Biological Modeling with Quiescent Phases

Quiescence or dormancy plays an important role in biological systems, from spore formation in bacteria to predator-prey cycles. In a mathematical framework, quiescence is modeled by diffusive coupling of the active dynamics to quiescent phases. Although coupling a given vector field to the zero field may appear simple at first glance, quiescent phases have biologically relevant effects which can be cast into rigorous mathematical formulations: permanence of stationary points, stabilization against oscillations and Hopf bifurcations, decrease in amplitude of periodic orbits. These features are common to ordinary and partial differential equations and delay equations and persist to some extent even for density-dependent transition rates. Applications range from tumor growth to engineered bacteria.