On the mechanical theory for biological pattern formation (Q1803798)

The pattern formation potential of the purely mechanical Oster-Murray models [see \textit{J. D. Murray} and \textit{G. F. Oster}, IMA J. Math. Appl. Med. Biol. 1, 51-75 (1984; Zbl 0611.92003)] in embryology is investigated. These consist of 3 nonlinear PDE's: conservation equations for cells and extracellular matrix, respectively, and a cell-matrix equation. Numerical solutions yield standing waves for the case of purely local cell-matrix interactions, whereas stationary stable patterns (in one and two dimensions) are achieved when biologically realistic long-range traction forces are included. Biological examples are discussed, among which are wound healing and formation of fingerprint patterns.