WHAT HAS BIOLOGY DONE FOR MATHEMATICS: SUCCESS STORIES

Abstract

Abstract. The long-time dynamics of ODE-PDE coupling models is a fascinating subject area. It is at very heart of understanding many important problems arising in the natural sciences. Heuristically, it is clear that the dynamics such coupled systems should depend drastically on the monotonicity properties of the ODE component. In the case, where this ODE component is "monotone", that is cannot produce internal stability (and all of the instability is driven by the coupling with the PDE component), one expects asymptotic compactness and the existence of finite-dimensional global attractor using the methods of infinite-dimensional dynamical systems. In contrast to that, in the case when ODE component is "nonmonotone", the ODE instability may produce asymptotic discontinuities and even may destroy the initially smooth spatial profile. Thus, in that case, the smoothing effect from the PDE component is not strong enough to suppress the development of discontinuities provided by the internal instabilities of the ODE component and as a result spatial discontinuities and extremely complicated spatial structures may appear.