Abstract
The theory of alternative stable states and tipping points has garnered a lot of attention in the last 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. As ecosystems are inherently spatial, it is important to understand the effects of incorporating spatial processes in models, and how those insights translate to the real world. Moreover, spatial ecosystem structures, such as vegetation patterns, are important in the prediction of ecosystem response in the face of environmental change. Models and observations from real savanna ecosystems and drylands have suggested that they may exhibit both tipping behavior as well as spatial pattern formation. Hence, in this paper, we use mathematical models of humid savannas and drylands to illustrate several pattern formation phenomena that may arise when incorporating spatial dynamics in models that exhibit tipping without resolving space. We argue that such mechanisms challenge the notion of large scale critical transitions in response to global change and reveal a more resilient nature of spatial ecosystems

Abstract
Climate tipping elements are large-scale subsystems of the Earth that may transgress critical thresholds (tipping points) under ongoing global warming, with substantial impacts on the biosphere and human societies. Frequently studied examples of such tipping elements include the Greenland Ice Sheet, the Atlantic Meridional Overturning Circulation (AMOC), permafrost, monsoon systems, and the Amazon rainforest. While recent scientific efforts have improved our knowledge about individual tipping elements, the interactions between them are less well understood. Also, the potential of individual tipping events to induce additional tipping elsewhere or stabilize other tipping elements is largely unknown. Here, we map out the current state of the literature on the interactions between climate tipping elements and review the influences between them. To do so, we gathered evidence from model simulations, observations, and conceptual understanding, as well as examples of paleoclimate reconstructions where multi-component or spatially propagating transitions were potentially at play. While uncertainties are large, we find indications that many of the interactions between tipping elements are destabilizing. Therefore, we conclude that tipping elements should not only be studied in isolation, but also more emphasis has to be put on potential interactions. This means that tipping cascades cannot be ruled out on centennial to millennial timescales at global warming levels between 1.5 and 2.0 ∘C or on shorter timescales if global warming surpassed 2.0 ∘C. At these higher levels of global warming, tipping cascades may then include fast tipping elements such as the AMOC or the Amazon rainforest. To address crucial knowledge gaps in tipping element interactions, we propose four strategies combining observation-based approaches, Earth system modeling expertise, computational advances, and expert knowledge.

Abstract
Climate response metrics are used to quantify the Earth's climate response to anthropogenic changes of atmospheric \COO. Equilibrium Climate Sensitivity (ECS) is one such metric that measures the equilibrium response to \COO\ 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.

Abstract
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 the crossing of a tipping point not necessarily leads to a full reorganization of the system. Instead, it might lead to a reorganization of only part of the spatial domain, limiting the impact of these events on the system's functioning.

Abstract
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. Such state dependencies cannot be fully captured with common linear regression techniques that relate feedback strengths linearly to changes in the global mean temperature. Hence, transient changes are difficult to track and it becomes easy to underestimate future warming this way. Here, we present a multivariate and spatial 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. The new framework is illustrated using the Community Earth System Model version 2 (CESM2).

Abstract
The concept of tipping points and critical transitions helps inform our understanding of the catastrophic effects that global change may have on ecosystems, Earth system components, and the whole Earth system. The search for early warning indicators is ongoing, and spatial self-organization has been interpreted as one such signal. Here, we review how spatial self-organization can aid complex systems to evade tipping points and can therefore be a signal of resilience instead. Evading tipping points through various pathways of spatial pattern formation may be relevant for many ecosystems and Earth system components that hitherto have been identified as tipping prone, including for the entire Earth system. We propose a systematic analysis that may reveal the broad range of conditions under which tipping is evaded and resilience emerges.

Abstract
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 CO2. 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.

Abstract
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.

