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

EVOLUTION OF PERIODICITY IN PERIODICAL

NICOLAS LEHMANN-ZIEBARTH,1,2 PAUL P. H EIDEMAN,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 USA 2Department of Zoology, University of Wisconsin-Madison, Madison, Wisconsin 53706 USA

Abstract. Periodical cicadas present numerous puzzles for biologists. First, their period is ®xed, with individuals emerging as adults precisely after either 13 or 17 years (depending on species). Second, even when there are multiple species of either 13- or 17-year cicadas at the same location, only one or rarely two broods (cohorts) co-occur, so that periodical adults appear episodically. Third, the 13- or 17-year periods of cicadas suggest there is something important about prime numbers. Finally, single broods can dominate large areas, with geographical boundaries of broods remaining generally stable through time. While previous mathematical models have been used to investigate some of these puzzles individually, here we investigate them all simultaneously. Unlike previous models, we take an explicitly evolutionary approach. Although not enough information is known about periodical cicadas to draw ®rm conclusions, the theoretical arguments favor a combination of predator satiation and nymph competition as being key to the evolution of strictly ®xed periods and occurrence of only one brood at most geographical locations. Despite ecological mechanisms that can select for strictly ®xed periods, there seem to be no plausible ecological mechanisms that select for periods being prime numbers. This suggests that the explanation for prime-numbered periods, rather than just ®xed periods, may reside in physiological or genetic mechanisms or constraints. Key words: Allee effects; evolution of periodicity; Magicicada; rock±paper±scissors competition; spatial dynamics.

INTRODUCTION cicadas only emerge at a given location once or rarely twice every 13 or 17 years, causing a strikingly epi- Since the discovery of the periodical cicadas, Mag- sodic pattern of species that, when present, are strik- icicada spp., in eastern North America some 300 years ingly noticeable.

Special Feature ago (Oldenburg 1666, Walsh and Riley 1868), biolo- Periodical cicadas present numerous biological puz- gists have been fascinated by their periodicity. There zles. What were the evolutionary forces that created are seven species of periodical cicadas that divide into synchrony in the emergence of broods? Not only is two categories: four species that live for 13 years, and emergence timed to be exactly 13 or 17 years, but three species that live for 17 years (Williams and Simon emergence in the spring occurs over a narrow window, 1995, Marshall and Cooley 2000). Generally, the geo- with most individuals emerging over a few days (Wil- graphical ranges of the 13- and 17-year cicadas are liams and Simon 1995), implying strong selection for nonoverlapping, with 13-year cicadas occurring to the the emergence of large numbers of cicadas together. south and west of 17-year cicadas. Periodical cicadas Why does only one brood typically dominate in a given emerge as a group in late spring, forming large and geographical location? This suggests that there are ad- noisy mating congregations for roughly a month, and vantages not only in emerging within a large brood, then the adults die. Because individuals emerge after but also in emerging periodically so that cicadas are exactly 13 or 17 years (depending on the species), the not present every year. And why are the periods of populations are divided into discrete cohorts, or broods, periodical cicadas both prime numbers? Since period that emerge together. In any one geographical location, length is evolutionarily labile, this seems hardly co- there is typically only one or rarely two broods of a incidental. The seven species actually consist of three given species, and when there are multiple species sets of sister species that are morphologically, behav- (which is often the case), their broods emerge in the iorally, and (in some cases [Martin and Simon 1988]) same years (Lloyd and Dybas 1966a, Dybas and Lloyd genetically distinct, with each of the three sets con- 1974, Williams and Simon 1995). Therefore, periodical taining one 17-year species, and one or two 13-year species (Simon et al. 2000, Cooley et al. 2003). The Manuscript received 25 October 2004; accepted 15 December difference among the species within these three sets is 2004; ®nal version received 11 February 2005. Corresponding Editor: A. A. Agrawal. For reprints of this Special Feature, see primarily period length, suggesting that period length footnote 1, p. 3137. can change evolutionarily between 13 and 17 relatively 3 Corresponding author. E-mail: [email protected] easily. 3200 December 2005 EMPIRICALLY MOTIVATED ECOLOGICAL THEORY 3201

These questions have a mathematical component, might suggest a very long-term effect of the previous and it was for this reason that periodical cicadas were emergence, although this pattern was far less striking selected as a topic for an undergraduate summer re- than the short-lived increase in abundance of other spe- search program involving all of the authors. The goal cies following emergence. of this program was to give undergraduate students Nymphs feed on xylem from the roots of a variety experience in research at the interface of biology and of deciduous tree species (White and Strehl 1978). In mathematics. Because the question of periodical cicada a study investigating underground survival, Karban periodicity is inherently both biological and mathe- (1997) found high nymph mortality in the ®rst two matical, it gave a compelling empirical problem to years following egg laying, but subsequent low mor- demonstrate the value of mathematics in the biological tality until emergence. Furthermore, early mortality sciences. Developing ecological theory to address spe- was apparently density dependent; from ®ve study ci®c empirical problems±the theme of this Special Fea- sites, the two with much higher density than the others ture±may be useful not just in research, but