Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

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

Stephen Baigenta1

a1 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK (


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.

(Received May 06 2010)

(Online publication November 01 2011)


  • Lotka–Volterra equations;
  • curvature-preserving flow;
  • competitive dynamics;
  • carrying simplex

2010 Mathematics subject classification

  • Primary 35B05;
  • 34K19;
  • Secondary 34C12;
  • 35B41;
  • 34C37