton 1988, Yoshimura 1997), because this wouldrequire too many unverifiable assumptions about thegenetic consequences of hybridization and the histor-ical pattern of the evolution of periodicity.

In the base model, we consider the evolution of phe-notypes that differ in how variable generation time isamong offspring. Because we do not know the geneticsunderlying the evolution of periodicity in protoperiodiccicadas, we assume that evolution is haploid, modelingfemale offspring that have the phenotype of their moth-ers. Although haploid evolutionary models may notcapture the genotypic evolutionary process, they none-theless elucidate the forces driving natural selection(Maynard-Smith 1974).

The base model is age structured, following the dy-namics of a phenotype p,v defined by the mean periodp of its life cycle and the variability v about this mean.Thus, let xp,v(q, t) denote the adult density of the qth (q5 1, . . . , p) brood of a p,v periodical cicada. The sur-vival and fecundity of emerged adults is given by

ay(t)x9 (q, t) 5 x (q, t)exp r 2 (1)p,v p,v 1[ ]1 1 bX (t)s

where (q, t) is the density of newborn nymphs, r1x9p,v

sets the maximum realized fecundity to exp(r1), y(t) isthe density of predators in the year of emergence, a isthe predator attack rate, Xs is the total number of adultsfrom all phenotypes emerging in year t, and b deter-mines the rate at which predators satiate with increasingtotal adult density. If b 5 0, there is no satiation (typeI functional response), while as b increases, satiationbecomes stronger (type II functional response).


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We assume that competition among nymphs occursas density-dependent mortality of newborn nymphs.Karban (1997) showed that most nymph mortality oc-curred within the first two years (and possibly earlier,since two years was the shortest of his sampling in-tervals), and spreading the density-dependent effectsof competition on nymph mortality over two yearsmakes little quantitative difference to the model. Thedensity of newborn nymphs surviving competition is

21x0 (q, t) 5 x9 (q, t)[1 1 d X (t) 1 d X (t)]p,v p,v o o L L (2)

where X0(t) is the total density of newborn nymphs,XL(t) is the total density of nymphs between the agesof 1 and T, and the effect of competition with newbornand older nymphs increasing with d0 and dL, respec-tively. In this formulation, we control the window ofnymph ages that have a competitive effect on newbornnymphs by setting T. Biologically, older nymphs mighthave less effect on newborn nymphs because oldernymphs feed lower in the soil and can become rela-tively inactive once they reach the size, but not age,required for emergence. Because only single broodsoccur in most geographical locations, however, it isdifficult to study the competitive effects between dif-ferent-aged nymphs, so we have no direct informationabout T. For the initial analyses we set T 5 2 so thatthe effect of the size of this competitive window canbe seen in the model results, although later analysesshow that T is likely to be larger.

Finally, to model phenotypes that differ in variabilityin the timing of emergence, we assume that for a givenphenotype p,v, a fraction of v offspring emerges at atime different from characteristic period p. These off-spring are placed into broods symmetrically before andafter the parental brood following a geometric distri-bution. Furthermore, we assume that once nymphs havesurvived their first year, there is no further mortality.Thus, the density of adults of brood q of phenotype p,vin year t 1 p is

1 2 nx (q, t 1 p) 5 x 0 (q, t)p,v p,v2

p21 1 2 nzi z1 n x 0 (q 1 i, t 1 i). (3)O p,v2i52(p21)

We model haploid evolution by assuming that withprobability P 5 (q, t) a small number of individ-w x9p,v

uals (specifically, a density of 0.01) ‘‘mutate’’ fromphenotype p,v1 to a phenotype p,v2, thus permittingmutations between phenotypes with different variabil-ity v but the same mean period p. The parameter wscales the probability of mutation relative to the pop-ulation density; larger populations are more likely toproduce a mutant. The value of w 5 0.01 is set low sothat mutations are relatively rare, occurring with prob-ability 0.1 in a population with density 10 (which istypical for dominant broods in our simulations).