also in also had higher mortality. This could be explained by education. competition among nymphs for food (Williams and Si- Below, we ®rst review the biology of periodical ci- mon 1995). Nymphs grow through ®ve instars, and cadas. We then give an overview of previous theoretical once they have reached the last nymph instar, growth models addressing periodical cicadas, showing that our is arrested even if this occurs several years before their understanding of the evolution of cicada's periodicity timed emergence at 13 or 17 years (White and Lloyd is far from complete. Using a mathematical model that 1975). Arrested growth suggests strong evolutionary incorporates and generalizes features from many pre- forces underlying the strict periodicity of emergence. vious models, we investigate different mechanisms that Occasionally, however, ``mistakes'' are made, in which could be responsible for the evolution of strict peri- case a 17-year cicada generally emerges at 13 years, odicity. We then investigate the more speci®c problem although rarer mistakes are made in which emergence of explaining prime-numbered periods. Finally, we de- is off by only one year (Williams and Simon 1995). velop a spatial model of periodical cicada dynamics, Several authors argue that the phenotypic differentia- Feature Special which gives a strong argument in favor of one of the tion between 13- and 17-year periods may be governed several mechanisms that could lead to cicada period- by a single, diallelic locus (Lloyd et al. 1983, Cox and icity. Carlton 1988), suggesting that there is a genetic switch mechanism between prime-numbered periods. STUDY SYSTEM The life cycle and key ecological features of peri- THEORETICAL APPROACH odical cicadas are reviewed in depth by Lloyd and Dy- Previous models bas (1966a, b) and Williams and Simon (1995); here we present an abbreviated overview to explain the con- Numerous theoretical models have been used to in- struction of models. The adult stage starts as a given vestigate mechanisms that could create cicada peri- brood of nymphs digs its way from the ground at night odicity. Here we review some of the key models that to emerge over a 7±10 d period (Heath 1968, Williams explore different facets of periodicity. et al. 1993). After mating in large congregations, fe- Hoppensteadt and Keller (1976) and Bulmer (1977) males lay up to 600 eggs. Nymphs then hatch, drop to investigated mechanisms that could drive periodicity. the forest ¯oor, and burrow to underground roots. The Both investigations demonstrated that a combination total adult life span is roughly 2±6 weeks. Adult mor- of nymph competition and predator satiation could lead tality from bird can be high (Karban 1984), to one or a few broods dominating other broods and although birds become satiated when emerging broods driving them to extinction. Nymph competition acts are large (Karban 1982, Williams et al. 1993). Ander- across broods, with high enough densities of one brood son (1977) and Nolan and Thompson (1975) both re- leading to the suppression of other broods in following corded an increase in ¯edging success of birds during years. This process is exacerbated by predator satiation years of cicada emergence, showing that cicadas rep- (Bulmer 1977). When there is predator satiation, large resent an important food source for some species of broods are favored over small broods, since by satiating birds. In a recent study, Koenig and Liebhold (2005) their predators large broods have higher per capita sur- analyzed 37 years of North American Breeding Bird vival. Thus, once a brood is diminished to levels at Count data for 24 species that potentially eat cicadas. which predator satiation is no longer strong, the brood Of these, 15 showed some population abundance re- will be extinguished. Similar effects could be produced sponse to cicada emergences, with one species (the via rather than predator satiation (May Red-headed Woodpecker) showing an 18% increase in 1979), and between-year effects of parasitism or pre- abundance in the year following emergence. Increases dation could serve like competition to cause high den- in abundance, however, tended to be short-lived, lasting sities of one brood to reduce the density of broods in 1±3 years. A few species had lower than average abun- following years (Bulmer 1977, Behncke 2000). These dance in the years preceding cicada emergence, which models all begin with the assumption that cicadas have Special Feature oehri hyiiilyeeg erapart. year 1 emerge emerge never years, initially will they 156 if phenotypes within together 14-year least and at 12- together while co-occur, emerge phenotypes will 13-year hybridize; and they to 12- likely prime-period example, more for that be if, is fact in explanation might this phenotypes of 1988). dif®culty Carlton and A (Cox in mates time few ``wrong'' with the numbers parental small at neither emerged of hence period and the phenotype had hybrids if period or determining length, mechanism ge- the experienced in hybrids breakdown hybridization if of netic either chance phenotypes; disfavored be the other would limits with this because hybridization favored, pe- are 13- Prime-numbered 12-year, riods co-occurred. (e.g., etc.) but 14-year, periods factor, year, strict some different by cy- established numerous Yoshimura life previously periodic strictly 1988, were that cles assumed Carlton is it Here and 1997). (Cox avoid to evolution hybridization of consequence the as periods prime biologi- are which of unlikely. both cally emergence, cicada to cou- tightly pled ®tness and generations periodic strictly hav- ing predators upon relies gen- cicadas mechanism prime-period erating this cicadas Nonetheless, di- to periods. leads prime or eventually with This lengthened year. period one by with minished predators ci- and allowed mutant cicadas introducing then by ``evolve'' (2002) to predators al. and with et cadas Markus periodic cohort. strictly single predators themselves the a be and (i.e., to period synchronized), assumed strict were was with population broods single entire have cicada as to restrict- time ed were