Abstract
In a rapidly changing world, quantifying ecosystem resilience is an important challenge. Historically, resilience has been defined via models that do not take spatial effects into account. These systems can only adapt via uniform adjustments. In reality, however, the response is not necessarily uniform, and can lead to the formation of (self‐organised) spatial patterns – typically localised vegetation patches. Classical measures of resilience cannot capture the emerging dynamics in spatially self‐organised systems, including transitions between patterned states that have limited impact on ecosystem structure and productivity. We present a framework of interlinked phase portraits that appropriately quantifies the resilience of patterned states, which depends on the number of patches, the distances between them and environmental conditions. We show how classical resilience concepts fail to distinguish between small and large pattern transitions, and find that the variance in interpatch distances provides a suitable indicator for the type of imminent transition. Subsequently, we describe the dependency of ecosystem degradation based on the rate of climatic change: slow change leads to sporadic, large transitions, whereas fast change causes a rapid sequence of smaller transitions. Finally, we discuss how pre-emptive removal of patches can minimise productivity losses during pattern transitions, constituting a viable conservation strategy.

Abstract
Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coeffcients. 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.

Abstract
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.

Abstract
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.

Abstract
Spatial self-organization of dryland vegetation constitutes one of the most promising indicators for an ecosystem’s proximity to desertification. This insight is based on studies of reaction–diffusion models that reproduce visual characteristics of vegetation patterns observed on aerial photographs. However, until now, the development of reliable early warning systems has been hampered by the lack of more in-depth comparisons between model predictions and real ecosystem patterns. In this paper, we combined topographical data, (remotely sensed) optical data, and in situ biomass measurements from two sites in Somalia to generate a multilevel description of dryland vegetation patterns. We performed an in-depth comparison between these observed vegetation pattern characteristics and predictions made by the extended-Klausmeier model for dryland vegetation patterning. Consistent with model predictions, we found that for a given topography, there is multistability of ecosystem states with different pattern wavenumbers. Furthermore, observations corroborated model predictions regarding the relationships between pattern wavenumber, total biomass, and maximum biomass. In contrast, model predictions regarding the role of slope angles were not corroborated by the empirical data, suggesting that inclusion of small-scale topographical heterogeneity is a promising avenue for future model development. Our findings suggest that patterned dryland ecosystems may be more resilient to environmental change than previously anticipated, but this enhanced resilience crucially depends on the adaptive capacity of vegetation patterns.

Abstract
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.

Abstract
In this master thesis we study patterns in animal populations that arise due to a density dependent movement speed v of one of the involved species. This new description of the movement leads to a Cahn Hilliard equation describing the evolution of the concentration of the animal specie in question. Our main interest is a modiffcation of the generally used standard predator-prey reactiondiffusion type of description of the evolution of two interacting species, where the standard diffusive movement of one of the species is replaced with this fast Cahn-Hilliard like movement. This leads to a fourth order slow-fast partial differential equation, which forms the system that will be the main object of study in this thesis: $$ {\partial m \over \partial t} = d_m \nabla\left( v\left[v+m {\partial v \over \partial m}\right] \nabla m + vm {\partial v \over \partial a} \nabla a - \kappa \nabla \Delta m\right) + \varepsilon H(m,a) \\ {\partial a \over \partial t} = \varepsilon d_a \Delta a + \varepsilon G(m,a) $$ In this thesis we first present an in-depth literature study of the general Cahn-Hilliard system focusing on the evolution - both short and long term - of solutions starting from an uniform state. Subsequently we will analyze the full population model, with the Cahn-Hilliard like movement, on an one-dimensional spatial domain via a weakly non-linear stability analysis, leading to a (real) Ginzburg-Landau equation as amplitude equation for variations from steady states of the model. All our findings will be applied to a system describing the interaction between mussels and algae. This analytic approach, supplemented by numerical simulations on the one-dimensional model, is then used to explain the occurrence and behaviour of patterns in mussel beds.

Abstract
In this bachelor thesis we study models for the elastic deformations of patterned quasi 2D sheets. The main ingredients for these models are elastic beams and we focus on four different models for elastic beams in this study. We start from the theory of elasticity and derive four beam models for both inextensible and extensible beams, by using the Euler Lagrangian equations. One of these models is the classic Euler-Bernoulli beam model, which is valid for elastic beams that are not compressible and have only small, weakly non-linear deflections. The other models are extensions of this model and are valid for strong nonlinear behaviour and/or compressible beams. With these models we analyze the compression and buckling of beams with various boundary conditions such as a pinned-pinned beam and a beam attached to two circular, freely rotary nodes. Finally we also present some simulations for both a single beam between two circular, freely rotary nodes and the elastic patterned sheets.

