Transport, reaction, and delay in mathematical biology, and the inverse problem for traveling fronts

K. P. Hadeler

Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 57-77

Résumé

Reaction-diffusion equations are the standard models for reacting and moving particles in ecology, cell biology, and other fields of Biology. In many situations, a more detailed description of the movements of particles or individuals is required. Then reaction-transport systems, reaction Cattaneo systems, and Kramers–Langevin approaches can be used. The typical limit solutions in unbounded domains are traveling fronts which pose new mathematical problems in the case of transport equations. The inverse problem for traveling fronts is a novel problem, which is investigated here in great detail. Additional features are delays, which typically lead to oscillations, and quiescent phases, which can be shown to stabilize against the onset of oscillations. In particular, neutral delay equations can be rigorously derived from first-order hyperbolic equations with appropriate boundary conditions modelling age structure. Multi-species systems lead to study various phenomena such as Turing instability, interaction of diffusion and delay, and cross diffusion.