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

We investigate the local bifurcations experienced by a time-discrete dynamical system from population biology when there is an attractor in an invariant 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 understood by examining the sign structure of Lyapunov exponents transverse to the invariant 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 generic scenarios associated with blowout bifurcations: (1) a codimension-2 bifurcation involving heteroclinic chaos and on-off intermittency and (2) a sequence of riddling bifurcations that cause asymptotic indeterminacy. An ingredient that both scenarios have in common is the fact that the "resident" species subspace contains multiple invariant sets with transverse Lyapunov exponents that do not change sign simultaneously. This simple model adds on a short list of archetypical systems that are needed to investigate the structure of blowout bifurcations. From a biological viewpoint, the results imply that mutual invasibility in a constant environment is neither a necessary nor a sufficient condition for coexistence.