Localised auxin peaks in concentration-based transport models for plants - Draelants, Delphine; Avitabile, Daniele; Vanroose, Wim (Royal Society, 2015)
We study the existence and bifurcation structure of stationary localised auxin spots in concentration-based auxin-transport models posed on one- and two-dimensional networks of plant cells. In regular domains with small active transport coefficient and no diffusion, the geometry of the cellular array encodes the peaks' height and location: asymptotic calculations show that peaks arise where cells have fewer neighbours, that is, at the boundary of the domain. We perform numerical bifurcation analysis for a concrete model available in literature and provide numerical evidence that the mechanism above remains valid in the presence of diffusion in both regular and irregular arrays. Using the active transport coefficient as bifurcation parameter, we find snaking branches of localised solutions, with peaks emerging from the boundary towards the interior of the domain. In one-dimensional regular arrays we observe oscillatory instabilities along the branch. In two-dimensional irregular arrays the snaking is slanted, hence stable localised solutions with peaks exist in a wide region of parameter space: the competition between active transport and production rate determines whether peaks remain localised or cover the entire domain.