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
This thesis describes how complex and real-world relevant analytical solutions can be found starting from a simple Nagumo problem posed on one or two-dimensional lattices. In Chapters 2 and 3 we... Show moreThis thesis describes how complex and real-world relevant analytical solutions can be found starting from a simple Nagumo problem posed on one or two-dimensional lattices. In Chapters 2 and 3 we will discuss solutions with periodic asymptotic values, where different types of fronts can be combined. In Chapter 4 we will discuss the existence of planar defect solutions, where we analyse travelling fronts possessing corners. Show less
In the thesis, `Patterns in natural systems’ the formation and evolution of patterns as solutions of several partial differential systems are studied. These mathematical systems model three... Show more In the thesis, `Patterns in natural systems’ the formation and evolution of patterns as solutions of several partial differential systems are studied. These mathematical systems model three different biological and ecological processes. First, the way that plankton concentrates in the water column, under the influence of light and nutrient availability. Second, how tumor cells invade their healthy surroundings when it is incorporated that tumor cells cannot survive in a very small concentration. Lastly, the phenomenon that vegetation in semi-deserts organizes in strikingly regular patterns is studied. The mathematical tools that are used in this thesis, mostly arise from asymptotic analysis and geometric singular perturbation theory. 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
