The trade-off between the need to obtain new knowledge and the need to use that knowledge to impr... more The trade-off between the need to obtain new knowledge and the need to use that knowledge to improve performance is one of the most basic trade-offs in nature, and optimal performance usually requires some balance between exploratory and exploitative behaviors. Researchers in many disciplines have been searching for the optimal solution to this dilemma.
Discrete and Continuous Dynamical Systems - Series B, 2011
In this article, we consider the dryland vegetation model proposed by Gilad et al [6,. This model... more In this article, we consider the dryland vegetation model proposed by Gilad et al [6,. This model consists of three nonlinear parabolic partial differential equations, one of which is degenerate parabolic of the family of the porous media equation , and we prove the existence of its weak solutions. Our approach based on the classical Galerkin methods combines and makes use of techniques, parabolic regularization, truncation, maximum principle, compactness. We observe in this way various properties and regularity results of the solutions.
Egu General Assembly Conference Abstracts, May 1, 2010
According to the habitat heterogeneity hypothesis spatial heterogeneity positively correlates wit... more According to the habitat heterogeneity hypothesis spatial heterogeneity positively correlates with species diversity. Numerous studies have confirmed this hypothesis in various ecological contexts, but little attention has been given to the origin of spatial heterogeneity and to possible heterogeneity-diversity feedbacks. In particular, the link between spatial heterogeneity induced by vegetation pattern formation, and species diversity has remained unexplored. In this presentation we describe a mathematical modeling approach for exploring this link in the context of water-limited vegetation, and apply it to woody-herbaceous systems. We first address the relation between vegetation-landscape diversity and resource diversity, and discuss mechanisms of species-diversity change. We show that woody patches can buffer species-diversity loss as a result of an aridity stress, and that species diversity can also be affected by woody-pattern changes at the landscape scale. We then describe the derivation of community-level properties, such as diversity-resource relations. Community-level properties are derived by extending the space over which biomass variables are defined to include trait subspaces, where different points represent distinct species. We demonstrate this approach with a simple example of a spatially uniform herbaceous community, choosing the tradeoff between investments in above and below-ground biomass as the axis that spans the trait subspace. We then extend this study to include the effect of a spatially localized woody patch on the diversity of herbaceous species. We conclude by delineating directions for further model studies and development.
In bistable systems, the stability of front structures often influences the dynamics of extended ... more In bistable systems, the stability of front structures often influences the dynamics of extended patterns. We show how the combined effect of an instability to curvature modulations and proximity to a pitchfork front bifurcation leads to spontaneous nucleation of spiral waves along the front. This effect is demonstrated by direct simulations of a FitzHugh-Nagumo (FHN) model and by simulations of order parameter equations for the front velocity and curvature. Spontaneous spiral-wave nucleation often results in a state of spatiotemporal disorder involving repeated events of spiral wave nucleation, domain splitting and spiral wave annihilation.
Patterned vegetation is characterisic for water-limited areas and both periodic and scale-free di... more Patterned vegetation is characterisic for water-limited areas and both periodic and scale-free distributions of vegetation patch sizes have been reported. Using a simple common mathematical modeling approach we study the physical and ecological conditions for the emergence of both types of patterns. By drawing an analogy between the vegetation model and a simpler inhibitor-activator model we discuss if scale-free patterns of dryland vegetation represent asymptotic patterns or more likely long transients.
We use the context of dryland vegetation to study a general problem of complex pattern forming sy... more We use the context of dryland vegetation to study a general problem of complex pattern forming systems - multiple pattern-forming instabilities that are driven by distinct mechanisms but share the same spectral properties. We find that the co-occurrence of such instabilities results in the growth of a single mode rather than two interacting modes. The interplay between the two mechanisms, which promote or counteract each other, compensates for the simpler dynamics of a single mode by inducing higher pattern diversity. Possible implications to biodiversity of ecosystems are discussed.
