Figure - available from: Nonlinearity
This content is subject to copyright. Terms and conditions apply.
A sample singular ε=0 bifurcation diagram in (a/m,c) parameter space. The solid green line indicates stripe solutions, while the solid purple line denotes the gap solutions. Vegetation-to-desert fronts are indicated by the dashed green line. Finally, desert-to-vegetation front solutions are given by the dashed and solid purple lines. Schematic depictions of the associated singular limit geometries are depicted in the insets, where the labels D and V denote the locations of the desert and vegetated equilibrium states, respectively. The precise bifurcation structure depends on the value of the parameter b; see section 2.4.
Source publication
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...
Similar publications
Pattern formation in clouds is a well-known feature, which can be observed almost every day. However, the guiding processes for structure formation are mostly unknown, and also theoretical investigations of cloud pattern are quite rare. From many scientific disciplines the occurrence of pattern in non-equilibrium systems due to Turing instabilities...
Airway smooth muscle cells (ASMCs) exist in a form of helical winding bundles within the bronchial airway wall. Such tubular tissue provides cells with considerable curvature as a physical constraint, which is widely thought as an important determinant of cell behaviors. However, this process is difficult to mimic in the conventional planar cell cu...
Exploiting data invariances is crucial for efficient learning in both artificial and biological neural circuits. Understanding how neural networks can discover appropriate representations capable of harnessing the underlying symmetries of their inputs is thus crucial in machine learning and neuroscience. Convolutional neural networks, for example,...
Wavelength selection in reaction--diffusion systems can be understood as a coarsening process that is interrupted by counteracting processes at a certain wavelength. We show that coarsening in mass-conserving reaction--diffusion systems is driven by self-amplifying mass transport between neighboring high-density domains. We derive a general coarsen...
In recent years, nonreciprocally coupled systems have received growing attention. Previous work has shown that the interplay of nonreciprocal coupling and Goldstone modes can drive the emergence of temporal order such as traveling waves. We show that these phenomena are generically found in a broad class of pattern-forming systems, including mass-c...
Citations
... Our second example is the logistic Klausmeier model introduced by Bastiaansen et al. (2019), ...
... The logistic term in (26) restricts the height of individual gaps, causing the gaps to grow in width instead; this was also observed in Bastiaansen et al. (2019) for (29) with parameter values (31). The dark green represents a vegetated state ( u ≈ 0.6), while the lighter yellow represents a bare state ( u ≈ 0). ...
... Over time, these patterns resemble radial fronts connecting the bare state to the vegetated state, similar to the axisymmetric and one-dimensional fronts studied in Bastiaansen et al. (2019); Byrnes et al. (2023);Carter et al. (2023). However, the geometry of the front interface is highly non-trivial, and so it is unclear whether these structure could be analysed using similar techniques. ...
Localised patterns are often observed in models for dryland vegetation, both as peaks of vegetation in a desert state and as gaps within a vegetated state, known as ‘fairy circles’. Recent results from radial spatial dynamics show that approximations of localised patterns with dihedral symmetry emerge from a Turing instability in general reaction–diffusion systems, which we apply to several vegetation models. We present a systematic guide for finding such patterns in a given reaction–diffusion model, during which we obtain four key quantities that allow us to predict the qualitative properties of our solutions with minimal analysis. We consider four well-established vegetation models and compute their key predictive quantities, observing that models which possess similar values exhibit qualitatively similar localised patterns; we then complement our results with numerical simulations of various localised states in each model. Here, localised vegetation patches emerge generically from Turing instabilities and act as transient states between uniform and patterned environments, displaying complex dynamics as they evolve over time.
... However, this model does not take other processes into account that may influence vegetation dynamics. Therefore, several extensions to the Klausmeier model have been proposed, that include: (i) the emergence of stationary patterns on flat terrains [25][26][27][28][29][30], (ii) the influence of inertial effects [28][29][30][31][32][33], (iii) the presence of secondary seed dispersal [32][33][34][35], (iv) the occurrence of toxicity compounds [36][37][38][39], and (v) a finite soil carrying capacity [40][41][42][43]. ...
... Inspired by the observations depicted in Figure 1, we investigate in this paper the existence of travelling pulses with a fixed wave speed C > 0. The existence of far-from-equilibrium pulse-type solutions has been studied in previous works, both in the context of specific models [40,46,52] and general classes of reaction-diffusion systems [53,54], using geometric singular perturbation theory (GSPT). We follow an analogous approach for our model (4) that includes autotoxicity. ...