We assume that predators experience density-depen-dent dynamics, with yearly fecundity of exp(r2), yearlysurvival of m, and a cicada-independent equilibriumpopulation density of K. In the presence of cicadas, thepredator density y(t) has dynamics given by

caX (t)sy(t 1 1) 5 y(t)exp r [1 2 hy(t)] 1 1 my(t)25 61 1 bX (t)s

(4)

where h 5 (r2 2log(1 2 m))/r2K is a scaling term, andc is the conversion rate of consumed cicadas into pred-ator offspring. Note that if c 5 0, the predator dynamicsare unaffected by cicada predation.

This model is sufficiently complex that simple an-alytical solutions do not exist. However, our goal is todetermine only whether different explanations of ci-cada periodicity are plausible. Therefore, we performedsimulations of particularly informative cases, using acomparative approach to hone our understanding ofdifferent mechanisms that could drive the evolution ofperiodicity in cicadas.

RESULTS

Strict-period broods

Before considering evolution, we first analyze themodel to determine factors that lead to the eliminationof broods when there is no variability in age at emer-gence (v 5 0). This is the problem addressed in nu-merous previous models (Hoppensteadt and Keller1976, Bulmer 1977, Behncke 2000) and serves to ex-pose the processes that underlie our later results onevolution.

We considered four cases, all of which have cicadaswith a strict 15-year life cycle. We selected a 15-yearlife cycle to be neutral, between 13 and 17 but not aprime number; choices other than 15 yield similar re-sults. Case I includes predator satiation (b . 0 in Eq.1) and between-brood nymph competition (dL . 0 andT 5 2 in Eq. 2) in which higher densities of 1- and 2-year-old nymphs decrease the survival of newbornnymphs. Simulations show that only a few of thebroods persist (Fig. 1A). The reason for this can beseen in Fig. 1B, in which the density of adults in gen-eration t 1 1 is plotted against the density of adultsin generation t that emerged 15 years previously. Theseplots were constructed by varying the density of adults(and hence newborns) in generation t while preservingthe density of 1- and 2-year-old underground nymphsand hence the strength of competition experienced bynewborns. The 15 separate lines in Fig. 1B correspondto the 15 broods, with lower lines indicating strongercompetition from previous broods. The sigmoidalshape of the lines shows the effect of predator satiation;if too few adults emerge, per capita predation is highand survival is low, leading to an Allee effect. For somebroods, the density of adults in generation t 1 1 isalways lower than the density in generation t, indi-

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FIG. 1. Brood elimination for case I (A, B) with predator satiation (b 5 1) and between-brood nymph competition (dL

5 0.2, Eq. 2), case II (C, D) with predator satiation (b 5 1) and predator reproduction in response to cicada predation (c 53, Eq. 4), case III (E, F) with predator satiation (b 5 1) and variation in initial cicada density, and case IV (G, H) withoutpredator satiation (b 5 0, K 5 6, Eq. 4) and between-brood nymph competition (dL 5 0.2). The top panel for each case givesthe adult population density for each of the 15 broods in each generation, and the bottom panel gives the number of adultspredicted in generation t 1 1 as a function of the number in generation t for t 5 8 (cases I–III) and t 5 40 (case IV).Parameter values not listed above for specific cases are: r1 5 3, d0 5 0.05, dL 5 0, a 5 0.35, r2 5 0.2, c 5 0, m 5 0.75,and K 5 35.

cating declining densities. For other broods, there is awindow of densities in generation t that lead to anincrease in density. Note that the number of broods thatpersist depends in part on the initial densities of broodsand the resulting transient dynamics that determinewhich broods dominate. This dependence on initialconditions makes any absolute conclusion about thenumber of persisting broods difficult, because there isa large role of transient dynamics in the initial estab-lishment of broods.

In case II, we assumed that there is no between-broodnymph competition (dL 5 0), but predator reproductionincreases in response to cicada predation (c . 0), thuscausing an increase in density that leads to higher pre-dation on subsequent broods (Fig. 1C). As a result, twobroods eliminate the others and create spikes in pred-ator densities corresponding to years of emergence(Fig. 1D). Somewhat surprisingly, although consis-tently, two adjacent broods persist, largely as a resultof the initial transient dynamics during which broodsuccess depends on initial densities.

