Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T01:20:27.157Z Has data issue: false hasContentIssue false

Convexity-preserving flows of totally competitive planar Lotka–Volterra equations and the geometry of the carrying simplex

Published online by Cambridge University Press:  01 November 2011

Stephen Baigent
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the flow generated by the totally competitive planar Lotka–Volterra equations deforms the line connecting the two axial equilibria into convex or concave curves, and that these curves remain convex or concave for all subsequent time. We apply the observation to provide an alternative proof to that given by Tineo in 2001 that the carrying simplex, the globally attracting invariant manifold that joins the axial equilibria, is either convex, concave or a straight-line segment.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Da Costa, F. P., Grinfeld, M., Mottram, N. J. and Pinto, J. T., Uniqueness in the Fréedericksz transition with weak anchoring, J. Diff. Eqns 246 (2009), 25902600.CrossRefGoogle Scholar
2.de Mottoni, P. and Schiaffino, A., Competition systems with periodic coefficients: a geometric approach, J. Math. Biol. 11 (1981), 319335.CrossRefGoogle Scholar
3.Doob, J. L., Measure theory, Graduate Texts in Mathematics, Volume 143 (Springer, 1994).Google Scholar
4.Hirsch, M. W., Systems of differential equations that are competitive or cooperative, III, Competing species, Nonlinearity 1 (1988), 5171.CrossRefGoogle Scholar
5.Mierczyński, J., Smoothness of unordered curves in two-dimensional strongly competitive systems, Appl. Math. (Warsaw) 25 (1999), 449455.CrossRefGoogle Scholar
6.Rockafellar, R. T., Convex analysis (Princeton University Press, 1997).Google Scholar
7.Tineo, A., On the convexity of the carrying simplex of planar Lotka–Volterra competitive systems, Appl. Math. Comput. 123 (2001), 93108.Google Scholar
8.Zeeman, E. C. and Zeeman, M. L., On the convexity of carrying simplices in competitive Lotka–Volterra systems. In Differential equations, dynamical systems and control science, Lecture Notes in Pure and Applied Mathematics, Volume 152, pp. 353364 (Dekker, New York, 1994).Google Scholar