The effect of holes on pattern formation in two species two dimensional reaction-diffusion systems

Abstract

Spatial patterns arise in a wide range of biological processes ranging from embryonic development to cell polarisation. Owing to the complexity of pattern formation, attempts to understand it often resorts to modelling by reaction-diffusion (RD) models, which is a model type consisting of coupled partial differential equations (PDE:s). In biological systems, it has been observed that some patterns form regions of high concentration (poles) close to regions in which the species relevant for pattern formation cannot enter (holes). In order to understand the impact of holes on RDmodels, and if RD-models can capture this behaviour with poles being confined on domains with holes, this thesis investigated two questions; (1) the impact of holes on pattern formation, (2) if poles by some strategy can be spatially confined on a domain containing holes. In order to answer these questions, two classical models, the Schnakenberg and Gierer-Meinhardt, were simulated on a two dimensional domain with zero, five, seven and 20 small densely packed circular holes. For solving the RD-systems numerically on such domains a finite element method (FEM) was implemented. The results suggested that on a domain with many holes poles have a tendency to accumulate close to, or directly in, the region dense in holes. The exact reason behind this behaviour is not known, but it might be due to the model species being confined in the diffusion-restricted region between holes. Regarding the control of pole formation, changing parameters outside the Turing space in a sub-region proved efficient for spatially confining poles to a specific region. Although potentially useful for recreating observed patterns, it should be noted that the usage of this method raises the question of why the parameter values are different in a sub-region.