We discuss an explicit-space model for vegetation dynamics in water-limited ecosystems. Under con... more We discuss an explicit-space model for vegetation dynamics in water-limited ecosystems. Under conditions of water stress, vegetation patterns such as spots, stripes and gaps spontaneously emerge. In this contribution, we focus on the interplay between water-vegetation feedbacks and rainfall intermittency, and we discuss an extended version of the model which allows for studying how evapotranspirations fluxes are modified by the presence of patterns.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015
We study resonant spatially periodic solutions of the Lengyel-Epstein model modified to describe ... more We study resonant spatially periodic solutions of the Lengyel-Epstein model modified to describe the chlorine dioxide-iodine-malonic acid reaction under spatially periodic illumination. Using multiple-scale analysis and numerical simulations, we obtain the stability ranges of 2:1 resonant solutions, i.e., solutions with wavenumbers that are exactly half of the forcing wavenumber. We show that the width of resonant wavenumber response is a non-monotonic function of the forcing strength, and diminishes to zero at sufficiently strong forcing. We further show that strong forcing may result in a π/2 phase shift of the resonant solutions, and argue that the nonequilibrium Ising-Bloch front bifurcation can be reversed. We attribute these behaviors to an inherent property of forcing by periodic illumination, namely, the increase of the mean spatial illumination as the forcing amplitude is increased.
A quasi-2-dimensional stationary spot in a disk-shaped chemical reactor is observed to bifurcate ... more A quasi-2-dimensional stationary spot in a disk-shaped chemical reactor is observed to bifurcate to an oscillating spot when a control parameter is increased beyond a critical value. Further increase of the control parameter leads to the collapse and disappearance of the spot. Analysis of a bistable activator-inhibitor model indicates that the observed behavior is a consequence of interaction of the front with the boundary near a parity breaking front bifurcation.
We present a detailed theory of the appearance of low dimensional dynamics in a hydrodynamic syst... more We present a detailed theory of the appearance of low dimensional dynamics in a hydrodynamic system. The system chosen has been subjected to a careful experimental study; it involves dynamics of surface waves in a cylinder of fluid which is oscillated vertically. We apply center manifold and normal form theories to derive low dimensional nonlinear evolution equations and an approximate solution of the original partial differential equations. All the major experimental findings are rationalized by the theory: the appearance of periodic and chaotic mixed mode states, the existence of an asymmetry between the interacting modes, the qualitative structure of the phase diagram in the vicinity of the critical point and the presence of enslaved stable modes in the surface wave patterns. We also present theoretical predictions concerning the nature of the periodic mixed mode state and the mechanism of the onset of chaos. The periodic dynamics involve a sequence of homoclinic bifurcations in which periodic orbits glue and disglue alternatively. At the bifurcation points, the periods of the orbits diverge to infinity. This behavior results from an interplay between two symmetric saddle points and a saddle focus. The transition to chaos involves two interfering mechanisms: period doubling and gluing bifurcations around a saddle focus. We show that the flow is reducible to a non-monotonic discontinuous map of the interval and derive an analytical form for that map. This form suggests that other routes to chaos, including uninterfered cascades of gluing bifurcation, may be observed in different experimental realizations of the system.
ABSTRACT Background/Question/Methods The principle of competitive exclusion states that the numbe... more ABSTRACT Background/Question/Methods The principle of competitive exclusion states that the number of competing species is restricted by the number of limiting resources. Through the years this principle elicited interest in ideas aiming to explain the contradicting observation in many systems, including vegetation, that the number of competing species is larger than the number of resources. These ideas are mostly based on heterogeneity of space and time, and non-linearity of the growth or mortality terms. Results/Conclusions We suggest that even in a uniform environment in space and in time, and linear growth and mortality functions of the different species, self emergent patterns may induce coexistence of two species competing on one resource. We show an example of this phenomena in a simple PDE vegetation model, and map the parameter space to identify regions of coexistence and other special behaviors induced by pattern formation such as total extinction and immunity to invasion of a better competitor.
Dryland landscapes are mosaics of patches that differ in resource concentration, biomass producti... more Dryland landscapes are mosaics of patches that differ in resource concentration, biomass production and species richness. Understanding their structure and dynamics often calls for the identification of key species that modulate the abiotic environment, redistribute resources and facilitate the growth of other species.
Large responses of ecosystems to small changes in the conditions—regime shifts—are of great inter... more Large responses of ecosystems to small changes in the conditions—regime shifts—are of great interest and importance. In spatially extended ecosystems, these shifts may be local or global. Using empirical data and mathematical modeling, we investigated the dynamics of the Namibian fairy circle ecosystem as a case study of regime shifts in a pattern-forming ecosystem. Our results provide new support, based on the dynamics of the ecosystem, for the view of fairy circles as a self-organization phenomenon driven by water–vegetation interactions. The study further suggests that fairy circle birth and death processes correspond to spatially confined transitions between alternative stable states. Cascades of such transitions, possible in various pattern forming systems, result in gradual rather than abrupt regime shifts.
