REACTION-DIFFUSION WAVES: CLASSICAL THEORYAND RECENT DEVELOPMENTS

摘要

Systematic studies of reaction-diffusion waves begin in the 1930s with the worksby Zeldovich - Frank-Kamenetskii in combustion theory and by Fisher andKolmogorov - Petrovskii - Piskunov in population dynamics. The theory ofreaction-diffusion waves is now well developed and includes detailed analysis ofthe scalar reaction-diffusion equations and monotone systems, of flame stabilityand nonlinear dynamics, of waves in excitable media, of reaction-diffusion-convection waves, and so on. After a short review of the theory of reaction-diffusion waves, some recent developments in the theory of flame propagationwill be presented. They concern some mathematical aspects of combustionwaves with the Lewis number different from 1 and reaction-diffusion-convectionwaves. Models in population dynamics with intra-specific competition will bediscussed. They provide a new mechanism of pattern formation in biology andcan be used to describe the emergence of biological species in the process ofevolution.