In case III, we eliminated any direct interaction be-tween broods by removing both between-brood com-petition (dL 5 0) and predator reproduction in responseto cicada predation (c 5 0), and selected initial den-sities of cicadas randomly from a uniform distribution

between 0 and 4 (Fig. 1E). Despite the absence of in-teractions between broods, only three broods persist.This is simply because the Allee effect created by pred-ator satiation requires an initial density of roughly 3.5for persistence (Fig. 1F). Although this is a trivial case,we use it to emphasize the importance of initial den-sities when there is predator satiation, and later wedemonstrate that this effect alone can lead to the evo-lution of periodicity.

Finally, in case IV we removed predator satiation (b5 0) but included between-brood nymph competition(dL . 0) in the absence of cicada-dependent predatorreproduction (c 5 0) (Fig. 1G). In this case, between-brood competition is sufficient to cause the persistenceof only a subset of broods, because between-broodcompetition decreases the population growth rate ofsome populations below the replacement rate (Fig. 1H).Thus, predator satiation is not essential for brood elim-ination, as also shown by Bulmer (1977).

Evolution of periodicity

All four cases analyzed above gave rise to broodelimination and dominance by just a few broods. Broodelimination (or at least strong reduction in density) isa necessary component of the evolution of strict pe-riodicity, because if all broods persist at high density,


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FIG. 2. Evolution of fixed periods for case I (A) with predator satiation (b 5 1) and between-brood nymph competition(dL 5 0.2), case II (B) with predator satiation (b 5 1) and predator reproduction in response to cicada predation (c 5 3),case III (C) with predator satiation (b 5 1) and variation in initial cicada density, and case IV (D) without predator satiation(b 5 0, K 5 6) and between-brood nymph competition (dL 5 0.2). For each case, all initial individuals had phenotype 15,0.2in which 20% of offspring emerge in less than or more than 15 years. Mutations to phenotypes 15,0.1 and 15,0 occurred atlow probability. Parameter values are as in Fig. 1; v is variability about the mean period, p.

density-dependent processes affecting cicada survivaland reproduction will not vary strongly with the timingof emergence. Can all four mechanisms causing broodelimination drive evolution of strict periodicity?

To investigate the evolution of periodicity, we con-sidered three phenotypes, each with a mean period of15 years, but differing in variability in period from v5 0.2 (20% of the brood having a period different from15 years) to v 5 0.1 and v 5 0 (strict period). Weassumed that initially the entire population consistedof phenotype p 5 15, v 5 0.2, and allowed the othertwo phenotypes to arise via mutations. We selectedinitial densities to be the same for all broods. None-theless, even if some broods were greatly reduced indensity, broods never went extinct as long as variable-period phenotypes persisted, because variability in gen-eration time always allowed the repopulation of broods.

In case I, between-brood nymph competition andpredator satiation rapidly led to the evolution of a strict-ly periodic phenotype (Fig. 2A, v 5 0) as the twophenotypes with variable generation times were ex-cluded. This occurred because between-brood nymph

competition and predator satiation caused the severereduction of some broods. As a result, the strictly pe-riodic phenotype was favored because it did not loseindividuals that emerged in years with reduced adultdensities and hence higher per capita predation rates.Similarly, case II leads to the severe reduction in den-sity of some broods and the consequent advantage tothe strictly periodic phenotype (Fig. 2B). Thus, pred-ator satiation and an increase in predator reproductionin response to cicada predation caused the evolution ofstrict periodicity.

For case III we assumed that the fecundity of adults(regardless of phenotype) emerging in the same yearwas a random variable; specifically, the value of r1 wasdrawn from a normal random variable each year. Thisresulted in variation in densities among broods, andwhen broods dropped below the threshold of the Alleeeffect, they would often continue to decline to verylow densities (Fig. 2C). This created an advantage forthe strictly periodic phenotype, even in the absence ofbetween-brood nymph competition and predator repro-duction in response to cicada predation. Finally, case

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IV shows that predator satiation is not essential for theevolution of strict periods provided between-broodcompetition can cause the severe reduction in densityof some broods (Fig. 2D).

In summary, in all cases in which some broods werereduced in density, evolution favored the strictly pe-riodic phenotype. While this highlights the central re-quirement of brood elimination for the evolution ofstrict periods, it does not give any insight into whichmechanism might in fact be responsible for the evo-lution of the strict periodicity of cicadas.

