Periodical Cicadas Research Articles

Avian Predation Pressure as a Potential Driver of Periodical Cicada Cycle Length

The extraordinarily long life cycles, synchronous emergences at 13- or 17-year intervals, and complex geographic distribution of periodical cicadas (Magicicada spp.) in eastern North America are a long-standing evolutionary enigma. Although a variety of factors, including satiation of aboveground predators and avoidance of interbrood hybridization, have been hypothesized to shape the evolution of this system, no empirical support for these mechanisms has previously been reported, beyond the observation that bird predation can extirpate small, experimentally mistimed emergences. Here we show that periodical cicada emergences appear to set populations of potential avian predators on numerical trajectories that result in significantly lower potential predation pressure during the subsequent emergence. This result provides new support for the importance of predators in shaping periodical cicada life history, offers an ecological rationale for why emergences are synchronized at the observed multiyear intervals, and may explain some of the developmental plasticity observed in these unique insects.

At the limits: habitat suitability modelling of northern 17‐year periodical cicada extinctions (Hemiptera: Magicicada spp.)

AbstractAimAdult periodical cicadas emerge as temporally isolated, synchronized multispecies communities (‘broods’) that are for the most part geographically contiguous and that fit together in jigsaw‐puzzle‐like fashion. Some year‐classes of 17‐year cicadas have become extinct within historical times. We investigate two general causes for these extinctions – anthropogenic habitat destruction and post‐glacial climate change.LocationPeriodical cicadas are confined to the eastern United States, east of the Great Plains. We document the locations of known periodical cicada extinctions in two broods of 17‐year cicadas in Connecticut, Rhode Island, and upstate New York, USA.MethodsUsing additional distributional records of 17‐year cicadas, we develop habitat suitability models for all 17‐year periodical cicadas, using data layers that reflect both ecological and anthropogenic factors.ResultsClimatological data layers related specifically to annual mean temperature and temperature during the warmest months make the greatest contributions to our models, and data layers most specifically related to deforestation and habitat fragmentation tend to make much smaller contributions. Two well‐documented extinct populations of periodical cicadas occurred in locations where these models predict relatively low habitat suitability for 17‐year cicadas.Main conclusionsOur results and other circumstantial evidence discount the importance of anthropogenic habitat destruction in explaining these particular extinctions.

Morphological conservatism in the foreleg structure of cicada hatchlings, Burmacicada protera n. gen., n. sp. in Burmese amber, Dominicicada youngi n. gen., n. sp. in Dominican amber and the extant Magicicada septendecim (L.) (Hemiptera: Cicadidae)

Two cicada hatchlings (Hemiptera: Cicadidae) in Burmese and Dominican amber are described as Burmacicada protera n. gen., n. sp. and Dominicicada youngi n. gen., n. sp., respectively. Although very similar in appearance, the two species can be separated by body contour, the nature of the process on the terminal antennomere and the shape and size of protrusions, teeth and spines on the forelegs. A comparison of the forelegs of the fossil hatchlings with those of an extant hatchling of the periodical cicada, Magicicada septendecim (L.), reveals a remarkable degree of morphological conservatism over 100 million years. A brief review of fossil cicadas is presented.

Resource pulses of dead periodical cicadas increase the growth of American bellflower rosettes under competitive and non-competitive conditions

This experiment investigates how pulsed nutrient resources interact with intraspecific competition to influence biomass production and nutrient use efficiency in the American bellflower (Campanulastrum americanum). The competitive environment of these plants was manipulated by growing plants alone or with neighboring conspecifics, and the occurrence of pulsed nutrient resources was manipulated through the addition of 17-year periodical cicada (Magicicada septendecim) carcasses in order to simulate naturally occurring pulsed resources in this system. The addition of cicada carcasses increased mean plant biomass by 61 % compared to non-supplemented plants, while competition decreased mean plant biomass by 44 % compared to plants grown without competition; these effects were additive. In comparison, nutrient use efficiency decreased in fertilized plants (cicada-supplemented plants showed 20 % greater foliage nitrogen concentrations compared to non-supplemented plants), but was not significantly affected by the plant’s competitive environment. In addition, cicada supplementation did not significantly increase the biomass asymmetry in competing pairs of plants. These results suggest that these plants increased their total nutrient uptake at a timescale commensurate with the pulsed increase in nutrient availability due to cicada carcass decomposition.

