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

Part of: Functional-differential and differential-difference equations Partial differential equations Qualitative theory

Published online by Cambridge University Press:  01 November 2011

Stephen Baigent
Stephen Baigent
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK (s.baigent@ucl.ac.uk)
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Abstract

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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.

Keywords

Lotka–Volterra equationscurvature-preserving flowcompetitive dynamicscarrying simplex

MSC classification

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 34K19: Invariant manifolds 34C12: Monotone systems 35B41: Attractors 34C37: Homoclinic and heteroclinic solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 53 - 63
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

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