same cicadas (2001), the Webb Unlike at emergence. reproduce they im- when positive pact a experience predators and predators, emerge with they when impact negative a experience cicadas oscillations. three-year or tend two- that show have dynamics to that creating and schedules predators two- fecundity or strict predators, driven of externally cycles either three-year prime on generating relies for numbers mechanism This non- prime-period and with periods. cicadas of prime levels hypothetical dominate broods predation to 3, tend high hence or escape that 2 18 frequently by and cicadas 10 divisible between not numbers are only Because three. the or are two either primes by predation divisible high years and in rates, abundances, This predator periods. high three-year to or leads two- either with ``quasi-cyclic'' of or dynamics cyclic idea having pred- species of an cicada consist that ators on assumed (2001) Building Webb (1977), numbers. Gould prime are periods place. ®rst the peri- single in how a evolve than a might how rather have odicity dominant, address hence become only and might They brood emergence cycle. at life age periodic ®xed same the 3202 nlgopo oesi agl ebl explaining verbal, largely is models of group ®nal A which in model stylized a built (2002) al. et Markus cicada why addressed has work theoretical Recent IOA EMN-IBRHE AL. ET LEHMANN-ZIEBARTH NICOLAS tbebudre eaaigbroods. separating apparently boundaries and stable sharp with emergence, cicada we spatial of the Finally, pattern addressing models numbers. mathematical no prime not of are know do (e.g., periods 1977) broods why Bulmer few address 1976, a Keller only and of Hoppenstaedt dominance the to factors leading address address that models explicitly and processes, models sur- evolutionary are Few periodicity incomplete. cicada prisingly underlie could that anisms estemxmmraie eudt oexp( to fecundity realized maximum ( the sets where ϭ p aiso phenotype a of namics selection natural driving 1974). forces (Maynard-Smith the elucidate not none- they theless may process, evolutionary models genotypic the evolutionary capture haploid moth- Although their of ers. phenotype the have modeling that haploid, offspring is female evolution that assume we protoperiodic cicadas, in periodicity genetics of the evolution is know the underlying not time do generation we variable Because offspring. how among in differ that notypes periodicity. of evolution the histor- of the pattern and the ical hybridization about of assumptions consequences would unveri®able genetic this many because too 1997), and require (Cox Yoshimura Age 1988, Ice last Carlton gen- the during historical periodicity not and of do eration hybridization We about periodicity. ideas of address evolution the driving main the dom- force as to others brood over model mechanism of one the pattern suggests ination of spatial the version prime. whether spatial being investigate a periods present of we ex- explanation Finally, to possible it modify a and then plore developing we model, After ``base'' periodicity. the analyzing of lead evolution might the that ad- to factors to multiple possible it simultaneously makes dress model amalgamated mod- cicada This existing els. of the collection of the many in amalgamates found features that model a constructed we ept ueosmdl,asesaottemech- the about answers models, numerous Despite ucinlrsos) hl as while response), functional I rmalpeoye mrigi year in emerging phenotypes all from ia n eudt feegdaut sgvnby given is adults emerged of fecundity and vival eoe togr(yeI ucinlresponse). functional II (type stronger becomes hs let Thus, oa dl est.If density. adult increasing with total satiate predators which at rate the mines h est fpeaosi h ero emergence, of year the in predators of density the h rdtratc rate, attack predator the fislf yl n h variability the and cycle life its of h aemdli g tutrd olwn h dy- the following structured, age is model base The periodicity, cicada of evolution the investigate To ntebs oe,w osdrteeouino phe- of evolution the consider we model, base the In 1,..., x x p,v Ј , p,v p,v Ј x ( p p,v ,t q, ,t q, ro fa of brood ) ( oe tutr n analysis and structure Model ,t q, stedniyo ebr nymphs, newborn of density the is ) ) ϭ eoeteautdniyo the of density adult the denote ) x ( b ,t q, p,v ϭ X p,v s )exp ,teei ostain(type satiation no is there 0, stettlnme fadults of number total the is endb h enperiod mean the by de®ned eidclccd.Tesur- The cicada. periodical [] 1 r clg,Vl 6 o 12 No. 86, Vol. Ecology, b Ϫ nrae,satiation increases, 1 v ϩ bu hsmean. this about ay t, bX ( and t ) s ( r t ) 1 ), b q y deter- ( h( th t a )is (1) r is q 1 December 2005 EMPIRICALLY MOTIVATED ECOLOGICAL THEORY 3203

We assume that competition among nymphs occurs We assume that predators experience density-depen- as density-dependent mortality of newborn nymphs. dent dynamics, with yearly fecundity of exp(r2), yearly Karban (1997) showed that most nymph mortality oc- survival of m, and a cicada-independent equilibrium curred within the ®rst two years (and possibly earlier, population density of K. In the presence of cicadas, the since two years was the shortest of his sampling in- predator density y(t) has dynamics given by tervals), and spreading the density-dependent effects caX (t) of competition on nymph mortality over two years y(t ϩ 1) ϭ y(t)exp r [1 Ϫ hy(t)] ϩϩs my(t) Ά·2 1 ϩ bX (t) makes little quantitative difference to the model. The s density of newborn nymphs surviving competition is (4) ϭ Ϫ Ϫ Љ ϭ Ј ϩ ϩ Ϫ1 where h (r2 log(1 m))/r2K is a scaling term, and xp,v(q, t) x p,v(q, t)[1 dX o o(t) dX L L(t)] (2) c is the conversion rate of consumed cicadas into pred- where X0(t) is the total density of newborn nymphs, ator offspring. Note that if c ϭ 0, the predator dynamics XL(t) is the total density of nymphs between the ages are unaffected by cicada predation. of 1 and T, and the effect of competition with newborn This model is suf®ciently complex that simple an- and older nymphs increasing with d0 and dL, respec- alytical solutions do not exist. However, our goal is to tively. In this formulation, we