Professor Karl Hadeler, Universität Tübingen
Title: Damping and excitation by quiescent or dormant phases
We consider differential equations where a vector field (the active phase) is diffusively coupled to the zero field (representing a quiescent phase). In a particle interpretation: Particles from different species switch between active and quiescent phases. Similarly we consider discrete time dynamical systems (coupled maps) where an 'active' map is coupled to the identity which in this case describes a quiescent phase. Quiescent phases may change the dynamical behavior drastically. There are two different scenarios depending on the rates with which particles enter or leave the quiescent phase. If these rates are the same (or about the same) for all species then quiescent phases act as damping and work against Hopf bifurcations, Neimark-Sacker and period doubling bifurcations. On the other hand, if these rates are rather different for different species, then a stable stationary point may be destabilized. If the exit time from the quiescent compartment is not exponentially distributed then quiescent phases lead to integral equations and delay equations. Predator-prey and epidemic models serve as examples.