In this work, an extension of the 1D Klausmeier model that accounts for the toxicity compounds is considered and the occurrence of travelling stripes is investigated. Numerical simulations are firstly conducted to capture the qualitative behaviours of the pulse-type solutions and, then, geometric singular perturbation theory is used to prove the existence of such travelling pulses by constructing the corresponding homoclinic orbits in the associated 4-dimensional system. A scaling analysis on the investigated model is performed to identify the asymptotic scaling regime in which travelling pulses can be constructed. Biological observations are extracted from the analytical results and the role of autotoxicity in travelling patterns is emphasized. Finally, the analytically constructed solutions are compared with the numerical ones, leading to a good agreement that confirms the validity of the conducted analysis. Numerical investigations are also carried out in order to gain additional information on vegetation dynamics.
... However, far less is known analytically concerning large amplitude or farfrom-onset planar vegetation patterns. A number of studies have considered existence and stability properties of banded vegetation (Bastiaansen et al. 2019;Carter and Doelman 2018;Doelman and van der Ploeg 2002;Sewalt and Doelman 2017), as well as desertification fronts (Carter et al. 2022;Fernandez-Oto et al. 2019), but less is known concerning far-from-onset radially symmetric vegetation patches. Prior work has considered small amplitude radial solutions (Hill 2021) and 1D simplifications (Jaibi et al. 2020). ...
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 solutions in the same model. In particular, we demonstrate the instability of spots of large radius by deriving an asymptotic relationship between a critical eigenvalue associated with the spot and a coefficient which encodes the sideband instability of a nearby stationary front. Furthermore, we demonstrate that spots are unstable to a range of perturbations of intermediate wavelength in the angular direction, provided the spot radius is not too small. Our results are accompanied by numerical simulations and spectral computations.
... Our second example is the logistic Klausmeier model introduced by Bastiaansen et al. [3], ...
... The logistic term in (2.26) restricts the height of individual gaps, causing the gaps to grow in width instead; this was also observed in [3] for one dimensional patterns. As a result, nearby gaps expand and coalesce, causing a collapse from a patterned state to a compact region of bare soil. ...
... Over time these patterns resemble radial fronts connecting the bare state to the vegetated state, similar to the axisymmetric and one dimensional fronts studied in [3,7,9]. However, the geometry of the front interface is highly nontrivial, and so it is unclear whether these structure could be analysed using similar techniques. ...
Localised patterns are often observed in models for dryland vegetation, both as peaks of vegetation in a desert state and as gaps within a vegetated state, known as `fairy circles'. Recent results from radial spatial dynamics show that approximations of localised patterns with dihedral symmetry emerge from a Turing instability in general reaction--diffusion systems, which we apply to several vegetation models. We present a systematic guide for finding such patterns in a given reaction--diffusion model, during which we obtain four key quantities that allow us to predict the qualitative properties of our solutions with minimal analysis. Our results are complemented by numerical simulations for various localised states in four well-established vegetation models, highlighting the universality of such solutions.
... In contrast, the analytical study of localized pattern formation for RD systems that combine diffusion, nonlinear reactions, and advection poses many new theoretical challenges (cf. [2,9,48,52]. The most common such RD models are chemotaxis-type systems, such as the prototypical Keller-Segel (KS) system [27,28], that are widely used to model how cells or bacteria direct their movements in response to an environmental chemical stimulus, such as observed in some foundational experiments (cf. ...
