One of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack... Show moreOne of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack of resources, vegetation self-organizes to sustain itself by forming large-scale spatial patterns. In this thesis, the underlying mathematical structure of these observed vegetation patterns is studied using partial differential equations models. The vegetation patterns are analyzed using techniques from geometrical singular perturbation theory and numerical simulations. Additionally, novel multi-front patterns are constructed that arise within one of the models studied. This interdisciplinary research allows for cross-fertilization of both mathematics and ecology. Show less
Vast, often populated, areas in dryland ecosystems face the dangers of desertification. Loosely speaking, desertification is the process in which a relatively dry region loses its vegetation -... Show moreVast, often populated, areas in dryland ecosystems face the dangers of desertification. Loosely speaking, desertification is the process in which a relatively dry region loses its vegetation - typically as an effect of climate change. As an important step in this process, the lack of resources forces the vegetation in these semi-arid areas to organise itself into large-scale spatial patterns. In this thesis, these patterns are studied using conceptual mathematical models, in which vegetation patterns present themselves as localised structures (for example pulses or fronts). These are analysed using mathematical techniques from (geometric singular) perturbation theory and via numerous numerical simulations. The study of these ecosystem models leads to new advances in both mathematics and ecology. Show less
In the first part of this thesis we study the geometry of folding patterns. Specifically, we focus on crease patterns consisting entirely of four-vertices; these are points where four fold... Show moreIn the first part of this thesis we study the geometry of folding patterns. Specifically, we focus on crease patterns consisting entirely of four-vertices; these are points where four fold lines come together. A single four-vertex is the simplest example of a foldable crease pattern that can be folded without bending the material in between the folds, and has a remarkable property: despite its single degree of freedom, it has two distinct folding motions. We make use of this property, and show how to design arbitrarily large four-vertex crease patterns, which can fold into two or more shapes. This is in contrast to other design methods, which produce patterns that can only fold into one specific shape. In the second part of this thesis, we study single four-vertices, and show a robust method to obtain four-vertices with three energy minima, which correspond to three different stable folded configurations. This too is in contrast to other experimental methods, which can only generate bistable vertices or patterns. Show less
In drylands, water is a crucial ingredient for the sustenance of vegetation. Due to climate change, dry areas are projected to become dryer, which puts the vegetation under increasing environmental... Show moreIn drylands, water is a crucial ingredient for the sustenance of vegetation. Due to climate change, dry areas are projected to become dryer, which puts the vegetation under increasing environmental pressure. If environmental conditions deteriorate, the amount of vegetation may become critical, beyond which the vegetation suddenly disappears. We study a phenomenological model - the extended Klausmeier model - which models the interaction between water and vegetation in drylands. In this spatially explicit model, due to drought, homogeneous vegetation transforms into a spatial pattern. We study different scenarios under which subsequent patterns form under decreasing rainfall conditions, eventually leading to a bare desert state. Show less
