arXiv subjects

Robbin Bastiaansen

Publications and source records attributed to Robbin Bastiaansen.

15 recordsLinked to original sources

arXiv2026

Optimisation of tipping pathways in a spatially heterogeneous world

In spatially extended systems, tipping does not necessarily lead to a uniform, abrupt transition typical of low-dimensional conceptual climate models. Instead, spatially structured forms of tipping can emerge, for example, through front propagation and pinning, coexistence between alternative states, and pattern formation. Tipping can therefore remain partial or spatially heterogeneous rather than affecting the entire system. Existing work on spatial tipping in conceptual models remains mostly descriptive, and a general framework to influence tipping dynamics through spatial interventions is still lacking. Here, we introduce a constrained optimisation framework that systematically identifies spatial interventions designed to maximise resilience or promote recovery of a desirable state, subject to dynamical and resource constraints. The framework is general and applicable to a broad class of spatially extended systems. We illustrate it using the one-dimensional Allen-Cahn equation with spatial heterogeneity, a minimal bistable model in which the key spatial tipping processes in gradient systems can be analysed explicitly. In this setting, optimisation acts by shifting local tipping thresholds and controlling front dynamics. Our results show that small-scale local, targeted modifications can determine large-scale system outcomes: optimal interventions can prevent tipping, induce recovery of the desirable state, or confine collapse to limited regions of the domain.

math.DS↗

arXiv2025

Early warning of critical transitions: distinguishing tipping points from Turing destabilizations

Current early warning signs for tipping points often fail to distinguish between catastrophic shifts and less dramatic state changes, such as spatial pattern formation. This paper introduces a novel method that addresses this limitation by providing more information about the type of bifurcation being approached starting from a spatially homogeneous system state. This method relies on estimates of the dispersion relation from noisy spatio-temporal data, which reveals whether the system is approaching a spatially homogeneous (tipping) or spatially heterogeneous (Turing patterning) bifurcation. Using a modified Klausmeier model, we validate this method on synthetic data, exploring its performance under varying conditions including noise properties and distance to bifurcation. We also determine the data requirements for optimal performance. Our results indicate the promise of a new spatial early warning system built on this method to improve predictions of future transitions in many climate subsystems and ecosystems, which is critical for effective conservation and management in a rapidly changing world.

math.DS↗

arXiv2025

The role of coupling and timescales for interacting tipping elements

Sudden and abrupt changes can occur in a nonlinear system within many fields of science when such a system crosses a tipping point and rapid changes of the system occur in response to slow changes in an external forcing. These can occur when time-varying inputs cross a bifurcation. If an upstream system loses stability in this way it may cause a downstream system influenced by it to tip, especially if the downstream system evolves on a much faster timescale, in what we call an accelerating cascade of tipping elements. In this paper, we identify the conditions on the coupling and timescales of the systems resulting in various types of tipping (cascade) responses. We also present a prototypical example of a unidirectionally coupled pair of simple tipping elements with hysteresis. This allows us to map out the various types of response as a function of system parameters and to link it to bifurcations of the underlying system that may have multiple timescales.

math.DS↗

arXiv2025

Early warning skill, extrapolation and tipping for accelerating cascades

We investigate how nonlinear behaviour (both of forcing in time and of the system itself) can affect the skill of early warning signals to predict tipping in (directionally) coupled bistable systems when using measures based on critical slowing down due to the breakdown of extrapolation. We quantify the skill of early warnings with a time horizon using a receiver-operator methodology for ensembles where noise realisations and parameters are varied to explore the role of extrapolation and how it can break down. We highlight cases where this can occur in an accelerating cascade of tipping elements, where very slow forcing of a slowly evolving ``upstream'' system forces a more rapidly evolving ``downstream'' system. If the upstream system crosses a tipping point, this can shorten the timescale of valid extrapolation. In particular, ``downstream-within-upstream'' tipping will typically have warnings only on a timescale comparable to the duration of the upstream tipping process, rather than the timescale of the original forcing.

