Abstract
For homogeneous hypercycles with 2, 3 or 4 substances the future behavior of its trajectories is easily understood, in fact any trajectory converges to an equilibrium point as t → +∞. In this paper we study the descent of the trajectories, i.e. their behavior as t → -∞. It turns out that this backward behavior is not as uniform as the forward behavior. In fact, depending on the initial points some α-limit sets are singletons while others consist of certain edges of the state simplex.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aulbach, B., Flockerzi, D., Knobloch, H. W.: Invariant manifolds and the concept of asymptotic phase. Cas. pest. mat. 111, 156–176 (1986)
Conley, C.: Isolated invariant sets and the Morse index. C.B.M.S. Reg. Conf. Ser. Math. 38, Am. Math. Soc., Providence (1978)
Eigen, M.: Selforganization of matter and the evolution of biological macromolecules. Die Natur-wissenschaften 58, 465–523 (1971)
Eigen, M., Schuster, P.: The hypercycle: a principle of natural selforganization. Berlin Heidelberg New York: Springer 1979
Esser, M.: Qualitative Analyse der Hyperzyklus-Gleichung. Diplomarbeit, Universität Würzburg 1986
Hofbauer, J., Sigmund, K.: Evolutionstheorie und dynamische Systeme: Mathematische Aspekte der Selektion. Berlin Hamburg: Parey 1984
Palmer, K. J.: Linearization near an integral manifold. J. Math. Anal. Appl. 51, 243–255 (1975)
Palmer, K. J.: Qualitative behavior of a system of ODE near an equilibrium point—A generalization of the Hartman-Grobman theorem. Preprint 372, Institut für Angewandte Mathematik, Universität Bonn 1980
Author information
Authors and Affiliations
Additional information
Work partially supported by Deutsche Forschungsgemeinschaft under grant number Au 66–1
Rights and permissions
About this article
Cite this article
Aulbach, B., Flockerzi, D. The past in short hypercycles. J. Math. Biology 27, 223–231 (1989). https://doi.org/10.1007/BF00276104
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00276104