Quadratic Two-Species Population Models
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.
KeywordsEquilibrium Point Hilbert Function Simple Closed Curve Cyclic Manner Unique Critical Point
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- 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
- Professor W.F. Lucas 334 Upson Hall Cornell University Ithaca, NY 14853Google Scholar
- Coleman, C.S., “Hilbert’s 16th Problem: How Many Cycles?”, MAA Workshop on Modules in Applied Mathematics, Cornell University, 1976.Google Scholar
- Coleman, S.C., “Quadratic Population Models: Almost Never Any Cycles”, MAA Workshop on Modules in Applied Mathe mathematics, Cornell University, 1976.Google Scholar