Existence of symmetric and asymmetric spikes for a crime hotspot model

Copyright @ 2014 Society for Industrial and Applied MathematicsWe study a crime hotspot model suggested by Short, Bertozzi, and Brantingham in [SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 462--483]. The aim of this work is to establish rigorously the formation of hotspots in this model representing concentrations of criminal activity. More precisely, for the one-dimensional system, we rigorously prove the existence of steady states with multiple spikes of the following types: (i) multiple spikes of arbitrary number having the same amplitude (symmetric spikes), and (ii) multiple spikes having different amplitude for the case of one large and one small spike (asymmetric spikes). We use an approach based on Lyapunov--Schmidt reduction and extend it to the quasilinear crime hotspot model. Some novel results that allow us to carry out the Lyapunov--Schmidt reduction are (i) approximation of the quasilinear crime hotspot system on the large scale by the semilinear Schnakenberg model, and (ii) estimate of the spatial dependence of the second component on the small scale which is dominated by the quasilinear part of the system. The paper concludes with an extension to the anisotropic case

A recipe for desert : analysis of an extended Klausmeier model

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 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.NWO ComplexityUBL - phd migration 201

Lines in the sand : behaviour of self-organised vegetation patterns in dryland ecosystems

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 - 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. NWO Mathematics of Planet EarthAnalysis and Stochastic