Tipping Points, Patterns & Pathways of Resilience

A ‘pattern’ can be the undulating surface of water waves driven by the wind, or the complex shape of a growing cancerous tumor. From a mathematical point of view, it is a ‘spatial structure’ whose evolution in time is governed by a dynamical system. The resilience of an ecosystem is a measure of its ability to cope with changing conditions. A non-resilient ecosystem is vulnerable against change, a small effect may trigger a ‘catastrophic breakdown’: driven by declining rainfall, a dryland may deteriorate and the vegetated ecosystem may ‘collapse’ into a desert. This is an example of a ‘tipping point’: an (eco)system ‘crashes’ from a preferred state into a much less attractive state when an external factor passes through a critical threshold. Tipping points are predicted to (potentially) occur in many climate subsystems – such as ocean circulation, melting ice sheets and ecosystems – and are considered as key factors that limit our understanding of the impact of climate change.
Together with ecologists Max Rietkerk and his group we have demonstrated that these two seemingly unrelated concepts are closely connected: pattern formation may offer an ecosystem under pressure a way out. Crucial to the idea of tipping is that a system can only be in a limited number of very different states, whereas systems exhibiting patterns have an abundance of typically very similar states – as may be intuitively clear from the image of rippling water. Instead of collapsing, a system capable of exhibiting patterns may follow a ‘pathway of resilience’ around the tipping point through the space of stable patterns. In this talk we'll discuss various ways in which ecosystems may form patterns -- and thus may evade tipping and be more resilient. By the nature of mathematics, we will also indicate how our understanding of fundamental mechanisms enables us to find evidence for tipping in the Dutch river system and to make predictions about the malignancy of invasive tumors. On the other hand, by the nature of applied mathematics, we will highlight that the ecological point of view fuels the need to develop novel mathematics.