Spatial Patterns in Nature:
An Entry-Level Introduction to Their Emergence and Dynamics
(SIAM DS23 Minitutorial)

Abstract

Spatially patterned structures play an important role in many scientific disciplines and appear on many different spatial scales. These range from cell-scale patterns in wound healing angiogenesis to vegetation patterning in ecosystems. Spatially heterogeneous systems may behave very differently from homogeneous systems. Based on the homogeneous point of view, vegetation patterns classically are interpreted as early warning signals for tipping, i.e. the climate change driven collapse of an ecosystem. Mathematical studies of (heterogeneous) patterns recently showed that patterned states increase the resilience of the ecosystem and may instead enable it to evade tipping. Mathematical insights in pattern formation thus have an increasing impact in an expanding scientific spectrum. However, the literature typically is technical and abstract: there is a lack of comprehensive material that makes these techniques accessible for non-specialists.

This minitutorial aims to fill this gap. We will provide an entry-level overview of modern pattern formation theory on the emergence of patterns, and the equilibrium and transient behaviour of patterned systems. We will not go into all the technical details, but instead focus on the explicit 'tools' and ideas that can help to study a concrete model or real system. We will also highlight the implications and consequences of pattern formation theory for modelers and data analysts.