title: A Turing-Hopf Bifurcation Scenario for Pattern Formation on Growing Domains.
abstract: In this paper, we study the emergence of different patterns that are formed on both static and growing domains and their bifurcation structure. One of these is the so-called Turing-Hopf morphogenetic mechanism. The reactive part we consider is of FitzHugh-Nagumo type. The analysis was carried out on a flat square by considering both fixed and growing domain. In both scenarios, sufficient conditions on the parameter values are given for the formation of specific space-time structures or patterns. A series of numerical solutions of the corresponding initial and boundary value problems are obtained, and a comparison between the resulting patterns on the fixed domain and those arising when the domain grows is established. We emphasize the role of growth of the domain in the selection of patterns. The paper ends by listing some open problems in this area.
authors.affiliation: Universidad Autónoma de Guerrero, C. Carlos E. Adame #54, Col. Garita, C.P. 39650, Acapulco, Gro., Mexico.
authors.forename: Jorge A
authors.initials: JA
authors.lastname: Castillo
authors.affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), Ciudad Universitaria, C.P. 04510, Del. Coyoacán, Ciudad de México, Mexico. faustinos403@gmail.com.
authors.forename: Faustino
authors.initials: F
authors.lastname: Sánchez-Garduño
authors.affiliation: IIMAS-FENOMEC, UNAM, Apartado Postal 20-726, Admon. No. 20, C.P. 01000, Del. Álvaro Obregón, Ciudad de México, Mexico.
authors.forename: Pablo
authors.initials: P
authors.lastname: Padilla
mesh_terms: D019521:Body Patterning; D003198:Computer Simulation; D055641:Mathematical Concepts; D008954:Models, Biological; D009024:Morphogenesis; D010363:Pattern Recognition, Automated
keywords: FitzHugh–Nagumo model; Growing domain; Spiral and turbulent-like patterns; Target; Turing–Hopf bifurcation