1. Functional diversity (FD) has become a principal concept for revealing mechanisms driving comm... more 1. Functional diversity (FD) has become a principal concept for revealing mechanisms driving community assembly and ecosystem function. Multiple assembly processes, including abiotic filtering, competition and multi-trophic relationships, operate simultaneously to structure FD. In water-limited plant communities, FD is likely to reflect trade-offs between drought resistance vs. disturbance resistance and competitive ability. 2. We propose a mathematical mechanistic model for understanding the organization and function of water-limited plant communities. The approach captures the interplay between abiotic filtering, below-and above-ground competition and disturbance. We exploit this powerful model to uncover mechanisms underlying changes in functional diversity along stress gradients. 3. Our approach links biomass production and FD to environmental conditions through plant resource capture ability. Functional groups are defined along a single trade-off axis according to investment in capturing light (shoot) vs. water (root). Species growth rate is determined dynamically by the species traits, water availability and grazing stress. We derive biomass production, functional diversity and composition along precipitation and grazing gradients. 4. Model's results revealed several regimes structuring FD along the precipitation gradient: 'Struggle for water' at low precipitation, 'competition for water' at intermediate precipitation and 'competition for light' at high precipitation. 5. We observed a shift in grazing effect on FD from negative at very low precipitation, to positive at higher precipitation. Unimodal FD–grazing intensity relationship was observed under high precipitation , while under low precipitation, FD decreased moderately with increasing grazing intensity. 6. Synthesis. Our model showcases how fundamental tradeoffs in plant traits may drive functional diversity and ecosystem function along environmental gradients. It offers a mechanism through which novel understandings can be obtained regarding the interplay between water stress, below-and above-ground competition and disturbance intensity and history. We discuss further model testing possibilities as well as required empirical work.
Environmental changes can affect the functioning of an ecosystem directly, through the response o... more Environmental changes can affect the functioning of an ecosystem directly, through the response of individual life forms, or indirectly, through interspecific interactions and community dynamics. The feasibility of a community-level response has motivated numerous studies aimed at understanding the mutual relationships between three elements of ecosystem dynamics: the abiotic environment, biodiversity and ecosystem function. Since ecosystems are inherently nonlinear and spatially extended, environmental changes can also induce pattern-forming instabilities that result in spatial self-organization of life forms and resources. This, in turn, can affect the relationships between these three elements, and make the response of ecosystems to environmental changes far more complex. Responses of this kind can be expected in dryland ecosystems, which show a variety of self-organizing vegetation patterns along the rainfall gradient. This paper describes the progress that has been made in understanding vegetation patterning in dryland ecosystems, and the roles it plays in ecosystem response to environmental variability. The progress has been achieved by modeling pattern-forming feedbacks at small spatial scales and up-scaling their effects to large scales through model studies. This approach sets the basis for integrating pattern formation theory into the study of ecosystem dynamics and addressing ecologically significant questions such as the dynamics of desertification, restoration of degraded landscapes, biodiversity changes along environmental gradients, and shrubland-grassland transitions.
