Prime Survival – Food, Sex and Death

David G. Roberts

(Image source: http://www.cicadamania.com/cicadas/cicadas-and-prime-numbers/)

Background Mathematicians have been fascinated by prime numbers for millennia. And, for great reason, prime numbers provide the building blocks for all of the natural numbers. Indeed, every positive number greater than one is itself a prime or composite number. Prime numbers are only divisible by themselves and one, whereas it is possible to disassociate composite numbers into a unique set of ‘other’ numbers. As it turns out, those ‘other’ numbers or factors, as they are known in mathematics, are always prime numbers. Eratosthenes of Cyrene (ca. 275 – 194 BC), a prominent Greek scholar, and at that time, head of the Alexandrian Library, made contributions to many fields of learning, including poetry, astronomy, history, geography and mathematics. The mathematical contributions for which he is most well-known include approximation of the circumference of Earth, and a method with which to distinguish between prime and composite numbers up to a designated limit. This method of finding prime numbers is known as the ‘sieve of Eratosthenes’17. However, at the time of Eratosthenes’ discovery, a biological- mathematical generator of prime numbers, even if only of the small ones close to zero, was already in existence elsewhere in the world23&24 (Figure 1), and probably in Greece too20. 6&12 Indeed, periodical cicadas provide a fascinating example of ‘primes in nature’ . “An insect is said to be periodical if the life cycle has a fixed length of k years (k > 1) and if adults do not appear every year but only every kth year”7. Biologists and mathematicians alike, have attempted to explain this phenomenon for decades, even suggesting the occurrence of periodical cicadas as one of only a few genuine applications of mathematics as explanatory of a natural phenomenon2,3,4,5&22.

Figure 1. A fossilised cicada wing from the Mesozoic Era, a period of geological time from about 252 to 66 million years ago, found in 24 Australia . Cicadas are a type of insect found on every continent in the world except Antarctica. Their quintessential songs have been playing for orders of magnitude longer than there have been humans on Earth to hear them24. With its Mediterranean climate, it would have been a very rare ancient Greek indeed, not to have heard the rampant love songs of cicadas8. While today cicada songs do not always make favourable impressions13, they seemingly captivated the ancient Greeks, who revered the little beasts. According to Rory B. Egan, an eminent Greek historian and languages expert turned cultural entomologist, cicada manifestations were many and varied through Greek antiquity, including in literature, the visual arts, philosophy, religion and scientific writing9. For example, they were considered a symbol of death and rebirth, even immortality, as inferred from the story of Tithonus, presumably due to their intermittent emergence and apparent birth from Earth, a consequence of their partially hidden lifecycle (refer below). Aristotle, the great Greek polymath, wrote about their lifecycle and periodic emergence in his book Historia Animaliumcited in 9. While other examples abound, a translation by Egan of a poem from Anacreontea, a collection of largely unknown authored works from between 100 BC and 600 AC, probably encapsulates best the feelings the ancient Greeks had for cicadas:

We know that you are royally blest Cicada when, among the tree-tops, You sip some dew and sing your song; For every single thing is yours That you survey among the fields And all the things the woods produce. The farmers’ constant company, You damage nothing that is theirs; Esteemed you are by every human As the summer’s sweet-voiced prophet. Muses love you, and Apollo too, Who’s gifted you with high pitched song. Old age does nothing that can wear you, Earth’s sage and song-enamoured son; You suffer not, being flesh-and-blood-less – 9 A god-like creature, virtually. In Australia, the songs of cicadas are a harbinger of summer, and then when in full chorus, a distinctive soundtrack, to which the overwhelming majority of people are familiar. Cicadas and their songs are ubiquitous, occurring all over the continent, in bushland, in cities and in suburbia21. This is perhaps not surprising given that at any one time, in any one location, there may be several different cicada species up in the trees (or underground in the soil…but wait…their songs are loud and distinctive, as too is their urine, which floods down from the canopies of trees …surely the latter cannot be true….?). Currently, 300 species are scientifically described and formally recognised, most of which belong to a 18, 19&21 biological superfamily called the Cicadoidea . The songs, which are only vocalised by male cicadas to attract females with which to mate, signify the final few weeks of life. It is a period in which there is a flurry of promiscuous, unadulterated sex. However, not long after it begins (or even before) they die (a Greek tragedy perhaps…?), either through natural senescence, fungal attack, to which many cicada species are highly susceptible, or predation by voracious vertebrate predators, namely birds, reptiles, and even fishes15,25&26. Yet, as much as their persistent singing (or deafening drone…?) and abundant shed nymphal skin on the ground or on the trunks of trees implies that they are eminently and conspicuously present (Figure 2), few people know anything about how they came to be up in the trees.

