Arjen Doelman

Leiden University | LEI · Mathematical Institute

Tipping occurs when a critical point is reached, beyond which a perturbation leads to persistent system change. Here, we present observational indications demonstrating presently ongoing noise‐tipping of a real‐world system. Noise in a river system is associated with the changing flow rate. In particular, we consider the upper Rhine River delta, wh...

In this article, we study a system of reaction-diffusion equations in which the diffusivities are widely separated. We report on the discovery of families of spatially periodic canard solutions that emerge from {\em singular Turing bifurcations}. The emergence of these spatially periodic canards asymptotically close to the Turing bifurcations, whic...

We consider a Gatenby--Gawlinski-type model of invasive tumors in the presence of an Allee effect. We describe the construction of bistable one-dimensional traveling fronts using singular perturbation techniques in different parameter regimes corresponding to tumor interfaces with, or without, acellular gap. By extending the front as a planar inter...

Reaction-diffusion models describing interactions between vegetation and water reveal the emergence of several types of patterns and travelling wave solutions corresponding to structures observed in real-life. Increasing their accuracy by also considering the ecological factor known as autotoxicity has lead to more involved models supporting the ex...

Grouping behaviour of prey animals is thought to be mainly driven by fear of predation and resource scarcity. Fear of predation often leads to small inter-individual distances, while resource scarcity leads to the opposite. Consequently, it is believed that the number of individuals in a group (group size) is an emergent property of the trade-off b...

We construct far-from-onset radially symmetric spot and gap solutions in a two-component dryland ecosystem model of vegetation pattern formation on flat terrain, using spatial dynamics and geometric singular perturbation theory. We draw connections between the geometry of the spot and gap solutions with that of traveling and stationary front soluti...

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 explicitl...

From 08-12 August, 2022, 32 individuals participated in a workshop, Stability and Fluctuations in Complex Ecological Systems, at the Lorentz Center, located in Leiden, The Netherlands. An interdisciplinary dialogue between ecologists, mathematicians, and physicists provided a foundation of important problems to consider over the next 5-10 years. Th...

Spatial self-organization of ecosystems into large-scale (from micron to meters) patterns is an important phenomenon in ecology, enabling organisms to cope with harsh environmental conditions and buffering ecosystem degradation. Scale-dependent feedbacks provide the predominant conceptual framework for self-organized spatial patterns, explaining re...

We consider a class of singularly perturbed 2-component reaction–diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems, such fronts can exhibit an instability...

We construct far-from-onset radially symmetric spot and gap solutions in a two-component dryland ecosystem model of vegetation pattern formation on flat terrain, using spatial dynamics and geometric singular perturbation theory. We draw connections between the geometry of the spot and gap solutions with that of traveling and stationary front soluti...

We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems , such fronts can exhibit an instability...

We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems, such fronts can exhibit an instability...

Localized patterns in singularly perturbed reaction–diffusion equations typically consist of slow parts, in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system, alternated by fast excursions, in which the solution jumps from one slow manifold to another, or back to the original slow manifold. In t...

Resilience to tipping points in ecosystems Spatial pattern formation has been proposed as an early warning signal for dangerous tipping points and imminent critical transitions in complex systems, including ecosystems. Rietkerk et al . review how ecosystems and Earth system components can actually evade catastrophic tipping through various pathways...

Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts -- in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system -- alternated by fast excursions -- in which the solution jumps from one slow manifold to another, or back to the original slow manifold...

We present the multicomponent functionalized free energies that characterize the low-energy packings of amphiphilic molecules within a membrane through a correspondence to connecting orbits within a reduced dynamical system. To each connecting orbit we associate a manifold of low energy membrane-type configurations parameterized by a large class of...

In this paper we consider the 2-component reaction–diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics (Gilad et al., 2004). The nonlinear structure of this model is more involved than other more conceptual models, such as the extended Klausmeier model, and the an...

In this paper we consider the 2-component reaction-diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics. The nonlinear structure of this model is more involved than other more conceptual models, such as the extended Klausmeier model, and the analysis a priori is mo...

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‐organi...

We present the multicomponent functionalized free energies that characterize the low energy packings of amphiphilic molecules within a membrane through a correspondence to connecting orbits within a reduced dynamical system. To each connecting orbits we associate a manifold of low energy membrane-type configurations parameterized by a large class o...

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 resultin...

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...

Ecosystems’ responses to changing environmental conditions can be modulated by spatial self-organization. A prominent example of this can be found in drylands, where formation of vegetation patterns attenuates the magnitude of degradation events in response to decreasing rainfall. In model studies, the pattern wavelength responds to changing condit...

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...

The Klausmeier equation is a widely studied reaction-diffusion-advection model of vegetation pattern formation on gently sloped terrain in semiarid ecosystems. We consider the case of constantly sloped terrain and study the formation of planar vegetation stripe patterns which align in the direction transverse to the slope and travel uphill. These p...