Vegetation gap patterns in arid grasslands, such as the "fairy circles" of Namibia, are one of na... more Vegetation gap patterns in arid grasslands, such as the "fairy circles" of Namibia, are one of nature's greatest mysteries and subject to a lively debate on their origin. They are characterized by small-scale hexagonal ordering of circular bare-soil gaps that persists uniformly in the landscape scale to form a homogeneous distribution. Pattern-formation theory predicts that such highly ordered gap patterns should be found also in other water-limited systems across the globe, even if the mechanisms of their formation are different. Here we report that so far unknown fairy circles with the same spatial structure exist 10,000 km away from Namibia in the remote outback of Australia. Combining fieldwork, remote sensing, spatial pattern analysis, and process-based mathematical modeling, we demonstrate that these patterns emerge by self-organization, with no correlation with termite activity; the driving mechanism is a positive biomasswater feedback associated with water runoff and biomass-dependent infiltration rates. The remarkable match between the patterns of Australian and Namibian fairy circles and model results indicate that both patterns emerge from a nonuniform stationary instability, supporting a central universality principle of pattern-formation theory. Applied to the context of dryland vegetation, this principle predicts that different systems that go through the same instability type will show similar vegetation patterns even if the feedback mechanisms and resulting soil-water distributions are different, as we indeed found by comparing the Australian and the Namibian fairy-circle ecosystems. These results suggest that biomass-water feedbacks and resultant vegetation gap patterns are likely more common in remote drylands than is currently known. drylands | spatial pattern | Triodia grass | Turing instability | vegetation gap P attern-formation theory (1) and the influence of Alan Turing's work on understanding biological morphogenesis (2) are increasingly recognized in environmental sciences (3). Vegetation patterns resulting from self-organization occur frequently in waterlimited ecosystems and, similar to Turing patterns, show pattern morphologies that change from gaps to stripes (labyrinths) to spots with decreasing plant-available moisture (4-6). The patterns may emerge on completely flat and homogeneous substrate and are induced by positive feedbacks between local vegetation growth and water transport toward the growth location.
It is shown that within any class of commuting one-body potentials a Hohenberg-Kohn type theorem ... more It is shown that within any class of commuting one-body potentials a Hohenberg-Kohn type theorem is satisfied with respect to an appropriately defined density. The Hohenberg-Kohn theorem for local potentials follows as a special case.
Physica A: Statistical Mechanics and its Applications, 1998
We present a new set of kinematic equations for front motion in bistable media. The equations ext... more We present a new set of kinematic equations for front motion in bistable media. The equations extend earlier kinematic approaches by coupling the front curvature with the order parameter associated with a parity breaking front bifurcation. In addition to naturally describing the core region of rotating spiral waves the equations can be be used to study the nucleation of spiral-wave pairs along uniformly propagating fronts. The analysis of spiral-wave nucleation reduces to the simpler problem of droplet, or domain, nucleation in one space dimension.
The trade-off between the need to obtain new knowledge and the need to use that knowledge to impr... more The trade-off between the need to obtain new knowledge and the need to use that knowledge to improve performance is one of the most basic trade-offs in nature, and optimal performance usually requires some balance between exploratory and exploitative behaviors. Researchers in many disciplines have been searching for the optimal solution to this dilemma.
Discrete and Continuous Dynamical Systems - Series B, 2011
In this article, we consider the dryland vegetation model proposed by Gilad et al [6,. This model... more In this article, we consider the dryland vegetation model proposed by Gilad et al [6,. This model consists of three nonlinear parabolic partial differential equations, one of which is degenerate parabolic of the family of the porous media equation , and we prove the existence of its weak solutions. Our approach based on the classical Galerkin methods combines and makes use of techniques, parabolic regularization, truncation, maximum principle, compactness. We observe in this way various properties and regularity results of the solutions.
Egu General Assembly Conference Abstracts, May 1, 2010
According to the habitat heterogeneity hypothesis spatial heterogeneity positively correlates wit... more According to the habitat heterogeneity hypothesis spatial heterogeneity positively correlates with species diversity. Numerous studies have confirmed this hypothesis in various ecological contexts, but little attention has been given to the origin of spatial heterogeneity and to possible heterogeneity-diversity feedbacks. In particular, the link between spatial heterogeneity induced by vegetation pattern formation, and species diversity has remained unexplored. In this presentation we describe a mathematical modeling approach for exploring this link in the context of water-limited vegetation, and apply it to woody-herbaceous systems. We first address the relation between vegetation-landscape diversity and resource diversity, and discuss mechanisms of species-diversity change. We show that woody patches can buffer species-diversity loss as a result of an aridity stress, and that species diversity can also be affected by woody-pattern changes at the landscape scale. We then describe the derivation of community-level properties, such as diversity-resource relations. Community-level properties are derived by extending the space over which biomass variables are defined to include trait subspaces, where different points represent distinct species. We demonstrate this approach with a simple example of a spatially uniform herbaceous community, choosing the tradeoff between investments in above and below-ground biomass as the axis that spans the trait subspace. We then extend this study to include the effect of a spatially localized woody patch on the diversity of herbaceous species. We conclude by delineating directions for further model studies and development.