The use of Citizen Scientists to Record and Map 13-Year Periodical Cicadas (Hemiptera: Cicadidae: Magicicada) in South Carolina

Citizen science is the process in which citizens are involved in science as researchers (Carr 2004). This can range from involvement in a collaborative effort from academic and government agencies in the assessment of a common environmental (community) concern to the management of common natural resources (Conrad & Hilchey 2011). Traditionally, the use of citizen scientists involved the collection of data on species detection and habitat. This is due to the difficulty of a single monitoring agency in obtaining accurate data for proper population assessment (Conrad & Hilchey 2011). Increasing concern regarding the effect of climate change and land development on biodiversity has raised interest in assessing evolutionary responses of plants and animals, particularly for species that are absent in some years but abundant in others (Heliovaara et al. 1994; Lepetz et al. 2009). Brood XIX of periodical cicadas (Hemiptera: Cicadidae: Magicicada) consists of 3 species (M. tredecassini, M. tredecim, M. tredecula) that emerge synchronously every 13 years. Historically, Brood XIX has been recorded to span along the United States east coast from Maryland to Georgia and in the Midwest from Iowa to Oklahoma (Simon 1988). Males and females emerge from the ground close to their host tree and eclose into their adult form. Adults are approximately 4 cm long with red eyes, orange-veined wings and black bodies (Fig. 1). They live above ground for approximately 4-6 wk. The males produce species-specific calls that attract both sexes in the forest canopy resulting in large aggregations in areas of high population densities. After mating, females disperse to nearby trees and lay eggs into slits cut into the branches. Nymphs hatch from eggs after 6-8 wk and drop to the ground where they feed on root xylem fluids for another 13 yr (White & Strehl 1978; Williams & Simon 1995). In South Carolina, the 1985 emergence began around 21 Apr in Abbeville and Saluda counties. Cold weather delayed the emergence in 1998 resulting in fewer sightings. Based on anecdotal and general accounts, periodical cicadas were suspected to emerge in approximately 25 South Carolina counties (Gorsuch 1998). The anticipated 2011 Brood XIX emergence provided a unique opportunity to apply the use of citizen scientists in obtaining a more accurate assessment of the species range in South Carolina. The aim of the project was to use voluntary internet reports of sightings to examine spatial and temporal patterns of periodical cicada emergence and document citizen’s impressions of the Brood XIX emergence. Three South Carolina state institutions established internet sites for citizens to report sightings (Clemson University: http://clemsoncicada.word-

6-Year Periodicity and Variable Synchronicity in a Mass-Flowering Plant

Periodical organisms, such as bamboos and periodical cicadas, are very famous for their synchronous reproduction. In bamboos and other periodical plants, the synchronicity of mass-flowering and withering has been often reported indicating these species are monocarpic (semelparous) species. Therefore, synchronicity and periodicity are often suspected to be fairly tightly coupled traits in these periodical plants. We investigate the periodicity and synchronicity of Strobilanthes flexicaulis, and a closely related species S. tashiroi on Okinawa Island, Japan. The genus Strobilanthes is known for several periodical species. Based on 32-year observational data, we confirmed that S. flexicaulis is 6-year periodical mass-flowering monocarpic plant. All the flowering plants had died after flowering. In contrast, we found that S. tashiroi is a polycarpic perennial with no mass-flowering from three-year individual tracking. We also surveyed six local populations of S. flexicaulis and found variation in the synchronicity from four highly synchronized populations (>98% of plants flowering in the mass year) to two less synchronized one with 11–47% of plants flowering before and after the mass year. This result might imply that synchrony may be selected for when periodicity is established in monocarpic species. We found the selective advantages for mass-flowering in pollinator activities and predator satiation. The current results suggest that the periodical S. flexicaulis might have evolved periodicity from a non-periodical close relative. The current report should become a key finding for understanding the evolution of periodical plants.

Permanence induced by life-cycle resonances: the periodical cicada problem

Periodical cicadas are known for their unusually long life cycle for insects and their prime periodicity of either 13 or 17 years. One of the explanations for the prime periodicity is that the prime periods are selected to prevent cicadas from resonating with predators with submultiple periods. This paper considers this hypothesis by investigating a population model for periodical predator and prey. The study shows that if the periods of the two periodical species are not coprime, then the predator cannot resist the invasion of the prey. On the other hand, if the periods are coprime, then the predator can resist the invasion of the prey. It is also shown that if the periods are not coprime, then the life-cycle resonance can induce a permanent system, in which no cohorts are missing in both populations. On the other hand, if the periods are coprime, then the system cannot be permanent.

