On the convexity of the carrying simplex of planar Lotka–Volterra competitive systems

摘要

In this paper we prove that a result by Zeeman and Zeeman (On the convexity of carrying simplices in competitive Lotka–Volterra systems, Differential equations, dynamical systems and control science, A festschift in honor to Laurence Markus, 1994, pp. 353–364) is partially wrong and we find necessary and sufficient conditions for which the carrying simplex of a planar competitive Lotka–Volterra system is twice continuously differentiable. In particular, the main result by Mierczynski (J. Differential Equations 111 (1994) 385–409) is sharp.