Pattern Formation on Progressively less Regular Networks

Abstract

Globalization and climate change have a great impact on the spreading of species and their resources. Shifting climate zones affect habitats and ecosystems, which in turn changes the effective distances within these systems. Interactions between different species and their resources that were previously local can now take place over greater distances. The Reaction diffusion equations that are used to model Turing instabilities in spatial ecosystems are usually posed on a two-dimensional plane. But in real life, the interactions are not necessarily local and domains can be more complicated, like a network. This thesis aims to bridge part of the gap between a two-dimensional spatial domain and a network by studying Turing instabilities from Gray-Scott/Klausmeier type reaction diffusion equations posed on progressively less regular networks. Analytic and numerical methods are combined to make the transition from a continuous model to discretised equations and from a two-dimensional to a three-dimensional domain. A linear stability analysis of the discretised reaction diffusion system near the Turing bifurcation leads to the derivation of a discrete dispersion relation that links the unstable eigenvectors of the full system to the eigenvalues of the discrete Laplacian. Findings show that unstable, increasingly coarse patterns are emerging on the progressively less regular networks. The results demonstrate that the unstable eigenvectors for the less regular three-dimensional networks often correspond to analytically derived eigenfunctions for smaller continuous two-dimensional domains. In the ecological setting this would imply that more connected ecosystems behave similar to smaller scaled ecosystems, and might be less resilient due to restricted patterning options.