Multiscale modeling and simulation of traveling waves in biology: a review.
W. Duncan Martinson, Helen M. Byrne, Philip K. Maini

Abstract. Traveling waves are a near-ubiquitous phenomenon in mathematical biology, and have been studied in the context of embryonic development, cancer growth, wound healing, and epidemiology. Fisher's equation is the prototypical example of a problem that admits a traveling wave solution; it is a partial differential equation that was proposed to describe the spread of an advantageous gene. Although past investigators formulated and analyzed models for such translationally invariant systems at a macroscopic scale of interest, recent work has focused on analyzing and simulating traveling wave behavior in greater detail. This paper reviews the different scales and approaches by which researchers can model and simulate wave-like behavior, using the Fisher equation as a pedagogic example. We discuss different algorithms for simulating traveling waves, from traditional finite difference approaches to more recent hybrid multiscale algorithms

Rend. Mat. Appl. (7) 40 (2019) 191-216; pdf