Phenotypes with multiple periods

In the preceding analyses, we considered only phe-notypes that have an expected period of 15 years. Toinvestigate both the evolution of strict periodicity andthe selection of period length, we considered pheno-types p,v for p 5 12–16 and v 5 0.2, 0.1, and 0. Weselected this range of periods to encompass 13 but noother prime number. Populations were started with allbroods of all v 5 0.2 phenotypes, and the v 5 0.1 and0 phenotypes were generated via mutations. While mu-tations could change the variability in life cycle length(v), we assumed that the expected period ( p) was notsubject to change. Therefore, selection of a period ptook the form of competition between phenotypes.

For case I with predator satiation and between-broodcompetition, evolution favored cicadas with strict pe-riods of 12 and 15 years (Fig. 3). Examination of themodel showed that 4 and 5 broods of the 12-year cicadaand 15-year cicadas persisted, respectively, and thesebroods emerged synchronously every three years. Thissynchronization of broods makes sense, because theresulting high density of adults from multiple pheno-types guarantees predator satiation. The selection ofbroods that emerge every three years is a consequenceof the form of between-brood competition. Because weassumed that 1- and 2-year-old nymphs have a com-petitive effect on newborn nymphs, this sets a timedelay of three years in the system, with broods follow-ing one and two years after a large brood sufferingcompetition. If the delay is shortened, so that only 1-year-old nymphs have a competitive effect on newbornnymphs, then broods of even-numbered period are fa-vored, and broods emerge every two years. Conversely,if the delay is lengthened to four years, 12- and 16-year cicadas are favored. Although we present onlycase I for predator satiation and between-brood nymphcompetition, the results are similar for the other cases;when brood elimination occurs, thus favoring evolutionof strict periodicity, the 13-year phenotype does notpersist but is supplanted by periods that allow syn-chronous emergence among broods.

Prime-numbered periods

As found in the last example, those factors that causebrood elimination and promote the evolution of strictperiodicity also promote synchronous emergence

among individuals with different periods, thus favoringperiods that have a small common denominator. Thismakes it difficult to explain the evolution of prime-numbered periods. In an attempt to find a scenario inwhich prime-numbered periods are selected, we startedwith the situation similar to case IV with between-brood competition, yet instead of having no predatorsatiation, we imposed very mild predator satiation. Toselect for prime numbers, we followed Webb (2001) insupposing that there are two types of predators, onewith a two-year and the other with a three-year lifecycle. To accentuate the periodicity of predation, weassumed (unrealistically) that predators only reproduceonce (in their second or third year, respectively) andonly feed on cicadas in the year they reproduce. Thisis the extreme case of age-dependent fecundity. In themodel, the cyclicity in predator dynamics is maintainedby increased reproduction caused by predation on ci-cadas. Our model differs from Webb (2001) in that heassumes cicadas have strict periods, whereas we allowphenotypes with variable periods, and explicitly askwhether both strict periodicity and prime-numbered pe-riodicity can evolve simultaneously.

The scenario we constructed does favor cicadas with13-year periods over other periods between 12 and 16(Fig. 4). Nonetheless, in the scenario with sufficientlyweak predator satiation to remove strong selection forsynchrony and resulting dominance of nonprime pe-riods, selection for fixed over variable periods is tooweak to allow the period to become fixed. We couldnot find parameter values that simultaneously selectedfor strict periodicity (v 5 0) and 13-year periods. Fur-thermore, we do not think that this mechanism favoringprime periods is likely, because it requires strong andsimultaneous two- and three-year cycles in predation.Most birds, the main predators of cicadas, breed in theirfirst year and show constant annual survivorship (Go-telli 1998), making strong two- or three-year signalsin predation unlikely.

Spatial patterns

So far we have focused on the evolution of strictperiodicity and prime-numbered periods, ignoring thestriking spatial pattern shown by cicadas in which sin-gle broods often dominate large contiguous areas, andin the rare cases in which nonsynchronized broodsoverlap geographically, they are separated by at leastfour years (Lloyd and Dybas 1966b). We have shownthat there are at least four mechanisms that could drivethe evolution of strict periods. Here we ask whetherthe observed geographical patterns suggest that one ofthese mechanisms is stronger than the others. Specif-ically, we mathematically test the suggestion of Wil-liams and Simon (1995) that the geographical distri-bution of periodical cicadas implies that nymph com-petition is important.

