Numerical analysis of pattern formation in auxin transport models
12 May 2016
UAntwerp, Campus Middelheim, A.143 - Middelheimlaan 1 - 2020 Antwerpen
PhD defence Delphine Draelants - Department of Mathematics and Computer Science
Knowledge about the growth process of plants and how it is affected by different stimuli is important because it can have an immediate impact on the productivity and can improve and secure the food supply for the dramatically increasing world population. Plant growth is a complex self-organizing process with feedback loops and interactions between different components over the whole tissue. An important research topic in the context of plant growth is the study of auxin transport. Auxin is a plant hormone that has a major influence on the growth process. The goal is to understand the auxin distribution patterns and to predict the effect of stimuli.
In this thesis we concentrate on the analysis of concentration-based auxin transport models. We use a dynamical systems approach with numerical continuation methods and bifurcation analysis to detect the solutions and their stability as a function of the parameters.
We developed an open source software package PyNCT, that contains these methods and calculates the solution landscape in function of different parameters. The toolbox, based on sparse linear algebra, allows the study of complex models applied on large tissues.
We also classify existing models and analyze the pattern formation mechanism behind them. Besides showing that these models are capable of forming patterns with auxin peaks, we reveal the exact pattern formation mechanism. We show that in realistic tissues peaks arise as a consequence of the geometry of the tissue. The models are capable of generating a multitude of stationary patterns with a variable number of auxin peaks that can be selected by different initial conditions or by quasi-static changes in the active transport parameter. We relate the occurrence of localized patterns to a snaking bifurcation structure, which is known to arise in a wide variety of nonlinear media, but has not yet been reported in plant models.
Besides these static tissues, we also set up a framework to study growing tissues over time.
All results give us more insight in the pattern formation mechanism behind auxin transport models and must be complemented with experimental research to enlarge our knowledge about the growth process of plants.