In bistable systems, the stability of front structures often influences the dynamics of extended ... more In bistable systems, the stability of front structures often influences the dynamics of extended patterns. We show how the combined effect of an instability to curvature modulations and proximity to a pitchfork front bifurcation leads to spontaneous nucleation of spiral waves along the front. This effect is demonstrated by direct simulations of a FitzHugh-Nagumo (FHN) model and by simulations of order parameter equations for the front velocity and curvature. Spontaneous spiral-wave nucleation often results in a state of spatiotemporal disorder involving repeated events of spiral wave nucleation, domain splitting and spiral wave annihilation.
Patterned vegetation is characterisic for water-limited areas and both periodic and scale-free di... more Patterned vegetation is characterisic for water-limited areas and both periodic and scale-free distributions of vegetation patch sizes have been reported. Using a simple common mathematical modeling approach we study the physical and ecological conditions for the emergence of both types of patterns. By drawing an analogy between the vegetation model and a simpler inhibitor-activator model we discuss if scale-free patterns of dryland vegetation represent asymptotic patterns or more likely long transients.
We use the context of dryland vegetation to study a general problem of complex pattern forming sy... more We use the context of dryland vegetation to study a general problem of complex pattern forming systems - multiple pattern-forming instabilities that are driven by distinct mechanisms but share the same spectral properties. We find that the co-occurrence of such instabilities results in the growth of a single mode rather than two interacting modes. The interplay between the two mechanisms, which promote or counteract each other, compensates for the simpler dynamics of a single mode by inducing higher pattern diversity. Possible implications to biodiversity of ecosystems are discussed.
We discuss an explicit-space model for vegetation dynamics in water-limited ecosystems. Under con... more We discuss an explicit-space model for vegetation dynamics in water-limited ecosystems. Under conditions of water stress, vegetation patterns such as spots, stripes and gaps spontaneously emerge. In this contribution, we focus on the interplay between water-vegetation feedbacks and rainfall intermittency, and we discuss an extended version of the model which allows for studying how evapotranspirations fluxes are modified by the presence of patterns.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015
We study resonant spatially periodic solutions of the Lengyel-Epstein model modified to describe ... more We study resonant spatially periodic solutions of the Lengyel-Epstein model modified to describe the chlorine dioxide-iodine-malonic acid reaction under spatially periodic illumination. Using multiple-scale analysis and numerical simulations, we obtain the stability ranges of 2:1 resonant solutions, i.e., solutions with wavenumbers that are exactly half of the forcing wavenumber. We show that the width of resonant wavenumber response is a non-monotonic function of the forcing strength, and diminishes to zero at sufficiently strong forcing. We further show that strong forcing may result in a π/2 phase shift of the resonant solutions, and argue that the nonequilibrium Ising-Bloch front bifurcation can be reversed. We attribute these behaviors to an inherent property of forcing by periodic illumination, namely, the increase of the mean spatial illumination as the forcing amplitude is increased.
A quasi-2-dimensional stationary spot in a disk-shaped chemical reactor is observed to bifurcate ... more A quasi-2-dimensional stationary spot in a disk-shaped chemical reactor is observed to bifurcate to an oscillating spot when a control parameter is increased beyond a critical value. Further increase of the control parameter leads to the collapse and disappearance of the spot. Analysis of a bistable activator-inhibitor model indicates that the observed behavior is a consequence of interaction of the front with the boundary near a parity breaking front bifurcation.
We present a detailed theory of the appearance of low dimensional dynamics in a hydrodynamic syst... more We present a detailed theory of the appearance of low dimensional dynamics in a hydrodynamic system. The system chosen has been subjected to a careful experimental study; it involves dynamics of surface waves in a cylinder of fluid which is oscillated vertically. We apply center manifold and normal form theories to derive low dimensional nonlinear evolution equations and an approximate solution of the original partial differential equations. All the major experimental findings are rationalized by the theory: the appearance of periodic and chaotic mixed mode states, the existence of an asymmetry between the interacting modes, the qualitative structure of the phase diagram in the vicinity of the critical point and the presence of enslaved stable modes in the surface wave patterns. We also present theoretical predictions concerning the nature of the periodic mixed mode state and the mechanism of the onset of chaos. The periodic dynamics involve a sequence of homoclinic bifurcations in which periodic orbits glue and disglue alternatively. At the bifurcation points, the periods of the orbits diverge to infinity. This behavior results from an interplay between two symmetric saddle points and a saddle focus. The transition to chaos involves two interfering mechanisms: period doubling and gluing bifurcations around a saddle focus. We show that the flow is reducible to a non-monotonic discontinuous map of the interval and derive an analytical form for that map. This form suggests that other routes to chaos, including uninterfered cascades of gluing bifurcation, may be observed in different experimental realizations of the system.
