On the Convexity of Carrying Simplices in Competitive Lotka-Volterra Systems

In [2] M.W. Hirsch proves that for a competitive Lotka-Volterra system on the positive cone in R ⁿ , there is a globally attracting Lipschitz submanifold that projects radially homeomorphically to the unit simplex Δ in R ⁿ (here Δ = {χ ϵ Rⁿ: χ_i ≥ 0, ∀i and ∑ i = 1 n x i = 1 } ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141237/34660f10-2a14-45fc-b42c-ebb9e7f6391e/content/inequ23_353_1.tif"/> · This hypersurface is called the carrying simplex, and denoted by Σ. In this paper we prove that for a two-dimensional competitive Lotka-Volterra system Σ is convex. We give an example to show that the analogous statement in three-dimensions is false, and announce some results relating the convexity of Σ to the dynamics on Σ in higher dimensions.