We analyze the existence, linear stability, and slow dynamics of localized 1D spike patterns for a Keller--Segel model of chemotaxis that includes the effect of logistic growth of the cellular population. Our analysis of localized patterns for this two-component reaction-diffusion (RD) model is based, not on the usual limit of a large chemotactic drift coefficient, but instead on the singular limit of an asymptotically small diffusivity $d_2=\epsilon^2\ll 1$ of the chemoattractant concentration field. In the limit $d_2\ll 1$, steady-state and quasi-equilibrium 1D multi-spike patterns are constructed asymptotically. To determine the linear stability of steady-state $N$-spike patterns we analyze the spectral properties associated with both the ''large'' ${\mathcal O}(1)$ and the ''small'' $o(1)$ eigenvalues associated with the linearization of the Keller--Segel model. By analyzing a nonlocal eigenvalue problem characterizing the large eigenvalues, it is shown that $N$-spike equilibria can be destabilized by a zero-eigenvalue crossing leading to a competition instability if the cellular diffusion rate $d_1$ exceeds a threshold, or from a Hopf bifurcation if a relaxation time constant $\tau$ is too large. In addition, a matrix eigenvalue problem that governs the stability properties of an $N$-spike steady-state with respect to the small eigenvalues is derived. From an analysis of this matrix problem, an explicit range of $d_1$ where the $N$-spike steady-state is stable to the small eigenvalues is identified. Finally, for quasi-equilibrium spike patterns that are stable on an ${\mathcal O}(1)$ time-scale, we derive a differential algebraic system (DAE) governing the slow dynamics of a collection of localized spikes. Unexpectedly, our analysis is rather closely related to the analysis of spike patterns for the Gierer--Meinhardt RD system.
... To represent the vegetation dynamics in drylands, we consider a modified version of the commonly used Klausmeier model (Bastiaansen et al., 2019;Eigentler, 2021;Klausmeier, 1999). Although this model has been largely used in the spatial context, we start with a nonspatial variant here to highlight the differences in behavior predicted from the spatial and the non-spatial version. ...
... Using 8(b), we have, Det(J)=(−1 − v 2 1,2 )(1 − 2hv 1,2 )w 1,2 v 1,2 + 2w 1,2 v 1,2 (1 − hv 1,2 )v 2 1,2 =−w 1,2 v 1,2 ((1 + v 2 1,2 )(1 − 2hv 1,2 ) − 2(1 − hv 1,2 )v 2 1,2 ) =−w 1,2 v 1,2 ((1 + v 2 1,2 ) − 2hv 1,2 (1 + v 2 1,2 ) − 2v 2 1,2 + 2hv 3 1,2 ) =w 1,2 v 1,2 (−1 + v 2 1,2 + 2hv 1,2 ) Again replacing w 1,2 v 1,2 using (8b), Det(J)= u 1 − hv 1,2 (−1 + v 2 1,2 + 2hv 1,2 ) E wv is stable when Tr(J(w, v)) < 0 and Det(J(w, v)) > 0. It is easy to see from above, Det(J) > 0 implies (−1 + v 2 1,2 + 2hv 1,2 ) > 0 or, v 1,2 > −h + (1 + h 2 ) and Det(J) < 0 implies v 1,2 < −h + (1 + h 2 ). It can be shown from (9) that v 1 < −h + √ 1 + h 2 and v 2 > −h + √ 1 + h 2 (see (Bastiaansen et al, 2019)). So the uniform state (u 2 , v 2 ) is stable and (u 1 , v 1 ) is unstable. ...
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.
... Furthermore, the present study also aims at elucidating the role played by inertia in the formation of bidimensional stationary patterns. This latter goal will be achieved by analyzing dynamics occurring in the context of dryland ecology, where the interactioncompetition between water and vegetation biomass gives rise to several fascinating patterns with different geometries [3,[26][27][28][29][30][31][32][33]. ...
... In detail, while P * 1,2 and P * 3,4 denote stripes that are perpendicular to the x-axis and z-axis, respectively, the points Q * 1,2,3,4 correspond to rhombic planform patterns. Owing to the common structure of the denominators of the coefficients appearing in (32), the stationary value of the pattern amplitude and their existence and stability character (33) are independent of inertial times. However, according to (31), the time evolution of the pattern amplitude is strictly ruled by inertia, so that hyperbolicity plays an active role in determining the transient regime from a uniform state toward a patterned state and between different patterned states. ...
... One interesting feature involves the possibility of nullifying hyperbolicity effects when the inertial times associated to the two species fully compensate each other, i.e. τ u = τ w ⇒ 1/γ 0 = d/µ 0 . According to (31), (32), inertial effects are related to the difference τ u − τ w , so that it is expected that, under this constraint, the time evolution of the pattern amplitudes becomes independent of inertial times, regardless of their values [18]. Numerical results carried out close to the parabolic limit (γ 0 = µ 0 = 10 5 and d = 500) and far from it (γ 0 = 1 and µ 0 = d = 500) confirm our prediction, as depicted in Fig. 6. ...
