Classification of quadratic carrying simplices in two-dimensional competitive Lotka–Volterra systems

Abstract

In a two-dimensional competitive Lotka–Volterra system on R2+ the carrying simplex, or edge E, is the invariant curve, homeomorphic to the unit interval, that attracts all non-zero orbits and carries the asymptotic dynamics (Hirsch M W 1998 Nonlinearity 1 51–71, Zeeman M L 1993 Dynam. Stab. Sys. 8 189–21). In general E has no analytic formula, but exceptionally E can be straight or quadratic. Here quadratic means contained in a non-degenerate conic. We classify the quadratic edges, and show that they belong to four families, one convex and the other three concave. In all cases the conic is a parabola. Under standardization each family becomes a one-parameter family. In the standard convex family all the edges and parabolas are different, whereas in each standard concave family all the edges and parabolas are the same. In the space of all systems those with straight edges are of codimension 1, and those with quadratic edges of codimension 2. We use these results in a subsequent paper (Zeeman E C and Zeeman M L Nonlinearity 15 2019) to show that higher-dimensional carrying simplices are generically determined by their edges.