Invariant Manifolds for Competitive Systems in the Plane

TL;DR

This paper establishes conditions for the existence of invariant curves near fixed points in competitive planar systems, aiding in understanding their stability and basins of attraction.

Contribution

It provides new criteria for invariant curve existence in competitive maps, applicable to hyperbolic and nonhyperbolic cases, enhancing analysis of difference equations.

Findings

Invariant curves emanate from fixed points under specified eigenvalue conditions.

Invariant curves are increasing and partition the region into invariant subsets.

Applications include determining basins of attraction for equilibria.

Abstract

Let $T$ be a competitive map on a rectangular region $\mathcal{R}\subset \mathbb{R}^2$, and assume $T$ is $C^1$ in a neighborhood of a fixed point $\bar{\rm x}\in \mathcal{R}$. The main results of this paper give conditions on $T$ that guarantee the existence of an invariant curve emanating from $\bar{\rm x}$ when both eigenvalues of the Jacobian of $T$ at $\bar{\rm x}$ are nonzero and at least one of them has absolute value less than one, and establish that $\mathcal{C}$ is an increasing curve that separates $\mathcal{R}$ into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. Several applications to planar systems of difference equations with non-hyperbolic equilibria are given.