We constructed a spatial model of cicada dynamicsby translating our base model (Eqs. 1–3) onto a spatial


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FIG. 3. For case I, evolution of fixed periods when multiple periods are possible. Initially all individuals were equallydistributed among phenotypes p 5 12–16, v 5 0.2. Random mutations occurred to phenotypes with v 5 0.1 and v 5 0 andthe same period p as the parental phenotype. Parameter values are as in Fig. 1 for case I, except that K was lowered from35 to 30; the combination of between-brood competition and K 5 35 drove the entire initial cicada population extinct.

grid of 50 3 50 cells with wrap-around (torus) bound-aries. The dynamics of cicadas and predators withineach cell were governed by the base model, and eachyear 5% of cicadas and 20% of the predators dispersedto one of the adjacent four cells before reproduction.Although this is a simplistic depiction of dispersal,more complex descriptions of dispersal are unlikely togive qualitatively different results. Cicada dispersalcreates low-density populations in cells adjacent to es-tablished populations, and these have low populationgrowth rates due to lack of predator satiation. For thevalues of predator carrying capacity K used previously

(Fig. 1), this caused the cicada population to go extinct.Therefore, for the spatial simulations we reduced K,the carrying capacity of predators in the absence ofcicadas. We only considered cicadas with a strict 13-year period, assuming that a strict period had previ-ously evolved. Finally, we initially populated space byrandomly selecting 13 cells on the grid and placing oneof the 13 broods in each.

To summarize the results of the spatial model oncethe grid is fully occupied, we report the average numberof broods occurring in each cell with a density abovethe threshold of 1% of the maximum density found

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FIG. 4. For the case with predators that have fixed two- and three-year life cycles, evolution of prime-numbered periods.Initially all individuals were equally distributed among phenotypes p 5 12–16, v 5 0.2. Random mutations occurred tophenotypes with v 5 0.1 and v 5 0 and the same period p as the parental phenotype. Parameter values are: b 5 0.02, r1 53, d0 5 0.05, dL 5 0.2, a 5 0.35, r2 5 0.2, c 5 0.2, and K 5 7, where K is the joint carrying capacity of both types ofpredators.


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FIG. 5. Spatial distribution of 13 cicada broods simulatedon a 50 3 50 grid when 5% of cicadas and 20% of predatorsmove from their natal cell each generation. (A) Competitionfrom only nymph age classes 1–6 (T 5 6, Eq. 2) affectsnewborn nymphs, and per capita competition is strong (dL 50.6). For clarity, the range of one brood is shaded black anda second is shaded gray. (B) Competition from all nymph ageclasses (T 5 12), with weaker per capita competition (dL 50.08). The 13 broods are numbered. Other parameter valuesare: b 5 1, r1 5 3, d0 5 0.05, a 5 0.35, r2 5 0.2, c 5 0, andK 5 15.

among cells. For the four cases described previously,there were 5.81 6 0.16, 3.70 6 0.34, 10.57 6 1.83,and 13 6 0 broods per cell (mean 6 SD of 10 simu-lations). These numbers change with changes in modelparameters; for example, increasing cicada dispersalfrom 5% increases the average number of broods atany location. In cases I–III, the spatial distribution ofbroods was limited by the presence of other broods,but in case IV all broods spread throughout the grid.Even for the case with the lowest average number ofbroods per cell, case II (with predator reproductiondepending on cicada predation), there were more thanthree broods on average per location, in contrast to theusual situation in nature of only one or rarely twobroods.

In the model, the only plausible way to reduce thenumber of broods at the same location is to increasethe length of time over which an emerging brood couldaffect subsequent broods. In the base model, the mostbiologically realistic way of generating the requisitelong-term effects is via nymph competition (Williamsand Simon 1995). Until now, we have assumed thatonly 1- and 2-year-old nymphs have a competitive ef-fect on newborn nymphs (T 5 2 in Eq. 2), yet it islikely that older nymphs also have a competitive effect.In contrast, it is unlikely that predators have a long-term effect (case II); this would require predator re-production caused by a brood emergence to generatehigh predator densities that are then maintained formany years. Although there is equivocal evidence thatthis is possible for a few bird species, the increase ofbird abundance due to cicada emergences is generallyshort-lived (Koenig and Liebhold 2005).

