Quiescence, excitability, and heterogeneity in ecological models

Introducing quiescent phases into dynamical systems and ecological models tends to stabilize equilibria against the onset of oscillations and also to lower the amplitudes of existing periodic orbits. However, these effects occur when all interacting species go quiescent with the same rates and return to activity with the same rates. On the other hand, if the species differ with respect to these rates, then an equilibrium may even be destabilized. At least in the case of two interacting species this bifurcation phenomenon is closely related to the well-known Turing instability. In particular, for two species it is true that an equilibrium can be destabilized by quiescent phases if and only if it is excitable in the Turing sense. These effects are thoroughly studied and exhibited at the example of classical ecological models and epidemic models. Similar effects occur in delay equations and reaction-diffusion equations. The effect of stabilization against oscillations by quiescent phases can be shown as a special realization of a general principle saying that spatial heterogeneity stabilizes. The results on local stability of stationary points can be extended to periodic orbits. In particular, a geometric argument on the flow along a periodic orbit explains why convex periodic orbits, as observed in numerical simulations, tend to shrink when quiescent phases are introduced.