nlin.CD↗

arXiv2025

Multi-front dynamics in spatially inhomogeneous Allen-Cahn equations

Recent studies of biological, chemical, and physical pattern-forming systems have started to go beyond the classic `near onset' and `far from equilibrium' theories for homogeneous systems to include the effects of spatial heterogeneities. In this article, we build a conceptual understanding of the impact of spatial heterogeneities on the pattern dynamics of reaction-diffusion models. We consider the simplest setting of an explicit, scalar, bi-stable Allen-Cahn equation driven by a general small-amplitude spatially-heterogeneous term $\varepsilon F(U,U_x,x)$. In the first part, we perform an analysis of the existence and stability of stationary one-, two- and $N$-front patterns for general spatial heterogeneity $F(U,U_x,x)$. In addition, we explicitly determine the $N$-th order system of ODEs that governs the evolution of the front positions of general $N$-front patterns to leading order. In the second part, we focus on a particular class of spatial heterogeneities where $F(U,U_x,x) = H'(x) U_x + H''(x) U$ with $H$ either spatially periodic or localised. For spatially periodic heterogeneities, we show that the fronts of a multi-front pattern will get `pinned' if the distances between successive fronts are sufficiently large, {\it i.e.}, the multi-front pattern is attracted to a nearby stable stationary multi-front pattern. For localised heterogeneities, we determine all stationary $N$-front patterns, and show that these are unstable for $N > 1$. We find instead slowly evolving `trains' of $N$-fronts that collectively travel to $\pm \infty$, either with slowly decreasing or increasing speeds.

math.DS↗

arXiv2024

Assessment of Abrupt Shifts in CMIP6 Models using Edge Detection

Past research has shown that multiple climate subsystems might undergo abrupt shifts, such as the Arctic Winter sea ice or the Amazon rainforest, but there are large uncertainties regarding their timing and spatial extent. In this study we investigated when and where abrupt shifts occur in the latest generation of earth system models (CMIP6) under a scenario of 1\% annual increase in CO$_2$. We considered 82 ocean, atmosphere, and land variables across 57 models. We used a Canny edge detection method to identify abrupt shifts occurring on yearly to decadal timescales, and performed a connected component analysis to quantify the spatial extent of these shifts. The systems analyzed include the North Atlantic subpolar gyre, Tibetan Plateau, land permafrost, Amazon rainforest, Antarctic sea ice, monsoon systems, Arctic summer sea ice, Arctic winter sea ice, and Barents sea ice. Except for the monsoon systems, we found abrupt shifts in all of these across multiple models. Despite large inter-model variations, higher levels of global warming consistently increase the risk of abrupt shifts in CMIP6 models. At a global warming of 1.5 $^\circ \text{C}$, six out of ten studied climate subsystems already show large-scale abrupt shifts across multiple models.

physics.ao-ph↗

arXiv2023

Rethinking tipping points in spatial ecosystems

The theory of alternative stable states and tipping points has garnered substantial attention in the last several decades. It predicts potential critical transitions from one ecosystem state to a completely different state under increasing environmental stress. However, typically, ecosystem models that predict tipping do not resolve space explicitly. Ecosystems being inherently spatial, it is important to understand the effects of spatial processes. In fact, it has been argued that spatial dynamics can actually help ecosystems evade tipping. Here, using a dryland and a savanna-forest model as example systems, we provide a synthesis of several mechanisms by which spatial processes can change our predictions of tipping in ecosystems. We show that self-organized Turing patterns can emerge in drylands that help evade tipping, but that (non-Turing) patterns driven by environmental heterogeneity are key to evasion of tipping in humid savannas. Since the ecological interactions driving the dynamics of these ecosystems differ from each other, we suggest that tipping evasion mechanisms in ecosystems may be connected to the key ecological interactions in a system. This highlights the need for further research into the link between the two in order to formulate better strategies to make ecosystems resilient to global change.