Abstract
In the current Anthropocene, there is a need to better understand the catastrophic effects that climate and land-use change may have on ecosystems, earth system components and the whole Earth system. The concept of tipping points and critical transitions contributes to this understanding. Tipping occurs in a system when it is forced outside the basin of attraction of the original equilibrium, resulting in a critical transition to an alternative, often less-desirable, stable state. The general belief and intuition, based on simple conceptual models of tipping elements (i.e. ordinary differential equations), is that tipping leads to reorganization of the full (sub)system. In this talk, I will review and explore tipping in conceptual, but spatially extended, and potentially spatially heterogeneous, models (i.e. partial differential equations). In these spatially explicit models, additional stable states can emerge that are not uniform in space, such as Turing patterns and coexistence states (part of the domain in one state, the rest in another state with a spatial interface or front between these regions), which can lead to a different tipping behaviour. In particular, in these systems a tipping point might lead only to a slight restructuring of the system or to a tipping event in which only part of the spatial domain undergoes reorganization, limiting the impact of these events on the system’s functioning.

Abstract
Wetenschappers waarschuwen dat klimaatverandering kan leiden tot kantelpunten. Onomkeerbare situaties waarbij een ecosysteem omklapt. Van savanne naar woestijn, of een warme golfstroom die ineens stilvalt. Maar volgens klimaatwetenschapper en wiskundige Robbin Bastiaansen zijn ecosystemen veerkrachtiger dan we denken. Wat doet de natuur om zo’n omslagpunt te voorkomen?

Abstract
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 period. In this talk, I will illustrate some potential problems and caveats when estimating ECS from transient simulations. I highlight ways in which very slow timescales may lead to poor estimation of ECS even if there is a seemingly good fit to linear response over moderate timescales. Moreover, such slow timescale processes might lead to late abrupt responses ('late tipping points') associated with a system's nonlinearities. These ideas are illustrated using simulations on a global energy balance model with dynamic albedo. Also the implications for estimating ECS for global climate models are discussed, highlighting that it is likely to remain difficult to make definitive statements about the simulation times needed to reach an equilibrium.

Abstract
In the current Anthropocene, it has become clear that human activity can have large effects on many different ecosystems, climate subsystems and on the global climate of our Earth. For instance, greenhouse gas emissions force the global climate, leading to the current global warming. On smaller scales, these global human-induced climatic changes can have devastating effects and cause, for example, melting of glaciers, desertification and extinction of species. Hence, in order to prepare for what is yet to come, it is therefore vital to understand how these complex nonlinear systems work and respond to external forcing, and how they might interact. In this talk, I will illustrate how techniques and insights from dynamical system theory help in answering these talk. In particular, I will show that climate subsystems and ecosystems do respond to climate changes on many different time scales, including the possibility of climate surprises in the form of so-called tipping points. Further, I will illustrate how in these non-linear systems fast and slow dynamics can both occur -- and how that might lead to sudden late abrupt climate change, but also might help us to study these systems.

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.

Abstract
Over the last decades, there is a growing concern about the crossing of tipping points in climate (sub)systems and ecosystems, which can lead to critical shifts towards less desirable alternative states of these systems. There are now also many theoretical studies, where tipping points are related to bifurcation in dynamical systems - often very simple models with few variables, timescales and/or no spatial component. By contrast, the real system is a large multi-scale system, meaning that there many temporal (and spatial) scales in play. In this talk, I will set the scene for the session and give an overview of the theory of tipping. Hereby I will focus on the importance of timescales. I will also show some situations in which time scale separation can either surprise or help us. For instance, the presence of very slow time scales can lead to arbitrarily late tipping point surprises, whereas time scale separation might allow us to study the cascading effects that the tipping of one system might have on another.

Abstract
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.