Figure 2: The shed nymphal skin of perhaps 100 or more cicadas, a sure sign that summer has arrived in Australia. Note the adult cicada on the trunk of the tree…Can you locate it? (Image source: http://www.smh.com.au/environment/animals/suicide-song-cicada- sex-racket-risks-death-for-chance-at-love-20131230-3036d)

All cicada species, irrespective of where in the world they are found, have a fascinating, multistage lifecycle. After mating in the treetops, female cicadas lay eggs in splits in tree stems, from which about a month later emerge wingless cicadas called nymphs. Nymphs disperse to the ground via gravity, and then burrow into the soil. Once underground, they tunnel through the soil until they find a plant’s roots. Then, using piercing sucking mouthparts, they attach to roots, from which they slurp sap out of the internal transport system (xylem and phloem). None of this is particularly exciting. What is remarkable is the length of time that some cicada nymphs spend underground attached to a plant’s roots (with occasional dispersion) – seventeen years! Yes, that is right, seventeen years sucking sap! That seems more like a prison sentence, than an existence! However, suddenly they all emerge en masse from the soil, every single last one of them, perfectly synchronised, shedding their subterranean skin (Figure 3), and then taking flight to the trees to eat, court (or be lured), mate and then die, completing their lifecycle18.

Figure 3: An Australian green grocer cicada, Cyclochila australasiae, emerging from its subterranean skin. (Image source: https://commons.wikimedia.org/wiki/File:Green_grocer_cicada_molting.jpg)

The most well-known periodical cicadas are the Magicicada species’ from North America, which spend either 13 or 17 years underground12,16&26. The eminent palaeontologist and natural historian Stephen J. Gould (1941 – 2002) hypothesised about the possible link between the mathematics of prime numbers and cicada periodical life-cycles. Gould stated: “I am most impressed by the timing of the cycles themselves. Why do we have 13 and 17 year cicadas, but no cycles of 12, 14, 15, 16, or 18? They are large enough to exceed the life cycle of any predator, but they are also prime numbers….Many potential predators have 2-5 year life cycles”11. So, are 13 and 17 year periodical cicadas in North America merely a coincidence? With many potential predators (refer above), of which any by chance may have matching reproductive cycles, which then would be a good year for cicadas to emerge and disperse to the treetops? Well, we know that prime numbers are number greater than one and only divisible by themselves and one; they cannot be further divided into smaller subsets of numbers. Therefore, any prime number year would limit contact with potential predators, irrespective of the years in which predators themselves appeared in nature. As Gould points out, many predators have lifecycles that track two to five year cycles11. Therefore, it seems that in choosing a prime life-cycle, some cicada species have arrived at an optimal strategy with which to avoid predators! So, how do cicadas distinguish the prime years? Well, they do not. They are cicadas! They know only about food and sex, and nothing at all about prime numbers! Essentially, it is a numbers game, played out over evolutionary time. The observed prime lifecycles are likely a result of Darwinian natural selection. Cicadas that emerged to complete their lifecycles in years divisible by many prime factors likely met their end at the hand of swarms of hungry predators. At this point, it is important to recall that for all living organisms, behaviours, traits, etcetera, are judged by whether or not they provide an ‘advantage’. Here, ‘advantage’ implies a greater than before chance of survival, and hence reproduction. Cicada species, which by chance acquired a gene mutation/s conveying long, prime- numbered life cycles, would have had a selective advantage. In other words, they fared best, survived the longest, and hence contributed the most offspring to the next generation. Simple right…? Possibly not so; while many cicada species have prime life-cycles, many 27 others do not; there are other competing hypotheses . So, what about Australian cicadas? As far as we know, most Australian cicada species make their way to the surface earlier than North America Magicicada species, although studying cicadas requires abundant patience, and almost all understanding of the basic biology of most of the 300 or so currently described species (not to mention the likely hundreds of other species that exist in Australia but are yet to be described or even discovered) is in its infancy. Indeed, much remains to be learnt14. Nevertheless, early observational studies suggests that some of the larger cicada species in south eastern Australia emerge in accordance with either 5- or 7-year cycles underground, for example, the green grocer Cyclochila australasi, and the black prince, Psaltoda plaga (Figures 4 & 5)10&18. On the west coast of Australia, around Perth, the Sandgrinder cicada, Arenopsaltria fullo, predominates, but not much is known about its basic biology (Figure 6). Let us now explore primes and the potential interaction between cicadas and their predators using the sieve of ‘Eratosthenes’.