nlin.PS↗

arXiv2022

Climate Response and Sensitivity: Timescales and Late Tipping Points

Climate response metrics are used to quantify the Earth's climate response to anthropogenic changes of atmospheric CO2. Equilibrium Climate Sensitivity (ECS) is one such metric that measures the equilibrium response to CO2 doubling. However, both in their estimation and their usage, such metrics make assumptions on the linearity of climate response, although it is known that, especially for larger forcing levels, response can be nonlinear. Such nonlinear responses may become visible immediately in response to a larger perturbation, or may only become apparent after a long transient. In this paper, we illustrate some potential problems and caveats when estimating ECS from transient simulations. We highlight ways that very slow timescales may lead to poor estimation of ECS even if there is seemingly good fit to linear response over moderate timescales. Moreover, such slow timescale might lead to late abrupt responses ("late tipping points") associated with a system's nonlinearities. We illustrate these ideas using simulations on a global energy balance model with dynamic albedo. We also discuss the implications for estimating ECS for global climate models, highlighting that it is likely to remain difficult to make definitive statements about the simulation times needed to reach an equilibrium.

physics.ao-ph↗

arXiv2021

Partial tipping in a spatially heterogeneous world

Many climate subsystems are thought to be susceptible to tipping - and some might be close to a tipping point. The general belief and intuition, based on simple conceptual models of tipping elements, is that tipping leads to reorganization of the full (sub)system. Here, we explore tipping in conceptual, but spatially extended and spatially heterogenous models. These are extensions of conceptual models taken from all sorts of climate system components on multiple spatial scales. By analysis of the bifurcation structure of such systems, special stable equilibrium states are revealed: coexistence states with part of the spatial domain in one state, and part in another, with a spatial interface between these regions. These coexistence states critically depend on the size and the spatial heterogeneity of the (sub)system. In particular, in these systems a tipping point might lead to a partial tipping of the full (sub)system, in which only part of the spatial domain undergoes reorganization, limiting the impact of these events on the system's functioning.

nlin.PS↗

arXiv2021

Projections of the Transient State-Dependency of Climate Feedbacks

When the climate system is forced, e.g. by emission of greenhouse gases, it responds on multiple time scales. As temperatures rise, feedback processes might intensify or weaken. Current methods to analyze feedback strength, however, do not take such state dependency into account; they only consider changes in (global mean) temperature and assume all feedbacks are linearly related to that. This makes (transient) changes in feedback strengths almost intangible and generally leads to underestimation of future warming. Here, we present a multivariate (and spatially explicit) framework that facilitates dissection of climate feedbacks over time scales. Using this framework, information on the composition of projected (transient) future climates and feedback strengths can be obtained. Moreover, it can be used to make projections for many emission scenarios through linear response theory. The new framework is illustrated using the Community Earth System Model version 2 (CESM2).

physics.ao-ph↗

arXiv2020

Multivariate Estimations of Equilibrium Climate Sensitivity from Short Transient Warming Simulations

One of the most used metrics to gauge the effects of climate change is the equilibrium climate sensitivity, defined as the long-term (equilibrium) temperature increase resulting from instantaneous doubling of atmospheric CO$_2$. Since global climate models cannot be fully equilibrated in practice, extrapolation techniques are used to estimate the equilibrium state from transient warming simulations. Because of the abundance of climate feedbacks - spanning a wide range of temporal scales - it is hard to extract long-term behaviour from short-time series; predominantly used techniques are only capable of detecting the single most dominant eigenmode, thus hampering their ability to give accurate long-term estimates. Here, we present an extension to those methods by incorporating data from multiple observables in a multi-component linear regression model. This way, not only the dominant but also the next-dominant eigenmodes of the climate system are captured, leading to better long-term estimates from short, non-equilibrated time series.

physics.ao-ph↗

arXiv2019

Modelling honey bee colonies in winter using a Keller-Segel model with a sign-changing chemotactic coefficient