Abstract
In the current Anthropocene, there is a need to better understand the catastrophic effects that climate and land-use change may have on ecosystems, earth system components and the whole Earth system. The concept of tipping points and critical transitions contributes to this understanding. Tipping occurs in a system when it is forced outside the basin of attraction of the original equilibrium, resulting in a critical transition to an alternative, often less-desirable, stable state. The general belief and intuition, based on simple conceptual models of tipping elements (i.e. ordinary differential equations), is that tipping leads to reorganization of the full (sub)system. In this talk, I will review and explore tipping in conceptual, but spatially extended, and potentially spatially heterogeneous, models (i.e. partial differential equations). In these spatially explicit models, additional stable states can emerge that are not uniform in space, such as Turing patterns and coexistence states (part of the domain in one state, the rest in another state with a spatial interface or front between these regions), which can lead to a different tipping behaviour. In particular, in these systems a tipping point might lead only to a slight restructuring of the system or to a tipping event in which only part of the spatial domain undergoes reorganization, limiting the impact of these events on the system's functioning.

Abstract
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 the crossing of a tipping point not necessarily leads to a full reorganization of the system. Instead, it might lead to a reorganization of only part of the spatial domain, limiting the impact of these events on the system’s functioning.

Abstract
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 the crossing of a tipping point not necessarily leads to a full reorganization of the system. Instead, it might lead to a reorganization of only part of the spatial domain, limiting the impact of these events on the system’s functioning.

Abstract
In the current Anthropocene, there is a need to better understand the catastrophic effects that climate and land-use change may have on ecosystems, earth system components and the whole Earth system. The concept of tipping points and critical transitions contributes to this understanding. Tipping occurs in a system when it is forced outside the basin of attraction of the original equilibrium, resulting in a critical transition to an alternative, often less-desirable, stable state. The general belief and intuition, based on simple conceptual models of tipping elements (i.e. ordinary differential equations), is that tipping leads to reorganization of the full (sub)system. In this talk, I will review and explore tipping in conceptual, but spatially extended, and potentially spatially heterogeneous, models (i.e. partial differential equations). In these spatially explicit models, additional stable states can emerge that are not uniform in space, such as Turing patterns and coexistence states (part of the domain in one state, the rest in another state with a spatial interface or front between these regions), which can lead to a different tipping behaviour. In particular, in these systems a tipping point might lead only to a slight restructuring of the system or to a tipping event in which only part of the spatial domain undergoes reorganization, limiting the impact of these events on the system’s functioning.

Abstract
In the current Anthropocene, there is a need to better understand the catastrophic effects that climate and land-use change may have on ecosystems, earth systems components and the whole Earth system. In this talk, I will give an overview of my efforts in this direction using techniques and insights from dynamical systems theory. First, I will talk about future global warming projections that can be made using global climate models and linear response theory. Second, I will highlight some of the limitations of such linear theory using conceptual models -- in particular, the presence of bifurcations (also called 'tipping points') might lead to highly nonlinear behaviour that includes large, rapid and irreversible changes. Finally, I will explain how the common nonlinear theory on tipping points (based on the theory of ordinary differential equations) might again be too simplistic to characterise the response of spatially extended systems (that can be modelled with partial differential equations) in which bifurcations might induce only small or local changes instead.

Abstract
N/A

Abstract
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 the crossing of a tipping point not necessarily leads to a full reorganization of the system. Instead, it might lead to a reorganization of only part of the spatial domain, limiting the impact of these events on the system's functioning.

Abstract
N/A

Abstract
In the current Anthropocene, there is a need to better understand the catastrophic effects that climate and land-use change may have on ecosystems, earth system components and the whole Earth system. The concept of tipping points and critical transitions contributes to this understanding. Tipping occurs in a system when it is forced outside the basin of attraction of the original equilibrium, resulting in a critical transition to an alternative, often less-desirable, stable state.

The general belief and intuition, based on simple conceptual models of tipping elements (i.e. ordinary differential equations), is that tipping leads to reorganization of the full (sub)system. In this talk, I will review and explore tipping in conceptual, but spatially extended, and potentially spatially heterogeneous, models (i.e. partial differential equations). In these spatially explicit models, additional stable states can emerge that are not uniform in space, such as Turing patterns and coexistence states (part of the domain in one state, the rest in another state with a spatial interface or front between these regions), which can lead to a different tipping behaviour. In particular, in these systems a tipping point might lead only to a slight restructuring of the system or to a tipping event in which only part of the spatial domain undergoes reorganization, limiting the impact of these events on the system’s functioning.

