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Periodical cicadas. (English) Zbl 1004.92040
From the introduction: Among the 1500 species of cicadas the magicicadas occupy a remarkable position. They have the longest observed lifetime among periodical insects. They appear synchronously in huge numbers and the adults behave rather foolhardily. All this has been a source of puzzlement and continued fascination to biologists for more than 300 years. To evolutionary biologists their extraordinary life cycle represents a continuing challenge to stake out a possible evolutionary path. A further source of puzzlement in the evolutionary concept is the observed 4-year acceleration of some broods in some areas in central USA. Finally, the fact that the period is a prime number led to speculations about adapted periodic predators.
We study these problems by means of mathematical models, as carried out previously by F.C. Hoppensteadt and J.B. Keller [Science 194, 335-337 (1976)], M.G. Bulmer [Am. Natural. 111, 1099-1117 (1977)] and R.M. May [Nature 277, 347-349 (1979)]. Hoppensteadt and Keller explain the synchronous appearance by a limitation of the underground habitat and predation. With an appropriate choice of parameters they establish stable synchronous solutions for periods $$L$$ of over 10 years. Apart from the fact that these parameters seem to have been chosen in a rather ad hoc way, the main disadvantage is the apparent lack of a suitable predator. Bulmer shows that synchronous appearance is possible if competition between different age classes is more severe than within the same age group. In May’s paper the discussion is more of a qualitative nature, though May suggests the fungus Massospora as a possible replacement for a predator.
In this paper we take up some of these suggestions and develop a number of models which are more closely orieted to the known biology of magicicadas. The key features of these models are: underground habitat limitation, stochastic variations in predation and habitat, competition and the influence of the fungus. The models are represented either in recursion form, like the Hoppensteadt-Keller model, or in its extended version, which is a Leslie matrix type model. The interaction is nonlinear, and stochastic effects are also considered. For these models we show the convergence to stable generation distribution. Conditions for synchronous or periodic solutions can be derived from these results.

##### MSC:
 92D50 Animal behavior 92D15 Problems related to evolution 92D40 Ecology 39A10 Additive difference equations 37N25 Dynamical systems in biology
##### Keywords:
periodicity; magicicadas; Leslie matrix; stochastic effects
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