control the window of determine only whether different explanations of ci- nymph ages that have a competitive effect on newborn cada periodicity are plausible. Therefore, we performed nymphs by setting T. Biologically, older nymphs might simulations of particularly informative cases, using a have less effect on newborn nymphs because older comparative approach to hone our understanding of nymphs feed lower in the soil and can become rela- different mechanisms that could drive the evolution of tively inactive once they reach the size, but not age, periodicity in cicadas. required for emergence. Because only single broods occur in most geographical locations, however, it is RESULTS dif®cult to study the competitive effects between dif- Strict-period broods Feature Special ferent-aged nymphs, so we have no direct information Before considering evolution, we ®rst analyze the about T. For the initial analyses we set T ϭ 2 so that model to determine factors that lead to the elimination the effect of the size of this competitive window can of broods when there is no variability in age at emer- be seen in the model results, although later analyses gence (v ϭ 0). This is the problem addressed in nu- show that T is likely to be larger. merous previous models (Hoppensteadt and Keller Finally, to model phenotypes that differ in variability 1976, Bulmer 1977, Behncke 2000) and serves to ex- in the timing of emergence, we assume that for a given pose the processes that underlie our later results on phenotype p,v, a fraction of v offspring emerges at a evolution. time different from characteristic period p. These off- We considered four cases, all of which have cicadas spring are placed into broods symmetrically before and with a strict 15-year life cycle. We selected a 15-year after the parental brood following a geometric distri- life cycle to be neutral, between 13 and 17 but not a bution. Furthermore, we assume that once nymphs have prime number; choices other than 15 yield similar re- survived their ®rst year, there is no further mortality. sults. Case I includes predator satiation (b Ͼ 0inEq. Thus, the density of adults of brood q of phenotype p,v Ͼ 1) and between-brood nymph competition (dL 0 and in year t ϩ p is T ϭ 2 in Eq. 2) in which higher densities of 1- and 2- year-old nymphs decrease the survival of newborn 1 Ϫ␯ x (q, t ϩ p) ϭ xЉ (q, t) nymphs. Simulations show that only a few of the p,v2 p,v broods persist (Fig. 1A). The reason for this can be pϪ1 1 Ϫ␯ seen in Fig. 1B, in which the density of adults in gen- ϩ␯͸ ͦiͦ Љ ϩ ϩ xp,v(q i, t i). (3) ␶ϩ iϭϪ(pϪ1) 2 eration 1 is plotted against the density of adults in generation ␶ that emerged 15 years previously. These We model haploid evolution by assuming that with plots were constructed by varying the density of adults ϭ Ј probability P wxp,v(q, t) a small number of individ- (and hence newborns) in generation ␶ while preserving uals (speci®cally, a density of 0.01) ``mutate'' from the density of 1- and 2-year-old underground nymphs phenotype p,v1 to a phenotype p,v2, thus permitting and hence the strength of competition experienced by mutations between phenotypes with different variabil- newborns. The 15 separate lines in Fig. 1B correspond ity v but the same mean period p. The parameter w to the 15 broods, with lower lines indicating stronger scales the probability of mutation relative to the pop- competition from previous broods. The sigmoidal ulation density; larger populations are more likely to shape of the lines shows the effect of predator satiation; produce a mutant. The value of w ϭ 0.01 is set low so if too few adults emerge, per capita predation is high that mutations are relatively rare, occurring with prob- and survival is low, leading to an . For some ability 0.1 in a population with density 10 (which is broods, the density of adults in generation ␶ϩ1is typical for dominant broods in our simulations). always lower than the density in generation ␶, indi- Special Feature idwo este ngeneration in a is densities there of broods, other window For densities. declining cating ( satiation predator with F) (E, III case 4), Eq. 3, ϭ 3204 occd rdto ( predation cicada to nrae nrsos occd rdto ( predation cicada to response in increases iiso iaa admyfo nfr distribution uniform a from randomly cicadas of sities yp optto ( competition nymph estab- initial the in broods. dynamics of transient lishment of is role there because large the dif®cult, a about broods persisting initial conclusion of absolute on number any dependence determine makes This that conditions dominate. dynamics broods transient which resulting broods of the densities initial and the on that part broods in of depends number persist the that Note density. in increase we rosb eoigbt ewe-ro com- between-brood both ( petition removing by broods tween brood densities. which initial during on dynamics depends result transient success a initial as largely the consis- persist, of broods although emergence adjacent surprisingly, two of tently, Somewhat years 1D). to pred- (Fig. in corresponding spikes densities create and ator others two result, the a eliminate As pre- broods 1C). higher (Fig. to broods subsequent leads on that dation density in increase an causing rdtrstain( satiation predator F .,E.2,cs I(,D ihpeao aito ( satiation predator with D) (C, II case 2), Eq. 0.2, h dl ouaindniyfrec fte1 rosi ahgnrto,adtebto ae ie h ubro adults of number the gives panel bottom the and generation, each in broods 15 the of generation each in for predicted density population adult the IG ncs I easmdta hr sn between-brood no is there that assumed we II, case In aaee ausntlse bv o pc® ae are: cases speci®c for above listed not values Parameter ncs I,w lmntdaydrc neato be- interaction direct any eliminated we III, case In and K .Bodeiiainfrcs A )wt rdtrstain( satiation predator with B) (A, I case for elimination Brood 1. . ϭ 35. d L ϭ )adpeao erdcini response in reproduction predator and 0) b ϭ d 0, c L ␶ϩ ϭ ϭ K ϭ ) u rdtrreproduction predator but 0), ) n eetdiiilden- initial selected and 0), safnto ftenme ngeneration in number the of function a as 1 ,E.4 n ewe-ro