Fig. 5 gives two illustrative examples of increasingthe length of time over which nymphs have a com-petitive impact on newborn nymphs. In the first (Fig.5A), nymphs have a competitive effect on newbornsup to the age of 6 (T 5 6 in Eq. 2), and we made thiscompetition strong so that broods readily excluded eachother; the mean number of broods per location was 1.5.In this scenario, however, complex spatial patternsarise, rather than the observed dominance of a givenbrood over a wide geographical region. Furthermore,the boundaries are not stationary; the spiral patternsrotate slowly through time. These spatial dynamics arethe result of a nontransitive hierarchy in competitivedominance. For example, a brood emerging in 1990 isdominant to a brood emerging in 1994, which is turnis dominant to a brood emerging in 1999. But the broodemerging in 1999 is dominant to the brood emergingin 2003, which is the same brood that emerged in 1990.This type of rock–paper–scissors hierarchy is knownto produce complex, nonstationary spatial patterns(Durrett and Levin 1998, Frean and Abraham 2001; D.S. Griffeath, personal communication).

In the second example (Fig. 5B), all nymphs have acompetitive effect on newborns (T 5 12 in Eq. 2). Inthis case, well-defined and stationary geographical dis-

tributions of broods emerge, with a mean number ofbroods per location of 1.09. This case more accuratelymimics reality where the geographical ranges of broodsare contiguous and the boundaries appear to be static.Thus, it is likely that all, or at least most, nymph agegroups are involved in competition. Even if we arewrong in assuming that nymph competition is the mostlikely process responsible for between-brood interac-tions, whatever factor creates between-brood interac-tions must act over the long term, so that broods emerg-ing 12 years previously have an effect on newborns.

DISCUSSION

Using a suite of models, we investigated possiblemechanisms for the evolution of strict periods in pe-

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riodical cicadas. We showed four cases that can leadto elimination of broods of a hypothetical cicada thathas a strictly periodic life cycle (Fig. 1): (case I) pred-ator satiation and between-brood nymph competition;(case II) predator satiation and predator reproductionin response to cicada predation; (case III) predator sa-tiation and random cicada population densities; and(case IV) between-brood nymph competition in the ab-sence of predator satiation. These cases that createbrood elimination also lead to the evolution of strictperiodicity from an initial cicada phenotype having var-iable emergence times (Fig. 2). Thus, there are multiplescenarios that can lead to the evolution of cicada pe-riodicity.

Which of these scenarios is most likely? The spatialdistribution pattern of cicadas, with locations domi-nated most often by a single brood even when thereare multiple species co-occurring (Cooley et al. 2003),suggests that a large brood emergence can suppresssubsequent broods for many years. This implicates be-tween-brood nymph competition as a potentially strongfactor explaining the current distribution pattern of ci-cadas (Williams and Simon 1995). Our spatial modelconfirms that strong spatial partitioning of broods canoccur when there is predator satiation and a competitiveeffect of all ages of nymphs on newborn nymphs. Ifbetween-brood nymph competition was strong in theevolutionary history of periodical cicadas, it wouldhave created strong selection for strictly periodic emer-gence. This suggests, albeit with several leaps of faith,that between-brood nymph competition and predatorsatiation were key processes to the evolution of cicadasperiodicity.

This does not, however, address why cicadas haveprime-numbered periods. The difficulty in generatingprime-numbered periods is that the evolution of peri-odicity requires selection for synchronized emergenc-es, and the same selective forces also select for syn-chronization among cicadas with different periods, andhence for periods with small common denominators.We were able to create a model that selects for primenumbers, although this required strong and simulta-neous two- and three-year cycles in predation rates.Furthermore, to reduce the force of selection for syn-chronization across phenotypes with different periods,we had to assume weak predator satiation—satiationthat was so weak that evolution of strict periodicity didnot occur. Thus, we do not think that our model givesa plausible explanation for prime-numbered periods,and we could not devise another explanation and modelthat generated prime numbers.