Figure 4. The green grocer cicada, Cyclochila australasiae, from southeastern Australia, probably has a 7-year periodical life cycle. (Image source: https://commons.wikimedia.org/wiki/File:Cyclochila_australasiae.jpg)

Figure 5: Psaltoda plaga, commonly known as the black prince, an Australian cicada species that may have a 7-year periodical life cycle. (Image source: https://commons.wikimedia.org/wiki/File:Black_Prince_Cicada_03.jpg)

Figure 6: The sandgrinder cicada, Arenopsaltria fullo, a large cicada found on the coast from the Murchison River south to Augusta, Western Australia. It is common in endangered Banksia Woodlands of the Swan Coastal Plain ecological community. (Image source: https://www.friendsofqueensparkbushland.org.au/arenopsaltria-fullo/) Teaching Resource: Prime survival – food, sex and death; number relationship investigations

Primes and the sieve of Eratosthenes Eratosthenes devised a way in which to distinguish between prime and composite numbers (Figure 7). His method is known as the sieve of Eratosthenes. To determine with the sieve of Eratosthenes all prime numbers between 1 and some initial upper limit, defined as n, serially list them, and then cross them off in accordance with the rules below:

First, cross off 1, which is a numerical unit, not a prime or composite number.

Then, let p equal 2, the smallest prime number in existence. All multiples of 2 (that is, 2p, 3p…) are composite numbers, they all can be reduced and written as products of prime numbers (e.g. 2 × 2 = 4, 3 × 2 = 6….). Therefore, cross off all multiples of p by counting in increments of p, beginning with 2p (i.e. 2p = 4) and ending at n. Then, discern the first integer > p = 2 that is not crossed off. Redefine p as this new number, which itself is a prime. For integers between 1 and 100, the next prime number is 3. Cross off multiples of p = 3, beginning with 2p (i.e. 2p = 6) and again ending at n (or until there are no numbers remaining). Continue this iteration until there is no such number > p. All numbers that remain in the list are prime numbers below the limit n.

Figure 7: Eratosthenes of Cyrene (ca. 275 – 194 BC). (Image source: https://en.wikipedia.org/wiki/Eratosthenes) 1. Use the grid (Figure 8) on the following page to determine all of the prime numbers between 1 and 100. Write them down.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Figure 8: A grid of the numbers 1 to 100.

The sieve and exploration of periodical cicada predators Consider the North American periodical cicadas, which spend either 13 or 17 years underground, and the possible Australian periodical cicadas, the green grocer (Cyclochila australasi) and black prince (Psaltoda plaga) (Figures 4 and 5), which may spend either 5- or 7- years underground. Now consider them in the context of the life cycle of their potential predators. 2. If an insect predator had a 1-year life cycle, how many generations of that insect would have to pass before it could feed on each of the different kinds of periodical cicadas listed above?

3. If a predator had a 2-year life cycle, how many generations would pass before it could feed on the cicadas?

4. What about predators with 3-year life cycles, how many generations would pass before they could eat cicadas?

Now, consider ‘cicada predators’ that produce new generations (complete their life cycle) every 1, 2, 3….9 generations. Using a different coloured pencil for each predator e.g. red for the 1-year predator, blue for 2-year predator, and green for the 5-year predator etc. place a diagonal strike through each year a particular predator produces a new generation (i.e. completes its life cycle) (e.g. Figure 9), from which a tally of the total number of predators for any particular year can be calculated (use the grid in Figure 10 to do this).

Figure 9. A grid of the numbers 1 to 10 with 1-, 2- and 5-year predators depicted (refer to text above for further detail). Construct a key adjacent to Figure 10 to show which predators emerge, and in which years, throughout the 100-year period. Use Table 1 (below) as a guide to consolidate the predator emergence patterns displayed in Figure 10, and then use the data to produce a frequency histogram graph (a page on which to produce your graph is provided below) of the number of predators present per year, for the first 25 years. First consider a simple assumption about the predator-prey relationship between ‘cicada predators’ and the ‘cicadas’; assume that the greater the number of predators that are present in any particular year, the greater the number of cicadas that will be eaten, and hence the fewer adult cicadas that there will be available to produce the next generation of cicadas. The implication is reduced reproductive output, putting those cicadas at a selective disadvantage (refer above to background).

