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Invariant manifolds for competitive discrete systems in the plane. (English) Zbl 1202.37027

Summary: 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 \(\bar x \in \mathcal R\). The main results of this paper give conditions on \(T\) that guarantee the existence of an invariant curve emanating from \(\bar x\) when both eigenvalues of the Jacobian of \(T\) at \(\bar x\) 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. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.

MSC:

37D10 Invariant manifold theory for dynamical systems
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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