Developmental Plasticity of Life-Cycle Length in Thirteen-Year Periodical Cicadas (Hemiptera: Cicadidae)

Abstract Speciation in periodical cicadas (Magicicada Davis) is closely tied to changes in life-cycle length, which presents a paradox because these organisms depend on emergence synchrony for survival. Recently proposed speciation models invoke developmental plasticity as a possible solution: Environmentally triggered “4-yr accelerations” occur in 17-yr cicadas, suggesting that canalization of induced plasticity could change 17-yr populations into temporally isolated 13-yr populations. However, the reverse shift, 13-yr cicadas emerging in 17 yr, has never been documented. We searched 4 yr after the normal emergence of a 13-yr brood (and in a year with no expected periodical cicada emergences anywhere) and found periodical cicadas active at 26 of 92 sites, with examples of all four 13-yr species. At one location, we found evidence of at least 1,724 cicadas per ha emerging. Few males were heard singing at most sites, so these off-schedule cicadas apparently did not survive long in the face of predation. We also found one 13-yr species singing 8 yr late within the range of a different 13-yr brood, suggesting an 8-yr delayed emergence or consecutive generations of 4-yr delayed cicadas. Developmental plasticity in life-cycle length seems to be similar in 13- and 17-yr cicadas—both types possess the ability to switch to the opposite life cycle and to emerge 1 yr early and/or late. The confirmation of a reverse life-cycle switch in 13- cicadas suggests improvements to theories of life-cycle evolution in Magicicada and strengthens the case for developmental plasticity in speciation.

Life Cycle Replacement by Gene Introduction under an Allee Effect in Periodical Cicadas

Periodical cicadas (Magicicada spp.) in the USA are divided into three species groups (-decim, -cassini, -decula) of similar but distinct morphology and behavior. Each group contains at least one species with a 17-year life cycle and one with a 13-year cycle; each species is most closely related to one with the other cycle. One explanation for the apparent polyphyly of 13- and 17-year life cycles is that populations switch between the two cycles. Using a numerical model, we test the general feasibility of life cycle switching by the introduction of alleles for one cycle into populations of the other cycle. Our results suggest that fitness reductions at low population densities of mating individuals (the Allee effect) could play a role in life cycle switching. In our model, if the 13-year cycle is genetically dominant, a 17-year cycle population will switch to a 13-year cycle given the introduction of a few 13-year cycle alleles under a moderate Allee effect. We also show that under a weak Allee effect, different year-classes (“broods”) with 17-year life cycles can be generated. Remarkably, the outcomes of our models depend only on the dominance relationships of the cycle alleles, irrespective of any fitness advantages.

Avian predators are less abundant during periodical cicada emergences, but why?

Despite a substantial resource pulse, numerous avian insectivores known to depredate periodical cicadas (Magicicada spp.) are detected less commonly during emergence years than in either the previous or following years. We used data on periodical cicada calls collected by volunteers conducting North American Breeding Bird Surveys within the range of cicada Brood X to test three hypotheses for this observation: lower detection rates could be caused by bird calls being obscured by cicada calls ("detectability" hypothesis), by birds avoiding areas with cicadas ("repel" hypothesis), or because bird abundances are generally lower during emergence years for some reason unrelated to the current emergence event ("true decline" hypothesis). We tested these hypotheses by comparing bird detections at stations coincident with calling cicadas vs. those without calling cicadas in the year prior to and during cicada emergences. At four distinct levels (stop, route, range, and season), parallel declines of birds in groups exposed and not exposed to cicada calls supported the true decline hypothesis. We discuss several potential mechanisms for this pattern, including the possibility that it is a consequence of the ecological and evolutionary interactions between predators of this extraordinary group of insects.

