Bifurcation analysis of population invasion: On-off intermittency and basin riddling

We investigate the local bifurcations experienced by a time-discrete dynami cal system from population biology when there is an attractor in an invaria nt subspace that loses stability. The system describes competition between two species in a constant environment; invariant subspaces; contain single- species attractors; the loss of stability of the attractor in one invariant subspace means that the corresponding species (i.e. the "resident" species ) becomes invadable by its competitor. The global dynamics may be understoo d by examining the sign structure of Lyapunov exponents transverse to the i nvariant subspace. When the transverse Lyapunov exponent (computed for the natural measure) changes from negative to positive on varying a parameter, the system experiences a so-called blowout bifurcation. We unfold two gener ic scenarios associated with blowout bifurcations: (1) a codimension-2 bifu rcation involving heteroclinic chaos and on-off intermittency and (2) a seq uence of riddling bifurcations that cause asymptotic indeterminacy. An ingr edient that both scenarios have in common is the fact that the "resident" s pecies subspace contains multiple invariant sets with transverse Lyapunov e xponents that do not change sign simultaneously. This simple model adds on a short list of archetypical systems that are needed to investigate the str ucture of blowout bifurcations. From a biological viewpoint, the results im ply that mutual invasibility in a constant environment is neither a necessa ry nor a sufficient condition for coexistence.