Abstract
In the current Anthropocene, there is the need to understand the catastrophic effects that climatic and land use changes may have on ecosystems. The concepts of resilience, tipping points and critical transitions contribute to this understanding. Historically, these concepts have been defined and studied via models that do not take spatial effects into account. These systems can only adapt via uniform adjustments. In reality, however, the response of an ecosystem is not necessarily uniform, and can instead lead to the formation of spatial structures, such as regular patterning of vegetation (e.g. so-called Turing patterns), or spatial interfaces between vegetation types or between vegetation and bare soil (e.g. so-called coexistence states). In this talk, I will give an overview of the more intricate dynamics of such spatial structures in spatially heterogeneous ecosystems, and highlight how the classic theory of resilience and tipping might need to be refined because of these dynamics. In particular, I will show how the crossing of a tipping point does not necessarily lead to a full reorganization of an ecosystem, but instead might lead to a reorganization in only part of the spatial domain, limiting the impact of these events on the ecosystem's functioning.

Abstract
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.

Abstract
De aarde warmt op. Vanaf de industrialisatie is de gemiddelde temperatuur op aarde al toegenomen met ongeveer 1,1 graden Celsius. Over de gevolgen hiervan wordt al meer dan een eeuw gespeculeerd en deze worden met het jaar concreter – en heftiger. Een aantal voorbeelden uit 2021: extreme hitte in Canada, hevige bosbranden in Griekenland en Turkije, een snel smeltende ijskap, en extreme neerslag met overstromingen in Limburg. Om deze negatieve gevolgen te beperken zijn er internationaal afspraken gemaakt om de opwarming van de aarde te beteugelen. Centraal staat hierin de doelstelling om deze opwarming te beperken tot (globaal gemiddeld) anderhalve graad Celsius ten opzichte van het pre-industriële tijdperk. Op het moment is het echter nog onzeker of deze afspraken wel genoeg zijn om onomkeerbare veranderingen aan ons klimaat af te wenden.

Deze onzekerheid komt voort uit de complexiteit van het klimaatsysteem. De gevolgen van klimaatverandering zorgen zelf namelijk weer voor een verdere verandering aan het klimaat. Als bijvoorbeeld het ijs op aarde smelt, zal er minder zonlicht weerkaatst worden en wordt het dus nóg warmer. Meer technisch gesproken: het klimaat is een niet-lineair dynamisch systeem. Tijdens deze voordracht wordt uiteengezet hoe dergelijke systemen wiskundig benaderd worden. In het bijzonder wordt stilgestaan bij het aanwezig zijn van zogenoemde kantelpunten in deze systemen: situaties wanneer een ogenschijnlijk kleine verandering (bijvoorbeeld een net ietsje hogere temperatuur) plots toch drastische effecten heeft op de toestand van het systeem (bijvoorbeeld het klimaat). Aan de hand van een aantal relatief simpele modellen wordt getoond hoe wiskunde centraal staat in het begrijpen van het klimaatsysteem, maar ook waarom het zo moeilijk is om een antwoord te geven op de vraag hoeveel graden teveel is.

Abstract
In the current Anthropocene, there is a need to better understand the catastrophic effects that climate and land-use change may have on ecosystems, earth system components and the whole Earth system. The concept of tipping points and critical transitions contributes to this understanding. Tipping occurs in a system when it is forced outside the basin of attraction of the original equilibrium, resulting in a critical transition to an alternative, often less-desirable, stable state. Many climate subsystems are thought to be susceptible to tipping -- such as the Amazonian rainforest, vegetation in semi-arid regions, cloud formation, turbidity in shallow lakes, ocean and atmospheric circulation, ice sheets and the Earth as a whole. The general belief and intuition, based on simple conceptual models of tipping elements (i.e. ordinary differential equations), is that tipping leads to reorganization of the full (sub)system. In this talk, I will review and explore tipping in conceptual, but spatially extended, and potentially spatially heterogeneous, models (i.e. partial differential equations). In these spatially explicit models, additional stable states can emerge that are not uniform in space, such as Turing patterns and coexistence states (part of the domain in one state, the rest in another state with a spatial interface or front between these regions), which can lead to a different tipping behaviour. In particular, in these systems a tipping point might lead only to a slight restructuring of the system or 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.