yp optto ( competition nymph between-brood and 4) Eq. 6, IOA EMN-IBRHE AL. ET LEHMANN-ZIEBARTH NICOLAS ␶ htla oan to lead that c Ͼ b ϭ ) thus 0), b )advraini nta iaadniy n aeI G )without H) (G, IV case and density, cicada initial in variation and 1) ϭ r )adpeao erdcini epnet iaapeain( predation cicada to response in reproduction predator and 1) 1 ϭ eosrt htti fetaoecnla oteevo- the we to later lead periodicity. can and of alone lution satiation, effect den- predator this initial that is of demonstrate importance there the when emphasize sities to case, it trivial a use is we 3.5 this Although roughly 1F). of (Fig. density persistence for initial an pred- requires by created satiation effect ator persist. Allee the in- broods because of simply three is absence This only the broods, Despite 1E). between (Fig. teractions 4 and 0 between 3, ϭ idct,bcuei l rosprita ihdensity, high pe- at strict persist broods of all evolution if the because is of riodicity, density) component in reduction necessary strong a least at Brood (or broods. few elimination a just by dominance and elimination ( erdcin( reproduction nto,a losonb umr(1977). Bulmer by shown elim- also brood as for essential ination, not is satiation of predator 1H). (Fig. Thus, between-brood rate rate replacement the growth because below populations population broods, some the of decreases subset competition a persistence only the cause of to suf®cient is competition brood d ial,i aeI ermvdpeao aito ( satiation predator removed we IV case in Finally, d L l orcssaaye bv aers obrood to rise gave above analyzed cases four All )bticue ewe-ro yp competition nymph between-brood included but 0) 0 Ͼ ϭ )i h bec fccd-eedn predator cicada-dependent of absence the in 0) 0.05, b ϭ ␶ d for )adbtenbodnmhcmeiin( competition nymph between-brood and 1) L c ϭ vlto fperiodicity of Evolution d ␶ϭ ϭ L 0, ϭ )(i.1) nti ae between- case, this In 1G). (Fig. 0) a .) h o ae o ahcs gives case each for panel top The 0.2). cssIII and I±III) (cases 8 ϭ 0.35, r 2 ϭ clg,Vl 6 o 12 No. 86, Vol. Ecology, 0.2, ␶ϭ c ϭ 0(aeIV). (case 40 0, m ϭ 0.75, c d ϭ b L December 2005 EMPIRICALLY MOTIVATED ECOLOGICAL THEORY 3205 pca Feature Special

FIG. 2. Evolution of ®xed periods for case I (A) with predator satiation (b ϭ 1) and between-brood nymph competition ϭ ϭ ϭ (dL 0.2), case II (B) with predator satiation (b 1) and predator reproduction in response to cicada predation (c 3), case III (C) with predator satiation (b ϭ 1) and variation in initial cicada density, and case IV (D) without predator satiation ϭ ϭ ϭ (b 0, K 6) and between-brood nymph competition (dL 0.2). For each case, all initial individuals had phenotype 15,0.2 in which 20% of offspring emerge in less than or more than 15 years. Mutations to phenotypes 15,0.1 and 15,0 occurred at low probability. Parameter values are as in Fig. 1; v is variability about the mean period, p. density-dependent processes affecting cicada survival competition and predator satiation caused the severe and reproduction will not vary strongly with the timing reduction of some broods. As a result, the strictly pe- of emergence. Can all four mechanisms causing brood riodic phenotype was favored because it did not lose elimination drive evolution of strict periodicity? individuals that emerged in years with reduced adult To investigate the evolution of periodicity, we con- densities and hence higher per capita predation rates. sidered three phenotypes, each with a mean period of Similarly, case II leads to the severe reduction in den- 15 years, but differing in variability in period from v sity of some broods and the consequent advantage to ϭ 0.2 (20% of the brood having a period different from the strictly periodic phenotype (Fig. 2B). Thus, pred- 15 years) to v ϭ 0.1 and v ϭ 0 (strict period). We ator satiation and an increase in predator reproduction assumed that initially the entire population consisted in response to cicada predation caused the evolution of of phenotype p ϭ 15, v ϭ 0.2, and allowed the other strict periodicity. two phenotypes to arise via mutations. We selected For case III we assumed that the fecundity of adults initial densities to be the same for all broods. None- (regardless of phenotype) emerging in the same year theless, even if some broods were greatly reduced in was a random variable; speci®cally, the value of r1 was density, broods never went extinct as long as variable- drawn from a normal random variable each year. This period phenotypes persisted, because variability in gen- resulted in variation in densities among broods, and eration time always allowed the repopulation of broods. when broods dropped below the threshold of the Allee In case I, between-brood nymph competition and effect, they would often continue to decline to very predator satiation rapidly led to the evolution of a strict- low densities (Fig. 2C). This created an advantage for ly periodic phenotype (Fig. 2A, v ϭ 0) as the two the strictly periodic phenotype, even in the absence of phenotypes with variable generation times were ex- between-brood nymph competition and predator repro- cluded. This occurred because between-brood nymph duction in response to cicada predation. Finally, case Special Feature ess u ssplne yprosta lo syn- allow that periods broods. among by not emergence supplanted does chronous is phenotype 13-year but the persist periodicity, strict of evolution favoring cases; thus other occurs, the elimination brood for when similar are only nymph results 16- between-brood present the and competition, and we satiation predator 12- Although for years, I favored. case four are to cicadas lengthened year is delay the Conversely, years. if two fa- every are emerge broods period and even-numbered vored, of broods 1- newborn suffering then only on nymphs, effect that brood competitive so a large shortened, have a nymphs is year-old after delay the years If two competition. time follow- broods and a with one system, sets the ing in this years com- nymphs, three a of newborn delay have on nymphs effect 2-year-old petitive and 1- we that Because assumed competition. consequence between-brood a of is form the years of of three selection every emerge The that satiation. pheno- broods multiple predator from guarantees the adults types of because density sense, high makes resulting broods of This years. synchronization three these every synchronously and emerged respectively, broods persisted, the cicada cicadas 12-year of the 15-year Examination of broods and 3). 