Global stability of discrete-time competitive population models

We develop practical tests for the global asymptotic stability of interior fixed points for discrete-time competitive population models. Our method constitutes the extension to maps of the Split Lyapunov method developed for differential equations. We give ecologically-motivated sufficient conditions for global stability of an interior fixed point of a map of Kolmogorov form. We introduce the concept of a principal reproductive mode, which is linked to a normal at the interior fixed point of a hypersurface of vanishing weighted-average growth. Our method is applied to establish new global stability results for 3-species competitive systems of May-Leonard type, where previously only parameter values for local stability was known