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 resultin...

Significance Today, vast areas of drylands in semiarid climates face the dangers of desertification. To understand the driving mechanisms behind this effect, many theoretical models have been created. These models provide insight into the resilience of dryland ecosystems. However, until now, comparisons with reality were merely visual. In this arti...

In this article, a general geometric singular perturbation framework is developed to study the impact of strong, spatially localized, nonlinear impurities on the existence, stability and bifurcations of localized structures in systems of linear reaction–diffusion equations. By taking advantage of the multiple-scale nature of the problem, we derive...

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...

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...

It has been observed in the Gierer-Meinhardt equations that destabilization mechanisms are rather complex when spatially periodic pulse patterns approach a homoclinic limit. In this paper we show that this holds in much broader generality. While decreasing the wave number k, the character of destabilization alternates between two kinds of Hopf inst...

In this paper, we introduce a novel simplification method for dealing with physical systems that can be thought to consist of two subsystems connected in series, such as a neuron and a synapse. The aim of our method is to help find a simple, yet convincing model of the full cascade-connected system, assuming that a satisfactory model of one of the...

Semiarid ecosystems form the stage for a plethora of vegetation patterns, a feature that has been captured in terms of mathematical models since the beginning of this millennium. To study these patterns, we use a reaction-advection-diffusion model that describes the interaction of vegetation and water supply on gentle slopes. As water diffuses much...

Theory suggests that gradual environmental change may erode the resilience of ecosystems and increase their susceptibility to critical transitions. This notion has received a lot of attention in ecology in recent decades. An important question receiving far less attention is whether ecosystems can cope with the rapid environmental changes currently...

In this manuscript, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, i.e., defined on R. This problem corresponds to analyzing a discontinuous and non-autonomous n-dimensional system, (Equation) under the assump...

We consider a two-component system of evolutionary partial differential equations posed on a bounded domain. Our system is pattern forming, with a small stationary pattern bifurcating from the background state. It is also equipped with a multiscale structure, manifesting itself through the presence of spectrum close to the origin. Spatial processes...

For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in a...

In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the Evans function of the full linear stability problem can be approximated by a product of a slow and a fast reduced Evans function, which correspond to properl...

In this paper we develop an extended center manifold reduction method: a methodology to analyze the formation and bifurcations of small-amplitude patterns in certain classes of multi-component, singularly perturbed systems of partial differential equations. We specifically consider systems with a spatially homogeneous state whose stability spectrum...

We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numeric...

The functionalized Cahn-Hilliard (FCH) free energy models interfacial energy in amphiphilic phase separated mixtures. Its minimizers encompass a rich class of morphologies with detailed inner structure, including bilayers, pore networks, pearled pores, and micelles. We address the existence and linear stability of α-single curvature bilayer structu...

In recent years, methods have been developed to study the existence, stability and bifurcations of pulses in singularly perturbed reaction–diffusion equations in one space dimension, in the context of a number of explicit model problems, such as the Gray–Scott and the Gierer–Meinhardt equations. Although these methods are in principle of a general...

The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with long-range inhibition of growth and survival is an essential prerequisite for pattern formation....

In this article, we analyze traveling waves in a reaction–diffusion-mechanics (RDM) system. The system consists of a modified FitzHugh–Nagumo equation, also known as the Aliev–Panfilov model, coupled bidirectionally with an elasticity equation for a deformable medium. In one direction, contraction and expansion of the elastic medium decreases and i...

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and s...

We rigorously derive multi-pulse interaction laws for the semi-strong interactions in a family of singularly-perturbed and weakly-damped reaction-diffusion systems in one space dimension. Most significantly, we show the existence of a manifold of quasi-steady N-pulse solutions and identify a "normal-hyperbolicity" condition which balances the asymp...

We consider inhomogeneous non-linear wave equations of the type utt=uxx+V′(u,x)−αututt=uxx+V′(u,x)−αut (α⩾0α⩾0). The spatial real axis is divided in intervals IiIi, i=0,…,N+1i=0,…,N+1 and on each individual interval the potential is homogeneous, i.e., V(u,x)=Vi(u)V(u,x)=Vi(u) for x∈Iix∈Ii. By varying the lengths of the middle intervals, typically o...

We study in detail the existence and stability of localized pulses in a Gierer-Meinhardt equation with an additional “low” nonlinearity. This system is an explicit example of a general class of singularly perturbed, two component reaction-diffusion equations that goes significantly beyond well-studied model systems such as Gray-Scott and Gierer-Mei...

Two pulse solutions play a central role in the phenomena of selfreplicating pulses in D reactiondiusion systems In this work we focus on the D GrayScott model as a prototype We carry out an existence and stability study for solutions consisting of two pulses moving apart from each other with slowly varying velocities In the various parameter regime...