We suspect that the mechanism explaining why pe-riods are prime numbered does not directly involveecological processes. One possibility is that ecologicalprocesses select for synchronous emergence (regard-less of period), and the physiological/genetic countingmechanisms needed to synchronize emergence con-strain periods to prime numbers. Although recent stud-

ies have shown that periodical cicadas count years bydetecting seasonal changes experienced by their hosttrees (Karban et al. 2001), the actual counting mech-anism is unknown. Two types of evidence suggest aphysiological/genetic explanation for prime periods.First, when 17-year cicadas make ‘‘mistakes,’’ theyemerge generally in 13 years (Williams and Simon1995). Second, ecological and genetic studies havedemonstrated close associations between 13- and 17-year species within sets of sister species that sharemany morphological and behavioral traits (Simon et al.2000); thus, there seems to be a genetic switch between13- and 17-year periods (Cox and Carlton 1991). Apossible explanation for this, albeit pure speculation,is that cicadas count long periods using only two short-term clocks: one that counts two years and the otherthat counts three years. Such a dual clock system wouldallow cicadas to identify years that are not divisible byeither 2 or 3, which for the range of years 12 to 18 arethe prime numbers 13 and 17. We can find no examplesof dual-clock counting mechanisms in other biologicalsystems, and therefore there is no precedent to supportthis speculation. Nonetheless, our difficulty in derivingecological scenarios that could lead to prime-numberedperiods suggests looking for nonecological explana-tions.

This project has given an example in which a system-specific model was used to formulate and explore pos-sible hypotheses about the processes underlying a strik-ing biological phenomenon. We cannot prove that anyof our results explain the evolution of periodicity incicadas. Furthermore, we surely did not uncover andinvestigate all of the possible explanations for cicadaperiodicity, even though we went to greater lengthsthan might be apparent in this article to find new ex-planations for periodicity, particularly prime-numberedperiodicity. Nonetheless, taken together the models al-lowed us to crystallize the processes that potentiallylead to periodicity. Our results confirm and expand onthose of Hoppensteadt and Keller (1976) and Bulmer(1977) that predator satiation and between-broodnymph competition are likely important in explainingperiodicity, although we believe that prime-numberedperiods are unlikely to have an ecological explanation.

ACKNOWLEDGMENTS

We thank Kelley Tilmon for comments on the manuscript,and Sandy Liebhold for sharing a manuscript. This projectwas funded by a U.S. National Science Foundation Interdis-ciplinary Training for Undergraduates in Biological andMathematical Sciences (UBM) grant to A. R. Ives and P. A.Milewski.

LITERATURE CITED

Anderson, T. R. 1977. Reproductive responses of sparrowsto a superabundant food supply. Condor 79:205–208.

Behncke, H. 2000. Periodical cicadas. Journal of Mathe-matical Biology 40:413–431.

Bulmer, M. G. 1977. Periodical insects. American Naturalist111:1099–1117.


Spec

ialFeatu

re

Cooley, J. R., C. Simon, and D. C. Marshall. 2003. Temporalseparation and speciation in periodical cicadas. BioScience53:151–157.

Cox, R. T., and C. E. Carlton. 1988. Paleoclimatic influencesin the evolution of periodical cicadas (Insecta: Homoptera:Cicadidae: Magicicada spp.). American Midland Naturalist120:183–193.

Cox, R. T., and C. E. Carlton. 1991. Evidence of geneticdominance of the 13-year life cycle in periodical cicadas(Homoptera: Cicadidae: Magicicada spp.). American Mid-land Naturalist 125:63–74.

Durrett, R., and S. Levin. 1998. Spatial aspects of interspe-cific competition. Theoretical Population Biology 53:30–43.

Dybas, H. S., and M. Lloyd. 1974. The habits of 17-yearperiodical cicadas (Homoptera: Cicadidae: Magicicadaspp.). Ecological Monographs 44:279–324.

Frean, M., and E. R. Abraham. 2001. Rock-scissors-paperand the survival of the weakest. Proceedings of the RoyalSociety of London, Series B 268:1323–1327.

