Coupled dynamics, quiescent states, and reaction transport equations

Many physical models are based on the idea that several non-linear effects act simultaneously and (hence) the vector fields enter the model equations in an additive way (as in reaction diffusion equations). The additivity can be justified by Trotter formulas or the fractional step approach where the simultaneous action is replaced by a periodic action with short period.

Here different actions are coupled diffusively, i.e., the transition between different vector fields is governed by Poisson processes. While for linear vector fields the Trotter and the Poisson approach lead to the same limiting equation (for short periods or large coupling rates), for non-linear vector fields the limiting equations are quite different.

Away from the limiting case, for moderate coupling rates, it may be quite difficult to extract the behavior of the coupled system from the behavior of its components. The problem may even be meaningless for arbitrary systems. However, there are several useful scenarios:

- Coupling closely related dynamics (caricature of coupled map lattice).

- Coupling transport equations to reaction equations (equivalent damped wave equations, stability of stationary states, travelling front problem, linear determinacy).

- Coupling spatial spread to an infection dynamics (justification of epidemic contact distributions in terms of spatial processes).

- Coupling an arbitrary dynamics to a quiescent state. Although from the modeling point of view this extension should act similar to a delay, the qualitative features are quite different. A general stability theorem can be shown.

(joint work with Thomas Hillen and Mark Lewis)