Figure 10. A grid of the numbers 1 to 10.

Table 1. Numbers of cicada predators per year, for predators with life cycles of between 1- and 9-years duration.

Year Tally Frequency

1 I 1

2 II 2

. . .

25 II 2

5. In which year/s would cicada emergence be particularly disadvantageous for cicada survival? In other words, which years have the greatest numbers of predators?

6. How many potential predators would cicadas have to evade if they emerged every 6, 8 or 18 years?

The cicadas being considered above do not emerge every 6, 8 or 18 years, they emerge either every 5 or 7 years (Australian species), or 13 or 17 years (North American species). Return to Figure 10 and use a new unique colour to outline the grid-boxes to show the emergence patterns of the cicadas, and their association with predators. Indicate cicada emergence on your graph too.

8. Is there an easier method with which to discover the potential predator life cycles that will coincide with cicada emergences? Firstly, write down what is meant by the terms ‘factor’ and ‘product’. Provide examples to illustrate your thinking.

9. Write each of the following numbers as a product of prime numbers:

12 14 24 36 40 76 84 90 368 748 960 1000

Rationale for prime survival and links to the Australian Curriculum I have designed an interdisciplinary resource that transcends disciplinary boundaries. It was inspired by my background and interests in ecology and evolutionary biology, but has a mathematical foundation in prime numbers. Indeed, ‘prime cicadas’ may represent one of the first ever prime number generators, even before the prominent ancient Greek mathematician Eratosthenes had considered ways in which to distinguish prime from composite numbers. They may also provide one of only a few genuine mathematical explanations of natural phenomena, albeit initially through chance, and then by the hand of evolution, facilitated by immense geological time. The resource is intended for use in the mathematics classroom as an introduction to ‘number patterns and properties’, including prime numbers and prime factors (ACMNA149)1 (Table 2), but could also be used as presented, or modified accordingly, for age-appropriate learning in the science classroom. For teachers, the implication here is, use as much or as little as required, depending on the student cohort. Areas where the resource may be useful include: biological life cycles, classification and evolution, all of which span years 7 to 10 of the Australian Curriculum1.

Table 2. Interconnection of ‘prime survival’ and the Australian Curriculum.

Curriculum Curriculum Description of curriculum content connections reference

Mathematics ACMNA149 Investigate index notation and represent whole

numbers as products of powers of prime numbers.

Science ACSSU111 Classification helps organise the diverse group of

organisms.

ACSSU112 Interactions between organisms, including effects of human activities can be represented by food chains and food webs.

ACSHE223 Science knowledge can develop through collaboration across the disciplines of science and the contributions of people from a range of cultures.

ACSIS129 Construct and use a range of representations, including graphs, keys and models to represent and analyse patterns or relationships in data using digital technologies as appropriate.

ACSIS130 Summarise data, from students’ own investigations and secondary sources, and use scientific understanding to identify relationships and draw conclusions based on evidence.

ACSSU176 Ecosystems consist of communities of interdependent organisms and abiotic components of the environment: matter and energy flow through these systems.

ACSSU184 Transmission of heritable characteristics from one

generation to the next involves DNA and genes.

ACSSU185 The theory of evolution by natural selection explains the diversity of living things and is supported by a range of scientific evidence.

References 1. Australian curriculum, Assessment and Reporting Authority (ACARA). (2015). The Australian Curriculum: Science. Retrieved from http://www.australiancurriculum.edu.au/ 2. Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223-238. 3. Baker, A. (2009). Mathematical explanation in science. British Society for the Philosophy of Science, 60, 611–633. 4. Baker, A. (2017). Mathematical spandrels. Australian Journal of Philosophy, 95, 779– 793. 5. Bangu, S. I. (2008). Inference to the best explanation and mathematical realism. Synthese, 160, 13–20. 6. BBC Earth. (2017, May 5). 17 year periodical cicadas – Planet Earth – BBC Earth [video file]. Retrieved from https://www.youtube.com/watch?v=EWr8fzUz-Yw