Tomorrow People

Tomorrow People Greg Hrbek (bio) The year they took San Francisco and my sister forever away, I was still a baby. But in 2038, I was eleven years old, going on twelve. My family lived, in those days, on a cul-de-sac named after an extinguished tribe of Native Americans, in a house nearly a century old, a split-level ranch from the days of roof antennas. Next door, in a house of nearly identical size and design, lived my best friend since kindergarten, a boy who'd learned during the course of our fifth-grade genealogy project that there'd been a lynching in his family tree. In tribute to that great-uncle, my friend had changed his name to Zebedee. Only I still called him Plaxico. Of the twelve families in our subdivision, five (including Plaxico's) were black; seven (counting mine) white. It was an aberration of racial demographics that, since I had moved there, eight years earlier, we'd had no brown-skinned neighbors—until Abdelkarim Hussein Mohamed al-Nasr became the son and heir of Franklin D. Banfelder, a widower who every summer would bestow ice pops upon the children of the neighborhood, the same way he'd bestowed chocolate, back in his days as a contract soldier, upon the children of the faraway homeland of Abdelkarim. Mr. Banfelder had left town a week after school ended, not telling anyone where he was going. No sooner did he disappear than the periodical cicadas tunneled out of their underground cells for the first time since the year 2021. One evening around sunset, we heard a whisper everywhere. The nymphs were coming out. Shuffling by the millions to the trunks of trees and crawling up to the leafy branches. They made no sound that night. By the next morning, they had started to molt. I discovered one in my window, bonded to the fine wires of the screen. The insect wasn't free of itself yet. The nymphal body had dehisced down the back, a divide [End Page 13] so clean it looked like the work of a surgeon; and the new body, white and waxen, was humping through the cleft. After the back, the head began to break open—the eyes diverged and new eyes came into view, looking directly into mine. By the time Mr. Banfelder's micro-compact reappeared, our world was dizzy with their noise. We'd heard a rumor. Mr. B had gone to Iraq. Gone by himself but come back with a kid. My friends and I rode over there, the tires of our bikes crushing a crust of shed exoskeletons; and when Mr. B answered the doorbell and saw the four of us on the porch, he got a look on his face like he'd been waiting all morning, and said: "Boys, there's someone I want you to meet. Dorian, Dean, Zeb, Damian . . . This is Abdelkarim." And there he was. The kid who was like us except he wasn't, because his skin wasn't white and it wasn't black. (It was, I would later determine, the shade of the crayon in my box called "tumbleweed.") "Does he speak English?" Dean asked. "Ask him," said Mr. B, lighting up a green. "He doesn't bite." "Okay. Do you speak English?" "Qalilan," the kid said. "That means 'a little bit' in Arabic." Then Mr. Banfelder spoke to the boy in his native tongue, nodded encouragingly, pointing to the four of us. Finally, the kid said: "I pledge allegiance . . . to the flag . . . of . . ." "Of . . . the United States . . ." "United States . . ." "Of . . ." "Of . . ." "America." "Amereeka." I wondered if they were going to recite the whole thing word by word, then maybe sing the national anthem. But Mr. Banfelder just nodded, as if very proud of the boy's effort. At long last, he said: "Ice pops, anyone?" We followed him into the kitchen, where there were now five chairs around the glass table instead of the usual four. "Take a load off," said Mr. B. "How have you boys been? I leave for a week and return to find a plague being visited. What the hell'd you do while...

Plant architecture affects periodical cicada oviposition behavior on native and non‐native hosts

Variation in plant quality provides a basis for oviposition site selection for a variety of insects. Of the plant traits that influence plant–insect interactions, plant architecture has received little attention despite its putative role in modulating oviposition behavior. In a common garden comprised of native and non‐native plant species, we assessed how host plant architecture and identity influenced the oviposition behavior of 17‐year periodical cicadas (Homoptera: Cicadidae: Magicicada). On each host, we quantified the availability of branches suitable for oviposition and compared those measures with the branches used by ovipositing cicadas. Using this approach, we determined how the structural attributes of plants (i.e. branch diameter, length and incline) affected oviposition site selection. We then related cicada oviposition preferences to offspring performance by quantifying egg hatching success. On each host species, cicadas selectively used broader and longer branches for oviposition, suggesting that branch architecture provides a basis for oviposition behavior irrespective of plant identity. Broader and longer branches were more abundant on native than on non‐native hosts in our study, contributing to greater oviposition loads among the native species. Egg hatching success was similar among native and non‐native hosts. However, it is possible that the use of native plants for oviposition could enhance offspring output because native hosts generally contained more viable eggs per egg nest and more egg nests per plant. While previous accounts of cicada oviposition preferences have focused on differences in oviposition loads among hosts, our evaluation of within‐host branch selection by ovipositing cicadas helps to clarify oviposition preferences at a higher resolution and demonstrates that plant architecture provides an important basis for oviposition behavior. Furthermore, because branch structure can differ substantially among host species, our results suggest that periodical cicadas may be sensitive to the changes in plant composition that often result from non‐native plant invasions.

