Quadratic Two-Species Population Models

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


We proceed now to look at a rather general model for the interaction of two biological species. Special cases of this model will represent Predator-Prey, Competitive and Mutualistic interactions. We will however be able to demonstrate a very general result for all such models. Specifically, we will show that the models almost never admit periodic (cyclic) solutions. This of course means that in terms of finding a model which describes an ecosystem which is known to behave in a cyclic manner, the model discussed below is not adequate.


Equilibrium Point Hilbert Function Simple Closed Curve Cyclic Manner Unique Critical Point 
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  1. The material in this section is drawn from two teaching modules by C.S. Coleman. At the present time, these are available from:Google Scholar
  2. Professor W.F. Lucas 334 Upson Hall Cornell University Ithaca, NY 14853Google Scholar
  3. Coleman, C.S., “Hilbert’s 16th Problem: How Many Cycles?”, MAA Workshop on Modules in Applied Mathematics, Cornell University, 1976.Google Scholar
  4. Coleman, S.C., “Quadratic Population Models: Almost Never Any Cycles”, MAA Workshop on Modules in Applied Mathe mathematics, Cornell University, 1976.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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