Interplay between exogenous and endogenous factors in seasonal vegetation oscillations.

A fundamental question in ecology is whether vegetation oscillations are merely a result of periodic environmental variability, or rather driven by endogenous factors. We address this question using a mathematical model of dryland vegetation subjected to annual rainfall periodicity. We show that while spontaneous oscillations do not exist in realistic parameter ranges, resonant response to periodic precipitation is still possible due to the existence of damped oscillatory modes. Using multiple time-scale analysis, in a restricted parameter range, we find that these endogenous modes can be pumped by the exogenous precipitation forcing to form sustained oscillations. The oscillations amplitude shows a resonance peak that depends on model parameters representing species traits and mean annual precipitation. Extending the study to bistability ranges of uniform vegetation and bare soil, we investigate numerically the implications of resonant oscillations for ecosystem function. We consider trait parameters that represent species with damped oscillatory modes and species that lack such modes, and compare their behaviors. We find that the former are less resilient to droughts, suffer from larger declines in their biomass production as the precipitation amplitude is increased, and, in the presence of spatial disturbances, are likely to go through abrupt collapse to bare soil, rather than gradual, domino-like collapse.

Coupled Lorenz oscillators near the Hopf boundary: Multistability, intermingled basins, and quasiriddling.

We investigate the dynamics of coupled identical chaotic Lorenz oscillators just above the subcritical Hopf bifurcation. In the absence of coupling, the motion is on a strange chaotic attractor and the fixed points of the system are all unstable. With the coupling, the unstable fixed points are converted into chaotic attractors, and the system can exhibit a multiplicity of coexisting attractors. Depending on the strength of the coupling, the motion of the individual oscillators can be synchronized (both in and out of phase) or desynchronized and in addition there can be mixed phases. We find that the basins have a complex structure: the state that is asymptotically reached shows extreme sensitivity to initial conditions. The basins of attraction of these different states are characterized using a variety of measures and depending on the strength of the coupling, they are intermingled or quasiriddled.