7. Bulmer, M. G. (1977). Periodical insects. American Naturalist, 111, 1099–1117. 8. Cicada Song. (n.d.). Songs of European cicadas. Retrieved from http://www.cicadasong.eu/cicadidae.html 9. Egan, R. (1994). Cicadas in ancient Greece: ventures in classical tettigology. Cultural Entomology Digest, 3, 20–26. 10. Emery, D. L., Emery, S. J., Emery, N. J., & Popple, L. W. (2005). A phonological study of the cicadas (Hemiptera: Cicadidae) in western Sydney, New South Wales, with notes on plant associations. Australian Entomologist, 32, 97–110. 11. Gould, S. J. (1977). Ever Since Darwin. Reflections in Natural History. Wrights Lane, London, England: Penguin Books. 12. Grant, P. R. (2005). The priming of periodical cicada life cycles. Trends in Ecology and Evolution, 20, 169–174. 13. Heydeman, S. (Producer) (2013, December 13). Sounds of cicadas not music to ears [audio podcast]. Retrieved from http://www.abc.net.au/radionational/programs/bushtelegraph/bt- december-13/5154262 14. Jones, A. (2015, February 9). Inside the lifecycle of Australian cicadas. Retrieved from http://www.abc.net.au/radionational/programs/offtrack/inside-the-lifecycle-of-australian- cicadas/6075298 15. Koenig, W. D., & Liebhold, A. M. (2013). Avian predation pressure as a potential driver of periodical cicada cycle length. American Naturalist, 181, 145–149. 16. Lehmann-Ziebarth, N., Heideman, P. P., Shapiro, R. A., Stoddart, S. L., Hsiaco, C. C. L., Stephenson, G. R., Milewski, P. A., & Ives, A. R. (2005). Evolution of periodicity in periodical cicadas. Ecology, 12, 3200–3211. 17. Merzbach, U. C., & Boyer, B. (1991). A history of mathematics (3rd ed.). Hoboken: John Wiley & Sons. 18. Moulds, M. S. (1990). Australian cicadas. Kensington: New South Wales University Press. 19. Moulds, M. S. (2012). A review of the genera of Australian cicadas (Hemiptera: Cicadoidea). Zootaxa, 3287, 1–262.

20. Pinto-Juma, G. A., Quartau, J. A., & Bruford, M. W. (2009). Mitochondrial DNA variation and the evolutionary history of the Mediterranean species of Cicada L. (Hemiptera, Cicadoidea). Zoological Journal of the Linnean Society, 155, 266–288. 21. Popple, L. W. (n.d.). A web guide to the cicadas of Australia. Retrieved from http://dr- pop.net/cicadas.htm 22. Rizza, D. (2011). Magicicada, mathematical explanation and mathematical realism. Erkenn, 74, 101–114. 23. Sota, T., Yamamoto, S., Cooley, J. R., Hill, K. B. R., Simon, C., & Yoshimura, J. (2013). Independent divergence of 13- and 17-y life cycles among three periodical cicada lineages. Proceedings of the National Academy of Science, 110, 6919–6924. 24. Tillyard, R. J. (1921). Mesozoic insects in Queensland. No. 8 Hemiptera Homoptera (contd). The genus Mesogereon; with a discussion of its relationship with the Jurassic Palaeontinidae. The Proceedings of the Linnaean Society of New South Wales, 46, 270– 284. 25. Tokue, K. & Ford, H. A. (2006). Influence of food and nest predation on the life histories of two large honey eaters. Emu, 106, 273–281. 26. Williams, K. S., & Simon, C. (1995). The ecology, behaviour, and evolution of periodical cicadas. Annual Review of Entomology, 40, 269–295. 27. Yoshimura, J. (1997). The evolutionary origins of periodical cicadas during ice ages. The American Naturalist, 149, 112–124.

Recommended reading for teachers 11. Gould, S. J. (1977). Ever Since Darwin. Reflections in Natural History. Wrights Lane, London, England: Penguin Books. 12. Grant, P. R. (2005). The priming of periodical cicada life cycles. Trends in Ecology and Evolution, 20, 169–174.

Recommended reading (and viewing plus listening) for students 6. BBC Earth. (2017, May 5). 17 year periodical cicadas – Planet Earth – BBC Earth [video file]. Retrieved from https://www.youtube.com/watch?v=EWr8fzUz-Yw 13. Heydeman, S. (Producer) (2013, December 13). Sounds of cicadas not music to ears [audio podcast]. Retrieved from http://www.abc.net.au/radionational/programs/bushtelegraph/bt- december-13/5154262 14. Jones, A. (2015, February 9). Inside the lifecycle of Australian cicadas. Retrieved from http://www.abc.net.au/radionational/programs/offtrack/inside-the-lifecycle-of-australian- cicadas/6075298 21. Popple, L. W. (n.d.). A web guide to the cicadas of Australia. Retrieved from http://dr- pop.net/cicadas.htm