Thermoregulation in honey bee colonies during winter is thought to be self-organised. We added mortality of individual honey bees to an existing model of thermoregulation to account for elevated losses of bees that are reported worldwide. The aim of analysis is to obtain a better fundamental understanding of the consequences of individual mortality during winter. This model resembles the well-known Keller-Segel model. In contrast to the often studied Keller-Segel models, our model includes a chemotactic coefficient of which the sign can change as honey bees have a preferred temperature: when the local temperature is too low, they move towards higher temperatures, whereas the opposite is true for too high temperatures. Our study shows that we can distinguish two states of the colony: one in which the colony size is above a certain critical number of bees in which the bees can keep the core temperature of the colony above the threshold temperature, and one in which the core temperature drops below the critical threshold and the mortality of the bees increases dramatically, leading to a sudden death of the colony. This model behaviour may explain the globally observed honey bee colony losses during winter.

q-bio.PE↗

arXiv2018

Pulse solutions for an extended Klausmeier model with spatially varying coefficients

Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coefficients. We rigorously establish existence of stationary pulse solutions by blending techniques from geometric singular perturbation theory with bounds derived from the theory of exponential dichotomies. Moreover, the spectral stability of these solutions is determined, using similar methods. It is found that, due to the break-down of translation invariance, the presence of spatially varying terms can stabilize or destabilize a pulse solution. In particular, this leads to the discovery of a pitchfork bifurcation and existence of stationary multi-pulse solutions.

math.AP↗

arXiv2018

Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems

In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. We consider a two-component reaction-diffusion-advection model of Klausmeier type describing the interplay of vegetation and water resources and the resulting dynamics of these patterns. We focus on the large advection limit on constantly sloped terrain, in which the diffusion of water is neglected in favor of advection of water downslope. Planar vegetation pattern solutions are shown to satisfy an associated singularly perturbed traveling wave equation, and we construct a variety of traveling stripe and front solutions using methods of geometric singular perturbation theory. In contrast to prior studies of similar models, we show that the resulting patterns are spectrally stable to perturbations in two spatial dimensions using exponential dichotomies and Lin's method. We also discuss implications for the appearance of curved stripe patterns on slopes in the absence of terrain curvature.

math.AP↗

arXiv2018

The dynamics of disappearing pulses in a singularly perturbed reaction-diffusion system with parameters that vary in time and space

We consider the evolution of multi-pulse patterns in an extended Klausmeier equation with parameters that change in time and/or space. We formally show that the full PDE dynamics of a $N$-pulse configuration can be reduced to a $N$-dimensional dynamical system describing the dynamics on a $N$-dimensional manifold $\mathcal{M}_N$. Next, we determine the local stability of $\mathcal{M}_N$ via the quasi-steady spectrum associated to evolving $N$-pulse patterns, which provides provides explicit information on the boundary $\partial\mathcal{M}_N$. Following the dynamics on $\mathcal{M}_N$, a $N$-pulse pattern may move through $\partial\mathcal{M}_N$ and `fall off' $\mathcal{M}_N$. A direct nonlinear extrapolation of our linear analysis predicts the subsequent fast PDE dynamics as the pattern `jumps' to another invariant manifold $\mathcal{M}_M$, and specifically predicts the number $N-M$ of pulses that disappear. Combining the asymptotic analysis with numerical simulations of the dynamics on the various invariant manifolds yields a hybrid asymptotic-numerical method describing the full process that starts with a $N$-pulse pattern and typically ends in the trivial homogeneous state without pulses. We extensively test this method against PDE simulations and deduce general conjectures on the nature of pulse interactions with disappearing pulses. We especially consider the differences between the evolution of irregular and regular patterns. In the former case, the disappearing process is gradual: irregular patterns loose their pulses one by one. In contrast, regular, spatially periodic, patterns undergo catastrophic transitions in which either half or all pulses disappear. However, making a precise distinction between these two drastically different processes is quite subtle, since irregular $N$-pulse patterns that do not cross $\partial\mathcal{M}_N$ typically evolve towards regularity.

math.AP↗