Photo: Michigan Department of Natural Resources

The cicadas of Brood X have emerged throughout much of the eastern United States.  In certain areas of Ann Arbor, Michigan, where Mathematical Reviews is  physically located, they have become quite loud.   They belong to the genus Magicicada of periodical cicadas that emerge either in 13-year cycles or in 17-year cycles.  The cycle lengths are prime numbers, which makes mathematicians wonder why.

Brood X cicadas have been in the news in the US because they are have emerged after 17 years as larvae living underground.   Many of the stories (such as this and this) mention that the cycles are prime numbers:  Magicicada septendecim having a 17-year cycle and Magicicada tredecim a 13-year cycle.   Intuitively, this sounds interesting, but just what are the details?  The point that is generally made is that these cycle lengths help keep the cicada broods out of sync with potential predators, who instead rely on some other prey.

In 1979, Robert M. May wrote about the periodicity of cicadas in Nature.  He nicely lays out the issues and the possible explanations.  He mentions that 13 and 17 are prime, but doesn’t dwell on that, instead focusing on the relatively long duration of the nymph stage.  May is interpreting the cicadas’ life cycle as a dynamical system with some parameters.  His hypothesis is that for these particular species, the important parameters are in some special intermediate range that prevents the cicadas from being wiped out by predators or falling into an annual life cycle.  May also lays out one of the difficulties in studying cicadas with such life cycles, writing,  “Whatever their significance in the world of pedators and parasitoids, 13 and 17 years are much longer than the time scale for most research grants and tenure decisions.”

Here are some mathematical results related to cicadas that can be found using MathSciNet.  One theme is using discrete dynamical systems based on nonlinear Leslie matrix models for semelparous populations, with magical cicadas being a motivating example.

After the excerpts from MathSciNet, there is a link to an engaging video, followed by an observation about being careful when searching “anywhere” in MathSciNet.

MR2501473
Cushing, J. M. (1-AZ)
Three stage semelparous Leslie models. (English summary)
J. Math. Biol. 59 (2009), no. 1, 75–104.
92D25 (37N25 39A10 92D40)

The author considers nonlinear Leslie matrix models for the dynamics of semelparous populations. Synchronous cycles describe temporally synchronized collections of age cohorts that appear in periodic outbreaks of periodical insects (such as cicadas). In mathematical models, destabilization of an extinction equilibrium and the subsequent occurrence of a global continuum branch of positive equilibria represent synchronous cycles as a bifurcation scenario. The inherent net reproductive number $R_0$ plays an important role in the bifurcation which occurs at $R_0=1$. Although bifurcation behaviors at $R_0=1$ have been well researched for two-dimensional semelparous Leslie models, not much is known for semelparous models in three or higher dimensions due to complexity in analyses. In this paper, a complete description of the bifurcation at $R_0=1$ is presented for the three-dimensional case under some monotonicity assumptions on the nonlinear interaction terms.

Following the introduction of preliminary results, the author investigates the dynamics on the boundary of $\Bbb{R}_3^{+}$. For the classification of dynamics, symmetry/asymmetry as well as strength in competition among inter-age classes play an important role. A single-class three-cycle is a fixed point on a positive coordinate axis $A_+^0$ which is one possible representation of synchronous cycles. The other type of synchronous cycles are called two-class three cycles, which correspond to a fixed point on a positive coordinate plane $P_+^0$. It is shown that under the monotonicity assumptions, single-class three-cycles are globally asymptotically stable (GAS) on $A_+^0$ if $R_0>1$. However, several alternative results can be obtained which depend on parameter constraints if $R_0>1$ but near to $1$: two-class three-cycles can occur on $P_+^0$. Interestingly, the dynamics in the interior of $\Bbb{R}_3^{+}$ is affected by the dynamics on boundaries. A heteroclinic cycle can occur. As summarized in the abstract, strong inter-age class competitive interactions promote oscillations with separated life-cycle stages, while weak interactions promote stable equilibrations with overlapping life-cycle stages. The methods used in this paper include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques. Several figures depicted by numerical computations are shown to illustrate the possible outcomes.