Feeding ecology and evidence for amino acid synthesis in the periodical cicada ( Magicicada)

The periodical cicadas of the genus Magicicada (including M. septendecim, M. cassini, and M. septendecula) have the longest juvenile life span of any insect, living underground for 13 or 17 years and feeding exclusively on root xylem fluids. Due to their inaccessible life cycles very little is known about cicada nutrition, despite the fact that members of Magicicada can achieve a very large biomass in woodland habitats east of the Mississippi and hence constitute a major part of the ecosystem where they occur in high densities. Live cicadas were collected at two sites in early June of 2004, during the emergence of Brood X (both M. septendecim and M. cassini were recovered). We used a combination of stable isotopic measurements (δ 15N and δ 13C) and multivariate statistical techniques to test for differences in resource acquisition among the cicada species and sexes collected at two locations within the 17-year periodical Brood X range. The amino acid constituents of cicada chitin and organs, plus xylem extracted from a deciduous sapling, were also analyzed. The data show that male and female cicadas have different carbon fractionations, which could reflect differential resource utilization due to oviposition in females. Several essential amino acids for the cicada were absent in xylem. Carbon-isotopic composition of all amino acids in the cicadas was distinctly different from the limited set measured in the xylem. Because of the differences in isotopic composition, we conclude that amino acids were synthesized de novo rather than incorporated directly, most likely produced by endosymbiotic bacteria.

The Effect of Periodical Cicadas on Growth of Five Tree Species in Midwestern Deciduous Forests

Abstract Periodical cicada emergences in the eastern United States are one of the largest biomass fluctuations in the world, providing a unique opportunity to study if a root parasite can affect broad-scale patterns of forest growth and succession. We used dendrochronology to examine the direct effect of Brood X (17 y cicada) and Brood XXIII (13 y cicada) periodical cicadas on 89 individual trees from five species (Acer saccharum, Fraxinus americana, Quercus palustris, Q. velutina and Sassafras albidum) in Indiana. Standard dendrochronological techniques were used to date the tree-ring samples and our chronologies ranged from 63–98 y in age, spanning from four to eight cicada emergences. We removed the main climate variable from each species chronology by conducting a regression analysis and using the residuals for the remainder of the analysis. Significant climate models were developed for all five species. Acer saccharum growth correlated highest with Jun. temperature (r = −0.392), Fraxinus americana ...

Body Temperature and Responses of Periodical Cicadas (Homoptera: Cicadidae) to a Potential Predator

Brood emergences of periodical cicadas are closely synchronized resulting in a mass emergence which represents an effective anti-predatory response (Lloyd and Dybas, 1966a, b). Not only does the mass emergence swamp the local populations of birds and mammals, but the long interval between emergences (13 or 17 yr) prevents a predator population from tracking these peaks in cicada populations. Furthermore, the chorusing of densely packed males produces a sound level that can deter birds from effective hunting (Simmons et al, 1971). Individual periodical cicadas are nonetheless vulnerable given the ease at which they can be approached (Beamer, 1931; Lloyd and Dybas, 1966b). Local avian predators take substantial numbers as evidenced by the increases in their populations in the year following a mass emergence (Koenig and Liebhold, 2005). Also, small mammals such as voles, mice and shrews show substantial increases of cicadas in their diet during a mass emergence (Hahus and Smith, 1990). Steward et al. (1988) reported that many individual cicadas of the 13-year species (Magicicada spp.) responded actively to an approaching predator by flying away, falling, or by squawking in males. The cicadas changed how they responded over the emergence period; fewer individuals remained stationary after the period of peak density compared with before, and more males squawked as a part of their response near the end than at the start of emergence. The present study was intended to determine how 17-year cicadas, Magicicada cassini Fisch (Homoptera: Cicadidae), would respond to a model avian predator and to compare their responses to that reported in the literature for 13-year cicadas. Even though a periodical cicada can fly when its body temperature is above 19 °C (Heath, 1967), its body temperature may affect its decision of when to escape by flying. Several species of lizards respond differently to an approaching predator at varying body temperatures (Huey, 1982; Cooper, 2000, 2006). Therefore, the body temperature of each cicada was recorded along with its response.

Twig to Root: Egg-Nest Density and Underground Nymph Distribution in a Periodical Cicada (Hemiptera: Magicicada septendecim (L.))

The egg-laying behavior of female insects could directly benefit the mother, her offspring, or a combination of both. Periodical cicada (Magicicada septendecim (L.)) females oviposit in twigs in the forest canopy, and newly hatched nymphs fall to the ground, where they spend 17 years feeding on tree roots. If nymph dispersal from their mother's oviposition site is limited, then female oviposition site selection could influence offspring fitness and survival. Here, we show that there is no correlation between egg-nest density in the forest canopy and nymph density directly below. Although the extent of nymph post-hatching dispersal remains unknown, our findings cast doubt on direct benefits to offspring as an explanation for female oviposition site choice.