5 and (Fig. 4 years that showed 15 model and pe- 12 strict of with riods cicadas favored evolution competition, phenotypes. between competition of form the took cicadas. evo- of the periodicity for strict responsible which the into be of insight lution fact any in of give might evolution not mechanism the does it for periods, elimination strict re- brood central of the pe- highlights quirement strictly this the While favored phenotype. riodic evolution density, in reduced 2D). density (Fig. in broods reduction between-brood some severe of provided the cause periods can competition strict the for essential of not is evolution satiation predator that shows IV 3206 eidct lopooesnhoosemergence synchronous promote strict of also evolution the periodicity promote and elimination brood pheno- considered we length, types period and To of periodicity years. selection strict 15 of the evolution of the period both expected investigate an have that notypes v te rm ubr ouain eesatdwt all with all no started of but were broods 13 Populations encompass number. to prime periods other of range this selected ujc ocag.Teeoe eeto faperiod a of selection Therefore, change. to subject ain ol hnetevraiiyi ieccelength cycle life in ( variability the change mu- could While tations mutations. via generated were phenotypes 0 ,w sue htteepce eid( period expected the that assumed we ), o aeIwt rdtrstainadbetween-brood and satiation predator with I case For were broods some which in cases all in summary, In sfudi h ateape hs atr htcause that factors those example, last the in found As phe- only considered we analyses, preceding the In p,v for hntpswt utpeperiods multiple with Phenotypes v p rm-ubrdperiods Prime-numbered ϭ ϭ . hntps n the and phenotypes, 0.2 21 and 12±16 v ϭ .,01 n .We 0. and 0.1, 0.2, IOA EMN-IBRHE AL. ET LEHMANN-ZIEBARTH NICOLAS v p ϭ a not was ) . and 0.1 p ekt lo h eidt eoe®e.W could ( We periodicity selected simultaneously ®xed. strict that for become values to too parameter period ®nd is not the periods allow variable to over pe- weak ®xed nonprime for of selection dominance riods, for resulting selection and strong remove synchrony suf®ciently to with satiation scenario predator the 16 weak in and Nonetheless, 12 between 4). periods (Fig. other over periods 13-year simultaneously. evolve can ask pe- riodicity prime-numbered explicitly and periodicity and strict both periods, whether variable allow we with he whereas phenotypes that periods, strict in have (2001) Webb cicadas ci- from assumes on differs model predation Our by cadas. caused reproduction increased maintained is by the dynamics In predator in This fecundity. cyclicity age-dependent reproduce. the of model, they case year extreme and the the respectively) in is year, cicadas third on or feed only second their (in we life once reproduce predation, three-year only predators of a that periodicity (unrealistically) with assumed the other one accentuate the predators, To and of cycle. types two-year two a are with there in (2001) that Webb followed supposing we To numbers, satiation. prime predator for predator between- mild select no very with having imposed IV of we satiation, case instead yet to competition, similar brood situation in the scenario started a with we prime- selected, ®nd are of to periods prime-numbered attempt evolution which an the In explain periods. to This numbered denominator. dif®cult common it small makes a have that favoring periods thus periods, different with individuals among oryas(ly n ya 1966 Dybas least broods and at (Lloyd nonsynchronized by years separated which four are in they geographically, cases overlap rare and areas, the contiguous large in sin- dominate which often in broods cicadas gle by shown pattern the spatial ignoring striking periods, prime-numbered and periodicity h cnrow osrce osfvrccdswith cicadas favor does constructed we scenario The el 98,mkn togto rtreya signals three-year or unlikely. two- predation strong in (Go- making survivorship 1998), annual constant telli show and their in year breed cicadas, ®rst of predators predation. main in the birds, cycles and Most strong three-year requires and it two- because simultaneous likely, is favoring periods mechanism this prime that think not do we thermore, ytasaigorbs oe Es ±)ot spatial a onto 1±3) (Eqs. model base our translating by com- nymph important. that distri- is implies petition geographical cicadas the periodical that of Wil- bution (1995) of Simon suggestion and the Specif- liams test others. mathematically the we than ically, stronger whether of is one ask mechanisms that we these suggest patterns Here geographical periods. observed strict the drive could of that evolution mechanisms four the least at are there that ofrw aefcsdo h vlto fstrict of evolution the on focused have we far So ecntutdasailmdlo iaadynamics cicada of model spatial a constructed We pta patterns Spatial v ϭ )ad1-erpros Fur- periods. 13-year and 0) clg,Vl 6 o 12 No. 86, Vol. Ecology, b .W aeshown have We ). December 2005 EMPIRICALLY MOTIVATED ECOLOGICAL THEORY 3207 pca Feature Special