Co-limitation of marine phytoplankton growth by light and nutrient, both of which are essential for phytoplankton, leads to complex dynamic behaviour and a wide array of coherent patterns. The building blocks of this array can be considered to be deep chlorophyll maxima, or DCMs, which are structures localized in a finite depth interior to the wate...

We consider a Josephson junction system installed with a finite length inhomogeneity, either of microresistor or of microresonator type. The system can be modelled by a sine-Gordon equation with a piecewise-constant function to represent the varying Josephson tunneling critical current. The existence of pinned fluxons depends on the length of the i...

1 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA 2 Mathematisch Instituut, Universiteit Leiden, PO Box 9512, 2300 RA Leiden, Netherlands 3 Department of Mathematics and Center for BioDynamics, Boston University, 111 Cummington Street, Boston, MA 02215, USA 4 Laboratory of Nonlinear Studies and Comput...

The three-component reaction-diffusion system introduced in [C. P. Schenk et al., Phys. Rev. Lett., 78 (1997), pp. 3781] has become a paradigm model in pattern formation. It exhibits a rich variety of dynamics of fronts, pulses, and spots. The front and pulse interactions range in type from weak, in which the localized structures interact only thro...

Co-limitation of marine phytoplankton by light and nutrient leads to complex dynamic behavior and a wide array of coherent patterns. The building blocks of this array can be considered to be deep chlorophyll maxima, or DCMs, which are structures localized in the vertical direction. From an ecological point of view, DCMs are evocative of a balance b...

In this paper we introduce a novel generic destabilization mechanism for (reversible) spatially periodic patterns in reaction-diffusion equations in one spatial dimension. This Hopf dance mechanism occurs for long wavelength patterns near the homoclinic tip of the associated Busse balloon (= the region in (wave number, parameter space) for which st...

We investigate the dynamics of weakly-modulated nonlinear wave trains. For reaction-diffusion systems and for the complex Ginzburg--Landau equation, we establish rigorously that slowly varying modulations of wave trains are well approximated by solutions to Burgers equation over the natural time scale. In addition to the validity of Burgers equatio...

Internal waves provide a source of energy for mixing in the deep sea. At locations with rough topography these internal waves often take the shape of oblique beams. We consider the question if these beams transport tracers and enhance mixing. We present proof, from theory and laboratory experiments, that internal wave beams indeed drive transport,...

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk etal. (PRL 78:3781–3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a pr...

Recently, it has been discovered that the dynamics of phytoplankton concentrations in an ocean exhibit a rich variety of patterns, ranging from trivial states to oscillating and even chaotic behavior [J. Huisman, N. N. Pham Thi, D. M. Karl, and B. P. Sommeijer (2006), Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll m...

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a par...

We consider a spatially non-autonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-$\pi$ Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The non-autonomous character is due to the presence of a discontinuity point, n...

In deze strategie voor de Nederlandse wiskunde komt een nieuwe generatie wiskundehoogleraren aan het woord. Zij representeren de onderzoekers die de komende jaren het wiskundeonderzoek in Nederland gaan trekken. Hier formuleren ze de voorwaarden die ze zich voor hun onderzoek wensen. Zo kunnen zij hun onderzoeksambities verwezenlijken. De commissie...

Dit masterplan is na uitgebreide consultatie opgesteld namens de academische wiskundige gemeenschap van Nederland. De Nederlandse wiskunde heeft dankzij de recente clustering van het onderzoek, de toenemende nationale samenwerking in het wetenschappelijk onderwijs en de door het ministerie van OCW aangekondigde hervormingen van het voortgezet onder...

Dynamics of complex systems is one of the program 5 themes in the NWO (Netherlands Organisation for Scientific Research) strategy for the years 2007-2011. The ambition of the current proposal is to initiate integrated activities in the field of complex systems within the Netherlands, to provide opportunities for the Dutch scientific community to st...

In this paper, we study the existence and stability of pulse solutions in a system with interacting instability mechanisms, which is described by a Ginzburg–Landau equation for an A-mode, coupled to a diffusion equation for a B-mode. Our main question is whether this coupling may stabilize solutions of the Ginzburg–Landau equation that are unstable...

We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the strongly localized activator pulses interact through the weakly localized inhibitor, the interaction is not tail-tail as in the weak inte...

We consider a spatially non-autonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-$\pi$ Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The non-autonomous character is due to the presence of a discontinuity point, n...

We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray–Scott system: u″=uv2, v″=v−uv2, which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray–Scott model. Specifically, we establish the existence, and study properties, of solution branches...

Bioremediation is a promising technique for cleaning contaminated soil. We study an idealized bioremediation model involving a substrate (contaminant to be removed), electron acceptor (added nutrient), and microorganisms in a one-dimensional soil column. Using geometric singular perturbation theory, we construct traveling waves (TW) corresponding t...

We study the appearance and the relevance of edge bifurcations in the stability problem associated to a front pattern in a certain class of (singularly perturbed) bi-stable reaction-diffusion equations.