Traveling waves in delayed reaction-diffusion equations in biology
Sergei Trofımchuk, 1 Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile, Vitaly Volpert, 3 INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne 69603, France
Abstract
This paper represents a literature review on traveling waves described by delayed reactiondiffusion (RD, for short) equations. It begins with the presentation of different types of equations arising in applications. The main results on wave existence and stability are presented for the equations satisfying the monotonicity condition that provides the applicability of the maximum and comparison principles. Other methods and results are described for the case where the monotonicity condition is not satisfied. The last two sections deal with delayed RD equations in mathematical immunology and in neuroscience. Existence, stability, and dynamics of wavefronts and of periodic waves are discussed.