Stable Points, Stable Cycles and Chaos

  • James C. Frauenthal
Part of the The Umap Expository Monograph Series book series (UMAP)


We proceed now to look at the dynamics of a population with a life history in which successive generations do not overlap one another. Some types of fishes, such as the salmon, as well as many kinds of insects live in this way. The parent generation leaves its eggs before it dies in the fall. The eggs then winter over, and the young emerge in the spring. The potential for bizarre population trajectories exists; consider for example the thirteen year periodic cicada. Our goal will be to try to discover the range of behavior which is possible.


Tuning Parameter Periodic Point Linearize Stability Analysis Librium Point Stable Point 


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  1. Li, T.Y. and J.A. Yorke, “Period Three Implies Chaos”, American Mathematical Monthly, Volume 82, pp. 985–992Google Scholar
  2. May, R.M., “Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles and Chaos”, Science, Volume 186, pp. 645–647, November 15, 1974.CrossRefGoogle Scholar
  3. May, R.M., “Simple Mathematical Models with Very Complicated Dynamics”, Nature, Volume 261, pp. 459–467, June 10Google Scholar
  4. Straffin, P.D., Jr., “Periodic Points of Continuous Functions”, Mathematics Magazine, Volume 51, pp 99–105, March, 1978.Google Scholar

Copyright information

© Education Development Center, Inc. 1979

Authors and Affiliations

  • James C. Frauenthal
    • 1
  1. 1.Applied Mathematics and StatisticsState University of New YorkStony BrookUSA

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