Abstract
When the climate system is forced, for example, by the emission of greenhouse gases, it responds on multiple time scales. As temperatures rise, feedback processes might intensify or weaken. For instance, when ice melts due to increased temperatures, less solar radiation gets reflected back to outer space. Such state dependencies cannot be fully captured with common linear regression techniques that relate feedback strengths linearly to changes in the global mean temperature. Hence, transient changes are difficult to track and it becomes easy to underestimate future warming this way. In this talk, I present a multivariate and spatial framework that facilitates the 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. The new framework is illustrated using the Community Earth System Model version 2.

Abstract
In the current Anthropocene, there is a need to better understand the catastrophic effects that climate and land-use change may have on ecosystems, earth system components and the whole Earth system. The concept of tipping points and critical transitions contributes to this understanding. Tipping occurs in a system when it is forced outside the basin of attraction of the original equilibrium, resulting in a critical transition to an alternative, often less-desirable, stable state. Many climate subsystems are thought to be susceptible to tipping -- such as the Amazonian rainforest, vegetation in semi-arid regions, cloud formation, turbidity in shallow lakes, ocean and atmospheric circulation, ice sheets and the Earth as a whole.

The general belief and intuition, based on simple conceptual models of tipping elements (i.e. ordinary differential equations), is that tipping leads to reorganization of the full (sub)system. In this talk, I will explore tipping in conceptual, but spatially extended, and potentially spatially heterogeneous, models (i.e. partial differential equations). In these spatially explicit models, additional stable states can emerge that are not uniform in space, such as Turing patterns and coexistence states (part of the domain in one state, the rest in another state with a spatial interface or front between these regions), which can lead to a different tipping behaviour. In particular, in these systems a tipping point might lead only to a slight restructuring of the system or 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.

Abstract
When the climate system is forced, for example, by the emission of greenhouse gases, it responds on multiple time scales. As temperatures rise, feedback processes might intensify or weaken. For instance, when ice melts due to increased temperatures, less solar radiation gets reflected back to outer space. Such state dependencies cannot be fully captured with common linear regression techniques that relate feedback strengths linearly to changes in the global mean temperature. Hence, transient changes are difficult to track and it becomes easy to underestimate future warming this way. In this talk, I present a multivariate and spatial framework that facilitates the 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. The new framework is illustrated using the Community Earth System Model version 2.

Abstract
As a response to climatic changes, many ecosystems self-organize into large-scale spatial patterns - for example, vegetation bands or patches emerging in the process of desertification. In mathematical models (singularly perturbed reaction-diffusion equations with parameters that may vary in time and space) these patterns present themselves as interacting localized structures (e.g. pulses). In this talk, I explain how the dynamics of these can be analyzed via (geometric singular) perturbation theory and numerical simulations. I will describe how their response to further environmental change is a combination of slow (migratory) pattern rearrangement (on a \(N\)-dimensional manifold \(\mathcal{M}_N\) ) and fast pattern-to-pattern transitions (from one manifold \(\mathcal{M}_N\) to another \(\mathcal{M}_M\), with \(M < N\)). This leads to a framework of interlinked phase portraits that describes the location (and possible annihilation) of the localized structures over time. Using this framework, I will indicate the effect ground topography can have on movement and stability of vegetation patterns. Furthermore, I will also highlight the differences between small and large pattern transitions and describe the dependency on the rate of climatic change: slow change leads to sporadic, large transitions, whereas fast change causes a rapid sequence of smaller transitions.

