Nicolás Verschueren van Rees, Ph.D

Title: Pattern formation in extended systems: from biologically inspired to conceptual models

Abstract. We will present some results in the investigation of three minimal dynamical systems models of relevance in biological processes: the polarization of a one-dimensional cell, wrinkling of elastic rings, and pattern formation on a finite disk.

In the first problem, we revisit the so-called wave-pinning model (Mori, Edelstein et al.), including two generic terms to break the hypothesis of total protein conservation to produce a polarized profile. The study of this generalized model reveals a connection between the Turing and wave-pinning mechanisms in establishing a polarization profile.

In the second problem, we consider a minimal model for the shape of an elastic ring on a plane. This model appears naturally in fluid dynamics, and it can also be regarded as a phenomenological model for shapes observed in arteries. Starting from the trivial circle solution, we investigate the possible solutions as a function of the parameters related to the tension and pressure on the ring. A wide variety of shapes are revealed and organized in a bifurcation diagram supporting buckling, wrinkling and localized states. Subsequently, we show that wrinkling solutions are universal, allowing an exact wrinkling solution for a family of models.

Lastly, we investigate the influence of a finite domain in the nascence and organization of patterns by considering the Swift-Hohenberg model with a cubic-quintic nonlinearity posed on a finite disk with Neumann boundary conditions. Linear stability analysis of the trivial solution reveals the existence of three qualitatively different unstable modes, namely radial, wall and multiarm modes. Using numerical continuation, we discover bidimensional generalizations of the snaking and ladders scenario in the radial and angular coordinates.

The investigation of these problem is undertaken using numerical (continuation and time integration) and analytical (linear and weakly non-linear analysis, inner layer asymptotics) tools. In each case, the results are likely to play a role in describing the mechanisms responsible for the spatial organization in the respective contexts.