ABSTRACT Background/Question/Methods The principle of competitive exclusion states that the numbe... more ABSTRACT Background/Question/Methods The principle of competitive exclusion states that the number of competing species is restricted by the number of limiting resources. Through the years this principle elicited interest in ideas aiming to explain the contradicting observation in many systems, including vegetation, that the number of competing species is larger than the number of resources. These ideas are mostly based on heterogeneity of space and time, and non-linearity of the growth or mortality terms. Results/Conclusions We suggest that even in a uniform environment in space and in time, and linear growth and mortality functions of the different species, self emergent patterns may induce coexistence of two species competing on one resource. We show an example of this phenomena in a simple PDE vegetation model, and map the parameter space to identify regions of coexistence and other special behaviors induced by pattern formation such as total extinction and immunity to invasion of a better competitor.
Dryland landscapes are mosaics of patches that differ in resource concentration, biomass producti... more Dryland landscapes are mosaics of patches that differ in resource concentration, biomass production and species richness. Understanding their structure and dynamics often calls for the identification of key species that modulate the abiotic environment, redistribute resources and facilitate the growth of other species.
Large responses of ecosystems to small changes in the conditions—regime shifts—are of great inter... more Large responses of ecosystems to small changes in the conditions—regime shifts—are of great interest and importance. In spatially extended ecosystems, these shifts may be local or global. Using empirical data and mathematical modeling, we investigated the dynamics of the Namibian fairy circle ecosystem as a case study of regime shifts in a pattern-forming ecosystem. Our results provide new support, based on the dynamics of the ecosystem, for the view of fairy circles as a self-organization phenomenon driven by water–vegetation interactions. The study further suggests that fairy circle birth and death processes correspond to spatially confined transitions between alternative stable states. Cascades of such transitions, possible in various pattern forming systems, result in gradual rather than abrupt regime shifts.
1. Functional diversity (FD) has become a principal concept for revealing mechanisms driving comm... more 1. Functional diversity (FD) has become a principal concept for revealing mechanisms driving community assembly and ecosystem function. Multiple assembly processes, including abiotic filtering, competition and multi-trophic relationships, operate simultaneously to structure FD. In water-limited plant communities, FD is likely to reflect trade-offs between drought resistance vs. disturbance resistance and competitive ability. 2. We propose a mathematical mechanistic model for understanding the organization and function of water-limited plant communities. The approach captures the interplay between abiotic filtering, below-and above-ground competition and disturbance. We exploit this powerful model to uncover mechanisms underlying changes in functional diversity along stress gradients. 3. Our approach links biomass production and FD to environmental conditions through plant resource capture ability. Functional groups are defined along a single trade-off axis according to investment in capturing light (shoot) vs. water (root). Species growth rate is determined dynamically by the species traits, water availability and grazing stress. We derive biomass production, functional diversity and composition along precipitation and grazing gradients. 4. Model's results revealed several regimes structuring FD along the precipitation gradient: 'Struggle for water' at low precipitation, 'competition for water' at intermediate precipitation and 'competition for light' at high precipitation. 5. We observed a shift in grazing effect on FD from negative at very low precipitation, to positive at higher precipitation. Unimodal FD–grazing intensity relationship was observed under high precipitation , while under low precipitation, FD decreased moderately with increasing grazing intensity. 6. Synthesis. Our model showcases how fundamental tradeoffs in plant traits may drive functional diversity and ecosystem function along environmental gradients. It offers a mechanism through which novel understandings can be obtained regarding the interplay between water stress, below-and above-ground competition and disturbance intensity and history. We discuss further model testing possibilities as well as required empirical work.
