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Transport-Reaction Systems: Higher-Dimensional Domains and a Qualitative Dichotomy

Hyperbolic transport-reaction systems are often used to model the finite speed movement and interaction of particles, bacteria or animals. The most known example is the Goldstein-Kac model (1974) and its connection to the telegraph equation. To the best of our knowledge, results in the literature are currently limited to one-dimensional domains and often focus on specific models. A question of particular interest is whether the additional spacial variable and transport can cause stable reactions to become unstable. If so, what are the patterns or phenomena occurring and how does the solution behave qualitatively? In the context of diffusion-driven transport, Alan Turing was the first to ask and answer this question. Patterns generated by hyperbolic transport-reaction systems on the other hand are less well understood. We study general linear transport-reaction systems on an arbitrary dimensional hypercube with periodic boundary conditions. In the first half of the talk, we show a weak spectral mapping theorem and demonstrate its application by deriving a convergence result for a previously proposed Goldstein-Kac model on a two-dimensional domain. In the second half of the talk, we restrict ourselves to the one-dimensional domain and study transport-driven instabilities. We introduce a certain class of so-called hyperbolic instabilities and prove a qualitative dichotomy: Transport-driven instabilities are either Turing patterns or increasingly oscillating hyperbolic instabilities. A new algebraic condition for the existence of Turing patterns is obtained as a side-product.