FIG. 3. For case I, evolution of ®xed periods when multiple periods are possible. Initially all individuals were equally distributed among phenotypes p ϭ 12±16, v ϭ 0.2. Random mutations occurred to phenotypes with v ϭ 0.1 and v ϭ 0 and the same period p as the parental phenotype. Parameter values are as in Fig. 1 for case I, except that K was lowered from 35 to 30; the combination of between-brood competition and K ϭ 35 drove the entire initial cicada population extinct. grid of 50 ϫ 50 cells with wrap-around (torus) bound- (Fig. 1), this caused the cicada population to go extinct. aries. The dynamics of cicadas and predators within Therefore, for the spatial simulations we reduced K, each cell were governed by the base model, and each the carrying capacity of predators in the absence of year 5% of cicadas and 20% of the predators dispersed cicadas. We only considered cicadas with a strict 13- to one of the adjacent four cells before reproduction. year period, assuming that a strict period had previ- Although this is a simplistic depiction of dispersal, ously evolved. Finally, we initially populated space by more complex descriptions of dispersal are unlikely to randomly selecting 13 cells on the grid and placing one give qualitatively different results. Cicada dispersal of the 13 broods in each. creates low-density populations in cells adjacent to es- To summarize the results of the spatial model once tablished populations, and these have low population the grid is fully occupied, we report the average number growth rates due to lack of predator satiation. For the of broods occurring in each cell with a density above values of predator carrying capacity K used previously the threshold of 1% of the maximum density found Special Feature 3, 3208 ntal l niiul eeeulydsrbtdaogphenotypes among distributed equally were individuals all Initially F predators. d hntpswith phenotypes IG 0 ϭ .Frtecs ihpeaosta ae®e w-adtreya iecce,eouino rm-ubrdperiods. prime-numbered of evolution cycles, life three-year and two- ®xed have that predators with case the For 4. . 0.05, d L v ϭ ϭ 0.2, . and 0.1 a ϭ 0.35, v ϭ n h aeperiod same the and 0 r 2 ϭ IOA EMN-IBRHE AL. ET LEHMANN-ZIEBARTH NICOLAS 0.2, c ϭ .,and 0.2, p K steprna hntp.Prmtrvle are: values Parameter phenotype. parental the as ϭ ,where 7, p ϭ K 12±16, stejitcryn aaiyo ohtpsof types both of capacity carrying joint the is v ϭ ..Rno uain curdto occurred mutations Random 0.2. clg,Vl 6 o 12 No. 86, Vol. Ecology, b ϭ 0.02, r 1 ϭ December 2005 EMPIRICALLY MOTIVATED ECOLOGICAL THEORY 3209 among cells. For the four cases described previously, there were 5.81 Ϯ 0.16, 3.70 Ϯ 0.34, 10.57 Ϯ 1.83, and 13 Ϯ 0 broods per cell (mean Ϯ SD of 10 simu- lations). These numbers change with changes in model parameters; for example, increasing cicada dispersal from 5% increases the average number of broods at any location. In cases I±III, the spatial distribution of broods 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 of broods per cell, case II (with predator reproduction depending on cicada predation), there were more than three broods on average per location, in contrast to the usual situation in nature of only one or rarely two broods. In the model, the only plausible way to reduce the number of broods at the same location is to increase the length of time over which an emerging brood could affect subsequent broods. In the base model, the most biologically realistic way of generating the requisite long-term effects is via nymph competition (Williams and Simon 1995). Until now, we have assumed that only 1- and 2-year-old nymphs have a competitive ef- fect on newborn nymphs (T ϭ 2 in Eq. 2), yet it is likely that older nymphs also have a competitive effect. Feature Special 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 generate high predator densities that are then maintained for many years. Although there is equivocal evidence that this is possible for a few bird species, the increase of bird abundance due to cicada emergences is generally short-lived (Koenig and Liebhold 2005). Fig. 5 gives two illustrative examples of increasing FIG. 5. Spatial distribution of 13 cicada broods simulated the length of time over which nymphs have a com- ona50ϫ 50 grid when 5% of cicadas and 20% of predators petitive impact on newborn nymphs. In the ®rst (Fig. move from their natal cell each generation. (A) Competition from only nymph age classes 1±6 (T ϭ 6, Eq. 2) affects 5A), nymphs have a competitive effect on newborns ϭ newborn nymphs, and per capita competition is strong (dL up to the age of 6 (T ϭ 6 in Eq. 2), and we made this 0.6). For clarity, the range of one brood is shaded black and competition strong so that broods readily excluded each a second is shaded gray. (B) Competition from all nymph age ϭ ϭ other; the mean number of broods per location was 1.5. classes (T 12), with weaker per capita competition (dL 0.08). The 13 broods are numbered. Other parameter values In this scenario, however, complex spatial patterns ϭ ϭ ϭ ϭ ϭ ϭ are: b 1, r1 3, d0 0.05, a 0.35, r2 0.2, c 0, and arise, rather than the observed dominance of a given K ϭ 15. brood over a wide geographical region. Furthermore, the boundaries are not stationary; the spiral patterns rotate slowly through time. These spatial dynamics are tributions of broods emerge, with a mean number of the result of a nontransitive hierarchy in competitive broods per location of 1.09. This case more accurately dominance. For example, a brood emerging in 1990 is mimics reality where the geographical ranges of broods are contiguous and the boundaries appear to be static. dominant to a brood emerging in 1994, which is turn Thus, it is likely that all, or at least most, nymph age is dominant to a brood emerging in 1999. But the brood groups are involved in competition. Even if we are emerging in 1999 is dominant to the brood emerging wrong in assuming that nymph competition is the most in 2003, which is the same brood that emerged in 1990. likely process responsible for between-brood interac- This type of rock±paper±scissors hierarchy is known tions, whatever factor creates between-brood interac- to produce complex, nonstationary spatial patterns tions must act over the long term, so that broods emerg- (Durrett and Levin 1998, Frean and Abraham 2001; D. ing 12 years previously have an effect on newborns. S. Griffeath, personal communication). In the second example (Fig. 5B), all nymphs have a DISCUSSION competitive effect on newborns (T ϭ 12 in Eq. 2). In Using a suite of models, we investigated possible this case, well-de®ned and stationary geographical dis- mechanisms for the evolution of strict periods in pe- Special Feature tanprost rm ubr.Atog eetstud- recent con- Although numbers. emergence prime to synchronize periods strain to needed counting physiological/genetic mechanisms