Reviewed by Shinji Nakaoka

MR2350059
Diekmann, Odo (NL-UTRE)Wang, Yi (PRC-HEF)Yan, Ping (FIN-HELS-MS)
Carrying simplices in discrete competitive systems and age-structured semelparous populations. (English summary)
Discrete Contin. Dyn. Syst. 20 (2008), no. 1, 37–52.
39A11 (37N25 92D25)

The authors consider a nonlinear Leslie (discrete time matrix) model for a semelparous population under the assumption that density dependence (through competition among age classes) is a monotone (specifically Beverton-Holt) function of a single weighted total population size. The main result of the paper concerning this model is the existence of a carrying simplex, which is then exploited to prove the existence of a heteroclinic cycle lying on the coordinate axes of the positive cone. Ecologically this cycle represents a dynamic in which generations are separated in time and each erupts periodically as cohorts age (periodical insects such as cicadas provide specific biological examples). To obtain this result, the others use a generalization of the well-known work of M. W. Hirsch on the existence of carrying simplexes in competitive systems. The authors modify the hypotheses of this generalized theory in order to obtain a new theorem that is more amenable to the Leslie model application.

Reviewed by J. M. Cushing

MR1761531
Behncke, Horst (D-OSNB-MI)
J. Math. Biol. 40 (2000), no. 5, 413–431.
92D25 (39A10)

A few mathematical models are developed to study the evolution of periodicity and synchronicity of magical cicadas. The key features include “underground habitat limitation, stochastic variations in predation and habitat, competition and the influence of the fungus”. The models are represented either in recursion form or with a Leslie matrix. For these models, the author shows convergence to stable generation distributions. Conditions for synchronous or periodic solution are also derived. “In addition, the intermediate evolutionary models are studied in order to show that cicada populations are capacity limited, which is tacitly assumed in the standard model.”

Reviewed by Anthony Leung

MR2523298
Gourley, Stephen A. (4-SUR)Kuang, Yang (1-AZS)
Dynamics of a neutral delay equation for an insect population with long larval and short adult phases. (English summary)
J. Differential Equations 246 (2009), no. 12, 4653–4669.
34K20 (34K40 92D40)

The authors study the stability of equilibria in a nonlinear autonomous neutral delay differential population model recently formulated by Bocharov and Hadeler via the reduction of a standard structured population model [G. A. Bocharov and K.-P. Hadeler, J. Differential Equations 168 (2000), no. 1, 212–237; MR1801352; K.-P. Hadeler, in The 8th Colloquium on the Qualitative Theory of Differential Equations, No. 11, 18 pp., Electron. J. Qual. Theory Differ. Equ., Szeged, 2008; MR2509170]. This model describes the intriguing dynamics of an insect population, such as periodical cicadas and flightless marine midges, with long larval and short adult phases. In the present paper, the study of the stability for the nonlinear neutral delay differential model is transformed into an appropriate non-neutral nonautonomous delay differential equation of unbounded delay. It is shown that the biologically meaningful solutions are always positive and bounded, provided that the time adjusted instantaneous birth rate at the time of maturation is less than 1. Global stability of the extinction and positive equilibria is also obtained by the method of iteration. In addition, the analysis in the paper reveals the fact that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow unboundedly regardless of the population death process.

Reviewed by Si Ning Zheng

MR1849824
Webb, G. F. (1-VDB)
The prime number periodical cicada problem. (English summary)
Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 3, 387–399.
92D25

Summary: “Mathematical models are presented to argue for the significance of prime number emergences of 13 year and 17 year periodical cicadas (Magicicada spp.). The prime number values arise as resonances of emergences with 2 and 3 year quasi-cycling predators. Predators with 2 and 3 year quasi-cycles are present due to their age dependent fecundity and mortality rates. Their quasi-cycles are enhanced by the predation of cicadas during emergences and thus exert significant influence on the cicada periodic life cycles.”

Aside

Searching for “cicada” using the Anywhere field in a publication search brings up some curious matches.

One such curious match is:

MR2720210
Ferguson, Andrew (4-WARW-MI); Jordan, Thomas (4-BRST); Shmerkin, Pablo (4-MANC-CDA)
The Hausdorff dimension of the projections of self-affine carpets. (English summary)
Fund. Math. 209 (2010), no. 3, 193–213.
28A80 (28A78)

If you check out the item in MathSciNet, you will be hard-pressed to find the word “cicada” in the title of the paper, the review text, or any of the normal places.  Publication searches in MathSciNet include a variety of possible fields from the database.  One such field is the institution.  It turns out that the address of the third author on this paper is:

4-MANC-CDA
Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA), School of Mathematics
University of Manchester
(merged with University of Science and Technology in Manchester UMIST)
Manchester M13 9PL
ENGLAND

The “cicada” that was found is the acronym for the author’s institution!  The moral: “Anywhere” really means anywhere when doing a publication search.