Summary
Several geometrical interpretations of Liapunov functions for non-linear ecological models are examined and their limitations pointed out. In particular the geometrical nonuniqueness of Liapunov functions is illustrated by displaying explicitly and comparing four different Liapunov functions for the symmetric competition model for two species. The main point is that considerable care must be taken in using the geometrical properties of an arbitrary Liapunov function as a guide to stability under perturbations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aiken, R., Lapidus, L.: The stability of interacting populations. Internat. J. Systems Sci. 4, 691–695 (1974)
Brauchli, H.: On the norm-dependence of the concept of stability. In: P. Germain and B. Nayroles, eds. Applications of methods of functional analysis to problems in mechanics. New York: Springer, 1976
Gilpin, M. E.: A Liapunov function for competition communities. J. Theoret. Biol. 44, 35–48 (1974)
Goh, B. S.: Nonvulnerability of ecosystems in unpredictable environments. Theoret. Population Biology 10, 83–95 (1976)
Hahn, W. Stability of motion. New York: Springer, 1967
Holling, C. S.: Resilience and stability of ecological systems. Ann. Rev. Ecol. Syst. 4, 1–24 (1973)
Inselberg, A., Dula, G.: The geometry of Liapunov functions. Amer. Math. Monthly 81, 1102–1104 (1974)
LaSalle, J., Lefschetz, S.: Stability by Liapunov's direct method. New York: Academic, 1961
Lewontin, R. C.: The meaning of stability. Brookhaven Symposium in Biology No. 22. National Bureau of Standards, Springfield, Va., 1969
Ludwig, D.: Persistence of dynamical systems. SIAM Review 17, 605–640 (1975)
MacArthur, R. H.: Species packing and competitive equilibrium for many species. Theoret. Population Biology 1, 1–11 (1970)
May, R. M.: Stability and complexity in model ecosystems. 2nd ed. Princeton, N.J.: Princeton Univ. Press, 1974
Slobodkin, L. B.: The growth and regulation of animal numbers. New York: Holt, Rinehart and Winston, 1961
Tuljapurkar, S. D., Semura, J. S. Stochastic instability and Liapunov stability. J. Math. Biology 8 (1979)
Vent'sel, A. D., Friedlin, M. I.: On small random perturbations of dynamic systems. Russian Math. Surveys. 25, 1–55 (1970)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tuljapurkar, S.D., Semura, J.S. Liapunov functions: Geometry and stability. J. Math. Biology 8, 25–32 (1979). https://doi.org/10.1007/BF00280583
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00280583