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In a rapidly changing world, quantifying ecosystem resilience is an important challenge. Historically, resilience has been defined via models that do not take spatial effects into account. These systems can only adapt via uniform adjustments. In reality, however, the response is not necessarily uniform, and can lead to the formation of (self-organised) spatial patterns - typically localised vegetation patches. Classical measures of resilience cannot capture the emerging dynamics in spatially self-organised systems, including transitions between patterned states that have limited impact on ecosystem structure and productivity. We present a framework of interlinked phase portraits that appropriately quantifies the resilience of patterned states, which depends on the number of patches, the distances between them and environmental conditions. We show how classical resilience concepts fail to distinguish between small and large pattern transitions, and find that the variance in interpatch distances provides a suitable indicator for the type of imminent transition. Subsequently, we describe the dependency of ecosystem degradation based on the rate of climatic change: slow change leads to sporadic, large transitions, whereas fast change causes a rapid sequence of smaller transitions. Finally, we discuss how pre-emptive removal of patches can minimise productivity losses during pattern transitions, constituting a viable conservation strategy.

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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 CO2. 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.

Abstract
When the climate system is forced by e.g. changes in atmospheric CO2, it responds to this change on multiple time scales, showing responses on time scales ranging from very short to very long. It is clear that the behavior over these time scales can be very different: for instance, as ice melts, the ice-albedo feedback becomes less and less important. Predominantly used climate projection methods, however, typically do not adequately take such state changes into account and are univariate: they only consider the global mean surface temperature -- assuming everything else is just linearly correlated to that. In this talk, I will show multivariate estimation techniques that are capable of tracking these state changes by incorporating additional observables into the analysis directly. This has two important advantages. First, such methods are better equipped to provide projections for the longer time scales (for instance, estimations of equilibrium climate sensitivity). Second, it makes it possible to estimate other observables directly -- without making assumptions on their relation to the global mean surface temperature -- which leads to better quantitative insights in how precisely the climate will change in the future (for instance, how climate feedback processes might change over time).

Abstract
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 CO2. 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.

Abstract
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 CO2. 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.

Abstract
One of the most used metrics to track the effects of climate change is the equilibrium climate sensitivity, formally defined as the long-term (equilibrium) temperature increase that results from an instantaneous doubling of atmospheric CO2. Since global climate models cannot be fully equilibrated, predictive techniques have been developed that estimate the equilibrium state from transient warming simulations. Because of the abundance of climate feedbacks -- that span a wide range of spatial and temporal scales -- it is hard to predict long-term behaviour from short time series; many of the predominantly used techniques, including the often used technique by Gregory et al (2004), are only capable of detecting the most dominant feedback, thus hampering their ability to give accurate long-term predictions. In this presentation, I will present work-in-progress on an extension to the "Gregory method", in which data from additional observables (i.e. effective global albedo and emissivity) is used to fit a linear (multi-component) system. This way, not only the dominant but also the next dominant dynamics of the climate system can be captured, which can lead to better predictions for the equilibrium climate sensitivity from shorter time series.

Abstract
De aarde warmt op; vanaf de industrialisatie is de gemiddelde temperatuur op aarde toegenomen (met een tempo dat erg snel is voor klimatologische begrippen) en is er sprake van klimaatverandering. Over de gevolgen hiervan wordt al meer dan een eeuw gespeculeerd en deze worden met het jaar concreter. Een aantal voorbeelden uit 2019: extreme droogte, extreme neerslag met overstromingen, toename in hoeveelheid en intensiteit van bosbranden, smeltende ijskappen, en heftigere stormen. Om deze negatieve gevolgen te beperken zijn er internationaal afspraken gemaakt om de opwarming van de aarde te beteugelen. Centraal staat hierin de doelstelling om deze opwarming te beperken tot twee graden Celsius (ten opzichte van het pre-industriële tijdperk). Op het moment is het echter nog onzeker of deze afspraken wel genoeg zijn; met andere woorden, misschien is twee graden wel te veel.

