![]() To obtain this result, the others use a generalization of the well-known work of M. 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). 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. 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. Several figures depicted by numerical computations are shown to illustrate the possible outcomes.ĭiekmann, Odo (NL-UTRE) Wang, Yi (PRC-HEF) Yan, Ping (FIN-HELS-MS)Ĭarrying simplices in discrete competitive systems and age-structured semelparous populations. The methods used in this paper include the theory of planar monotone maps, average Lyapunov functions, and bifurcation theory techniques. 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. 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.įollowing the introduction of preliminary results, the author investigates the dynamics on the boundary of $\Bbb$ is affected by the dynamics on boundaries. 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. The inherent net reproductive number $R_0$ plays an important role in the bifurcation which occurs at $R_0=1$. ![]() 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. Synchronous cycles describe temporally synchronized collections of age cohorts that appear in periodic outbreaks of periodical insects (such as cicadas). The author considers nonlinear Leslie matrix models for the dynamics of semelparous populations. One theme is using discrete dynamical systems based on nonlinear Leslie matrix models for semelparous populations, with magical cicadas being a motivating example.Īfter the excerpts from MathSciNet, there is a link to an engaging video, followed by an observation about being careful when searching “anywhere” in MathSciNet. Here are some mathematical results related to cicadas that can be found using MathSciNet. 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.” 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 is interpreting the cicadas’ life cycle as a dynamical system with some parameters. 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. He nicely lays out the issues and the possible explanations. May wrote about the periodicity of cicadas in Nature. 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. 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. The cycle lengths are prime numbers, which makes mathematicians wonder why.īrood X cicadas have been in the news in the US because they are have emerged after 17 years as larvae living underground. They belong to the genus Magicicada of periodical cicadas that emerge either in 13-year cycles or in 17-year cycles. ![]() In certain areas of Ann Arbor, Michigan, where Mathematical Reviews is physically located, they have become quite loud. ![]() The cicadas of Brood X have emerged throughout much of the eastern United States. Photo: Michigan Department of Natural Resources
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