In this paper the formation of rhombic and hexagonal Turing patterns in bidimensional hyperbolic reaction-transport systems is addressed in the context of dryland ecology. To this aim, exploiting the guidelines of the Extended Thermodynamics Theory, a hyperbolic extension of the classical modified Klausmeier vegetation model for flat arid environments is here proposed. This framework accounts explicitly for the inertial effects associated with the involved species and allows a more suitable description of transient dynamics between a spatially-homogeneous steady-state towards a patterned configuration. To characterize the emerging Turing patterns, linear stability analysis on the uniform steady-states is firstly addressed. Then, to determine the Stuart-Landau equations ruling the time evolution of the pattern amplitudes close to onset, multiple-scale weakly nonlinear analysis is performed for the two abovementioned geometries. Finally, to validate the analytical predictions as well as to provide more insights into the spatio-temporal evolution of the resulting pattern in both transient and stationary regimes, numerical simulations have been also performed.
... Such as Jones [25] and Yanagida [42] proved that the fast pulses are stable for 0 < a < 1/2 by defining the Evans function [1,14]. Carter et al. [4] verified that the fast pulses obtained in [5] is nonlinearly stable for a ∈ [0, 1/2), where a can be taken as a small parameter by applying exponential dichotomies [7,8] and Lin's method [2,30,38]. ...
... In this paper, our investigation will be based on the results of Shen and Zhang [40], and discuss the stability of the travelling pulses obtained there, by the exponential dichotomy [2,4,7,8,24,39] and Lin's method [2,4,30,38]. ...
... In this paper, our investigation will be based on the results of Shen and Zhang [40], and discuss the stability of the travelling pulses obtained there, by the exponential dichotomy [2,4,7,8,24,39] and Lin's method [2,4,30,38]. ...
For a coupled slow--fast FitzHugh--Nagumo(FHN) equation derived from a reaction-diffusion-mechanics (RDM) model, Holzer, Doelman and Kaper in 2013 studied existence and stability of the travelling pulse, which consists of two fast orbit arcs and two slow ones, where one fast segment passes the unique fold point with algebraic decreasing and two slow ones follow normally hyperbolic critical curve segments. Shen and Zhang in 2019 obtained existence of the travelling pulse, whose two fast orbit arcs both exponentially decrease, and one of the slow orbit arcs could be normally hyperbolic or not at the origin. Here we characterize both nonlinear and spectral stabilities of this travelling pulse.
... However, far less is known analytically concerning large amplitude or far-from-onset planar vegetation patterns. A number of studies have considered existence and stability properties of banded vegetation [2,3,10,38], as well as desertification fronts [4,12], but less is known concerning far-from-onset radially symmetric vegetation patches. Prior work has considered small amplitude radial solutions [18] and 1D simplifications [19]. ...
... However, to our knowledge, no rigorous studies exist concerning large amplitude radially symmetric vegetation patches in a dryland ecosystem model. We consider the model introduced in [2] U t = ∆U + a − U − U V 2 V t = δ 2 ∆V − mV + U V 2 (1 − bV ), (1.1) which is a modification of the dryland ecosystem model originally proposed by Klausmeier [20]. This model also coincides with that studied in [11], in the case of a single species. ...
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 solutions in the same model. In particular, we demonstrate the instability of spots of large radius by deriving an asymptotic relationship between a critical eigenvalue associated with the spot and a coefficient which encodes the sideband instability of a nearby stationary front. Furthermore, we demonstrate that spots are unstable to a range of perturbations of intermediate wavelength in the angular direction, provided the spot radius is not too small. Our results are accompanied by numerical simulations and spectral computations.
... Especially in models of drylands, many patterned states have been analyzed, including coexistence states [37,61], although spatial heterogeneities are not often taken into account, with few exceptions [62,63]. However, the models considered are typically more complicated than (4) and more advanced mathematical techniques are used to analyze them that are beyond the scope of this article. ...
... However, dynamics of such models can be much more difficult. For instance, models with 2D or 3D spatial domains, the spatial interfaces can have complicated structures, and in systems with multiple components might also undergo bifurcations [37,61,74,75]. ...
... This equation can be tackled by using a method similar to the one employed in [61,62]. For this, we first differentiate (A.6) with respect to ξ to obtain ...
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.