the (regard- and period), emergence of synchronous less involve for select directly ecological that processes not is possibility does One processes. numbered ecological prime are riods numbers. prime model generated and explanation that another periods, devise not gives prime-numbered could model we for our and that explanation think plausible not a do did we periodicity Thus, strict occur. of satiationÐsatiation evolution not that predator weak so weak was assume that to had syn- periods, we different for with selection rates. phenotypes of across predation force chronization the in reduce cycles simulta- to three-year Furthermore, and and prime strong for two- required selects neous that this model although a create denominators. numbers, to common able small were with We syn- periods for for select and hence periods, also different with forces cicadas selective among chronization same the emergenc- and synchronized peri- es, for of selection evolution requires the odicity that is generating periods in prime-numbered dif®culty The periods. prime-numbered cicadas of evolution periodicity. the predator to processes and key were competition satiation faith, nymph of between-brood leaps several that with albeit would suggests, emer- This periodic it gence. strictly for cicadas, selection strong periodical created have the If of in nymphs. history strong newborn evolutionary was on competition nymphs nymph of between-brood ages all of competitive a effect and can satiation predator broods is there of when partitioning model occur spatial spatial strong Our that 1995). con®rms Simon ci- and of (Williams pattern cadas distribution current the explaining strong factor potentially a suppress be- as competition implicates can nymph This tween-brood years. emergence many brood for broods large there subsequent a when 2003), that even al. et brood suggests (Cooley co-occurring single species a domi- multiple by are locations often with most cicadas, nated of pattern distribution pe- cicada of evolution the to riodicity. lead can multiple that are there scenarios Thus, 2). strict (Fig. times of emergence var- iable having evolution create phenotype cicada the that initial an to from cases periodicity lead These also elimination satiation. brood predator ab- of and the in sence densities; competition nymph population between-brood IV) cicada (case sa- random predator III) and (case predation; tiation reproduction cicada predator to and response in satiation predator competition; II) pred- nymph I) (case between-brood (case 1): and (Fig. that satiation cycle cicada ator life hypothetical periodic strictly lead a a of can has broods that of cases elimination four to showed We cicadas. riodical 3210 essetta h ehns xliigwype- why explaining mechanism the that suspect We have cicadas why address however, not, does This spatial The likely? most is scenarios these of Which IOA EMN-IBRHE AL. ET LEHMANN-ZIEBARTH NICOLAS eid r nieyt aea clgclexplanation. ecological an have to unlikely prime-numbered are that between-brood periods believe explaining we and in although important periodicity, likely satiation are competition predator Bulmer nymph and that (1976) Keller (1977) on and expand Hoppensteadt and potentially of con®rm that those results Our processes periodicity. the to crystallize lead to us al- models lowed ex- the together new taken ®nd Nonetheless, to periodicity. article prime-numbered lengths particularly this periodicity, greater for in planations apparent to be went might we cicada than for though explanations even and possible periodicity, uncover the in not of periodicity all did of investigate surely evolution we the Furthermore, explain cicadas. any results that our prove of cannot We phenomenon. biological strik- a ing underlying pos- processes explore the and about hypotheses formulate sible to used was model speci®c explana- nonecological for tions. looking suggests prime-numbered periods to lead could that scenarios deriving ecological in dif®culty support our to Nonetheless, precedent speculation. no this is there therefore biological and other are systems, in examples 18 mechanisms no to ®nd counting 12 can dual-clock We years of 17. of and range 13 the numbers prime for the which by divisible 3, not or are 2 that either years other identify would the to system clock cicadas and dual allow years a Such two years. three counts counts that that short- one two only clocks: using term periods speculation, A long count pure 1991). cicadas albeit that Carlton is this, and for (Cox explanation periods possible 17-year between switch and genetic share a 13- be that to seems there species al. thus, et 2000); sister (Simon traits behavioral of and morphological sets many 17- within and have species 13- studies year between Simon genetic associations and close and (Williams demonstrated ecological years Second, they 13 1995). ``mistakes,'' in make generally cicadas emerge 17-year a periods. when suggest prime First, evidence for of explanation mech- types counting Two physiological/genetic actual unknown. the is 2001), anism host al. their et (Karban by by trees experienced years changes count cicadas seasonal periodical detecting that shown have ies enk,H 00 eidclccds ora fMathe- of Journal cicadas. Periodical 2000. H. sparrows of Behncke, responses Reproductive 1977. R. T. Anderson, umr .G 97 eidcliscs mrcnNaturalist American insects. Periodical 1977. G. M. Bulmer, ahmtclSine UM rn oA .Ie n .A. P. and and Ives R. Biological A. in to Milewski. grant Undergraduates (UBM) Sciences Interdis- for Mathematical Foundation project Science Training This National manuscript. U.S. ciplinary a a by sharing funded for was Liebhold Sandy and hspoethsgvna xml nwihasystem- a which in example an given has project This aia Biology matical Condor supply. food superabundant a to 111 etakKle imnfrcmet ntemanuscript, the on comments for Tilmon Kelley thank We :1099±1117. 40 A L :413±431. CKNOWLEDGMENTS ITERATURE C ITED clg,Vl 6 o 12 No. 86, Vol. Ecology, 79 :205±208. December 2005 EMPIRICALLY MOTIVATED ECOLOGICAL THEORY 3211

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