Thermal responses of periodical cicadas: within and between brood parity (Hemiptera: Cicadidae: <em>Magicicada</em> spp.)

Thermal responses of periodical cicadas: within and between brood parity (Hemiptera: Cicadidae: Magicicada spp.) Allen F Sanborn1, James E Heath21Department of Biology, Barry University, Miami Shores, FL, USA; 2Department of Physiology, University of Illinois, Urbana, IL, USAAbstract: Measurements of the thermal responses of insects have been used to determine thermal adaptation to various environments. We have compared the thermal responses (minimum flight temperature, maximum voluntary tolerance temperature, and heat torpor temperature) of five periodical cicada species (Magicicada septendecim, M. cassinii, M. septendecula, M. tredecassini, and M. neotredecim) to test the hypothesis that the thermal responses are similar within and between broods due to the similar environmental conditions experienced by each brood. Cicadas of Brood XIII were collected in 1973, 1990, and 2007 in northern Illinois. Comparisons are made to data from specimens collected during the 1987 Brood X and the 1998 Brood XIX emergences in southeastern Tennessee, and the 1989 Brood XXIII emergence in central Illinois. The maximum voluntary tolerance (an upper thermoregulatory temperature) and the heat torpor temperatures do not differ between the five species examined. Only one species (M. neotredecim) differs statistically in the minimum flight temperature from the other species and one species pair (M. septendecula and M. neotredecim) differs in heat torpor temperature. There were a few examples of between brood statistical differences in the minimum flight temperature (M. tredecassini) and heat torpor temperature (M. tredecassini, M. cassinii) and one species pair differed significantly in heat torpor temperature. The data show that thermal responses generally do not change statistically in different emergences of the same brood or over large portions of a species range and the thermal responses of all species are similar.Keywords: Magicicada, thermal responses, temperature adaptation, cicada

Effects of oviposition by periodical cicadas on tree growth

Periodical cicadas ( Magicicada spp.) occur at very high densities and synchronously emerge from underground every 13 or 17 years. During the emergence, adults lay eggs in tree branches, causing significant damage; however, the long-term impact of this damage is unknown. We conducted two large-scale field studies during the 2004 emergence of one brood (Brood X) to measure the growth of trees in relation to oviposition damage by periodical cicadas. In the first experiment, we netted areas to exclude cicadas from plots in 15 early successional forests and then measured trunk circumference for 3 years on more than 4000 trees of 52 species. In this experiment, oviposition had no detectable effect on the growth rates of trees. In the second study, we measured oviposition on 12 common tree species across six sites. We then measured the annual growth rings of these trees for 3 years after the emergence. In this experiment, oviposition was correlated with a slightly reduced growth in the emergence year and following year when the data were analyzed together, but when tree species were examined individually there were no clear effects of oviposition on tree growth. These data suggest cicada oviposition has little effect on the radial growth of trees, particularly in comparison to other factors.

Mathematical Explanation in Science

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss potential future directions of research for each side in the debate over the existence of abstract mathematical objects. 1. Introduction: Mathematical Explanation2. Indispensability and Explanation3. Is the Mathematics Indispensable to the Explanation? 3.1. Object-level arbitrariness3.2. Concept-level arbitrariness3.3. Theory-level arbitrariness4. Is the Explanandum ‘Purely Physical’?5. Is the Mathematics Explanatory in Its Own Right?6. Does Inference to the Best Explanation Apply to Mathematics? 6.1. Leng's first argument6.2. Leng's second argument6.3. Leng's third argument7. Conclusions

Allee effect in the selection for prime-numbered cycles in periodical cicadas

Periodical cicadas are well known for their prime-numbered life cycles (17 and 13 years) and their mass periodical emergences. The origination and persistence of prime-numbered cycles are explained by the hybridization hypothesis on the basis of their lower likelihood of hybridization with other cycles. Recently, we showed by using an integer-based numerical model that prime-numbered cycles are indeed selected for among 10- to 20-year cycles. Here, we develop a real-number-based model to investigate the factors affecting the selection of prime-numbered cycles. We include an Allee effect in our model, such that a critical population size is set as an extinction threshold. We compare the real-number models with and without the Allee effect. The results show that in the presence of an Allee effect, prime-numbered life cycles are most likely to persist and to be selected under a wide range of extinction thresholds.