Global stability of discrete-time competitive population models

We develop practical tests for the global 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 competitive map of Kolmogorov form, and show how these conditions can simplify when a carrying simplex is known to exist. We introduce the concept of a principal reproductive mode, which is linked to a left eigenvector for the dominant eigenvalue of a positive matrix, which in turn is linked to a normal vector, at an interior fixed point, to a hypersurface of vanishing weighted-average growth, and also to a normal to the carrying simplex when present. A connection with permanence is also discussed. As examples of applications, we first take two well-understood planar models, namely the Leslie-Gower and the May-Oster models, where our method confirms global stability results that have previously been established, through, for example, properties of monotone maps. We also apply our methods to establish new global stability results for 3-species competitive systems of May-Leonard type, giving detailed descriptions of the parameter ranges for which the models have globally stable interior fixed points