Gotelli, N. J. 1998. A primer of ecology. Sinauer Associates,Sunderland, Massachusetts, USA.

Gould, S. J. 1977. Ever since Darwin: reflections in naturalhistory. Norton, New York, New York, USA.

Heath, J. E. 1968. Thermal synchronization of emergence inperiodical ‘‘17-year’’ cicadas (Homoptera, Cicadidae,Magicicada). American Midland Naturalist 80:440–448.

Hoppensteadt, F. C., and J. B. Keller. 1976. Synchronizationof periodical cicada emergences. Science 194:335–337.

Karban, R. 1982. Increased reproductive success at high den-sities and predator satiation for periodical cicadas. Ecology63:321–328.

Karban, R. 1984. Opposite density effects of nymphal andadult mortality for periodical cicadas. Ecology 65:1656–1661.

Karban, R. 1997. Evolution of prolonged development: a lifetable approach to periodical cicadas. American Naturalist150:446–461.

Koenig, W. D., and A. M. Liebhold. 2005. Effects of peri-odical cicada emergences on abundance and synchrony ofavian populations. Ecology 86:1873–1882.

Lloyd, M., and H. S. Dybas. 1966a. The periodical cicadaproblem. I. Population ecology. Evolution 20:133–149.

Lloyd, M., and H. S. Dybas. 1966b. The periodical cicadaproblem. II. Evolution. Evolution 20:466–505.

Lloyd, M., G. Kritsky, and C. Simon. 1983. A simple Men-delian model for 13- and 17-year life cycles of periodical

cicadas, with historical evidence of hybridization betweenthem. Evolution 37:1162–1180.

Markus, M., O. Schulz, and E. Goles. 2002. Prey populationcycles are stable in an evolutionary model if and only iftheir periods are prime. ScienceAsia 28:199–203.

Marshall, D. C., and J. R. Cooley. 2000. Reproductive char-acter displacement and speciation in periodical cicadas,with description of a new species, 13-year Magicicada neo-tredecim. Evolution 54:1313–1325.

Martin, A. P., and C. Simon. 1988. Anomalous distributionof nuclear and mitochondrial DNA markers in periodicalcicadas. Nature 336:237–239.

May, R. M. 1979. Periodical cicadas. Nature 277:347–349.Maynard-Smith, J. 1974. Evolution and the theory of games.

Cambridge University Press, Cambridge, UK.Nolan, V., and C. F. Thompson. 1975. Occurrence and sig-

nificance of anomalous reproductive activities in two NorthAmerican non-parasitic cuckoos, Coccyzus spp. Ibis 117:496–503.

Oldenburg, H. 1666. Some observations of swarms of strangeinsects and the mischiefs done by them. PhilosophicalTransactions of the Royal Society, London 1.

Simon, C., J. Tang, S. Dalwadi, G. Staley, J. Deniega, andT. R. Unnasch. 2000. Genetic evidence for assortative mat-ing between 13-year cicadas and sympatric ‘‘17-year ci-cadas with 13-year life cycles’’ provides support for al-lochronic speciation. Evolution 54:1324–1336.

Walsh, B. D., and C. V. Riley. 1868. Potato bugs. AmericanEntomologist 1:21–29.

Webb, G. F. 2001. The prime number periodical cicada prob-lem. Discrete and Continuous Dynamical Systems–SeriesB 1:387–399.

White, J., and M. Lloyd. 1975. Growth rates of 17-and 13-year periodical cicadas. American Midland Naturalist 94:127–143.

White, J., and C. E. Strehl. 1978. Xylem feeding by periodicalcicada nymphs on tree roots. Ecological Entomology 3:323–327.

Williams, K. S., and C. Simon. 1995. The ecology, behavior,and evolution of periodical cicadas. Annual Reviews ofEntomology 40:269–295.

Williams, K. S., K. G. Smith, and F. M. Stephen. 1993. Emer-gence of 13-yr periodical cicadas (Cicadidae: Magicicada):phenology, mortality, and predation. Ecology 74:1143–1152.

Yoshimura, J. 1997. The evolutionary origins of periodicalcicadas during ice ages. American Naturalist 149:112–124.

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