Deze onzekerheid komt voort uit de complexiteit van het klimaatsysteem. De gevolgen van klimaatverandering zorgen zelf namelijk weer voor een verdere verandering aan het klimaat. Als bijvoorbeeld het ijs op aarde smelt, zal er minder zonlicht weerkaatst worden en wordt het dus nóg warmer. Meer technisch gesproken: het klimaat is een niet-lineair dynamisch systeem. Tijdens deze voordracht wordt uiteengezet hoe dergelijke systemen wiskundig benaderd worden. In het bijzonder wordt stilgestaan bij het aanwezig zijn van zogenoemde kantelpunten in deze systemen: situaties wanneer een ogenschijnlijk kleine verandering (bijvoorbeeld een paar graden) plots toch drastische effecten heeft op de toestand van het systeem (bijvoorbeeld het klimaat). Aan de hand van een aantal relatief simpele modellen wordt getoond hoe wiskunde centraal staat in het begrijpen van het klimaatsysteem, maar ook waarom het zo moeilijk is om een antwoord te geven op de vraag of twee graden te veel is.

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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 to organise itself into large-scale patterns. The behaviour of these patterns can be analysed using (conceptual) reaction-(advection)-diffusion models, in which these patterns present themselves as localized structures (e.g. as pulse solution). In this talk, first I will present the results of a comparison between conceptual model and real vegetation pattern characteristics. Subsequently, I will explain how the found multistability leads to novel adaptation mechanisms, which can be understood via a mathematical analysis of the dynamics of (disappearing) semi-strong interacting pulses in an ecosystem model with parameters that (may) vary in time and space

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Vast, often populated, areas in dryland ecosystems face the dangers of desertification. As an important step in this process, the lack of resources forces the vegetation in these regions to organise itself in large-scale patterns, such as banded vegetation patterns. Under continued drought, more and more vegetation bands will disappear. The central question to answer is how to minimize biomass loss under these worsening climatological circumstances. In this talk, I will give a summary of a recent mathematical study on banded vegetation patterns in a simple dryland ecosystem model that can help to shed light on this issue. Moreover, I present preliminary results of numerical simulations that give insight in several possible vegetation maintenance strategies and their short-term and long-term advantages and disadvantages.

Abstract
Thermoregulation in honey bee colonies during winter is thought to be self-organised. We added mortality of individual 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 honey bee 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.

Abstract
Today, an increasing amount of vegetation patterns in semi-arid climates face the dangers of desertification. Since the availability of aerial photography in the 1940s, it is known that vegetation in these regions forms patterns when water becomes more scarce. Many models have been created that try to explain and describe the existence, creation and disappearance of these patterns. However, results of these theoretical studies are rarely compared to reality - and comparisons often only consider the visual appearance. Nowadays, data about wavenumber, biomass and migration speed of vegetation patterns becomes more readily available. Therefore more thorough comparisons are possible only now. In this talk, I show that this new data aligns well with predictions of reaction-advection-diffusion models. Furthermore, I discuss why this suggests that vegetation patterns are more resilient and more adaptive than is often believed.

Abstract
The availability of aerial photography around the 1940s led to the discovery of vegetation patterns in semi-arid climates. Since then many theoretical models have been created to explain and describe these patterns. Studies of these models required the use of advanced mathematics and physics behind fluid dynamics. However, the results of these studies were rarely compared to reality - and comparisons often were only visual. Nowadays, data about vegetation patterns becomes more readily available and thorough comparisons are therefore only now possible. In this talk, the theoretical predictions based on reaction-(advection)-diffusion models will be explained and compared to data of vegetation patterns in Somaliland. The similarities between the two show an important spread in a wavenumbers of these vegetation patterns. Moreover, this observation suggest multistability of vegetation patterns, which indicates that patterns in semi-arid climates might be more resilient than is often believed.

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Today, an increasing amount of vegetation patterns in semi-arid climates face the dangers of desertification. To understand the driving mechanics behind this effect many ecosystem models have been created over the last years - both very simplistic ones and very realistic ones. The current consensus indicates the importance of water availability for the survivability (and movement) of vegetation. In turn the water availability is greatly influenced by the topography.

Often, simple models ignore the topographical details whereas advanced models can only be studied by means of numerical PDE simulations. In this talk, we present one way to add topographical effects in a simplified ecosystem model (extended-Klausmeier) that results in a heterogeneous PDE that we can still study analytically for some cases. Specifically we study the topographical effects on localized vegetation pulses, i.e. one-dimensional patches. This leads to new observations, including pulses that move downhill.

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