Environmental changes can affect the functioning of an ecosystem directly, through the response o... more Environmental changes can affect the functioning of an ecosystem directly, through the response of individual life forms, or indirectly, through interspecific interactions and community dynamics. The feasibility of a community-level response has motivated numerous studies aimed at understanding the mutual relationships between three elements of ecosystem dynamics: the abiotic environment, biodiversity and ecosystem function. Since ecosystems are inherently nonlinear and spatially extended, environmental changes can also induce pattern-forming instabilities that result in spatial self-organization of life forms and resources. This, in turn, can affect the relationships between these three elements, and make the response of ecosystems to environmental changes far more complex. Responses of this kind can be expected in dryland ecosystems, which show a variety of self-organizing vegetation patterns along the rainfall gradient. This paper describes the progress that has been made in understanding vegetation patterning in dryland ecosystems, and the roles it plays in ecosystem response to environmental variability. The progress has been achieved by modeling pattern-forming feedbacks at small spatial scales and up-scaling their effects to large scales through model studies. This approach sets the basis for integrating pattern formation theory into the study of ecosystem dynamics and addressing ecologically significant questions such as the dynamics of desertification, restoration of degraded landscapes, biodiversity changes along environmental gradients, and shrubland-grassland transitions.
Vegetation gap patterns in arid grasslands, such as the "fairy circles" of Namibia, are one of na... more Vegetation gap patterns in arid grasslands, such as the "fairy circles" of Namibia, are one of nature's greatest mysteries and subject to a lively debate on their origin. They are characterized by small-scale hexagonal ordering of circular bare-soil gaps that persists uniformly in the landscape scale to form a homogeneous distribution. Pattern-formation theory predicts that such highly ordered gap patterns should be found also in other water-limited systems across the globe, even if the mechanisms of their formation are different. Here we report that so far unknown fairy circles with the same spatial structure exist 10,000 km away from Namibia in the remote outback of Australia. Combining fieldwork, remote sensing, spatial pattern analysis, and process-based mathematical modeling, we demonstrate that these patterns emerge by self-organization, with no correlation with termite activity; the driving mechanism is a positive biomasswater feedback associated with water runoff and biomass-dependent infiltration rates. The remarkable match between the patterns of Australian and Namibian fairy circles and model results indicate that both patterns emerge from a nonuniform stationary instability, supporting a central universality principle of pattern-formation theory. Applied to the context of dryland vegetation, this principle predicts that different systems that go through the same instability type will show similar vegetation patterns even if the feedback mechanisms and resulting soil-water distributions are different, as we indeed found by comparing the Australian and the Namibian fairy-circle ecosystems. These results suggest that biomass-water feedbacks and resultant vegetation gap patterns are likely more common in remote drylands than is currently known. drylands | spatial pattern | Triodia grass | Turing instability | vegetation gap P attern-formation theory (1) and the influence of Alan Turing's work on understanding biological morphogenesis (2) are increasingly recognized in environmental sciences (3). Vegetation patterns resulting from self-organization occur frequently in waterlimited ecosystems and, similar to Turing patterns, show pattern morphologies that change from gaps to stripes (labyrinths) to spots with decreasing plant-available moisture (4-6). The patterns may emerge on completely flat and homogeneous substrate and are induced by positive feedbacks between local vegetation growth and water transport toward the growth location.
It is shown that within any class of commuting one-body potentials a Hohenberg-Kohn type theorem ... more It is shown that within any class of commuting one-body potentials a Hohenberg-Kohn type theorem is satisfied with respect to an appropriately defined density. The Hohenberg-Kohn theorem for local potentials follows as a special case.
Physica A: Statistical Mechanics and its Applications, 1998
We present a new set of kinematic equations for front motion in bistable media. The equations ext... more We present a new set of kinematic equations for front motion in bistable media. The equations extend earlier kinematic approaches by coupling the front curvature with the order parameter associated with a parity breaking front bifurcation. In addition to naturally describing the core region of rotating spiral waves the equations can be be used to study the nucleation of spiral-wave pairs along uniformly propagating fronts. The analysis of spiral-wave nucleation reduces to the simpler problem of droplet, or domain, nucleation in one space dimension.
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Papers by Ehud Meron
on the dynamics of the ecosystem, for the view of fairy circles as a self-organization phenomenon driven by water–vegetation interactions. The study further suggests that fairy circle birth and death processes correspond to spatially confined transitions between alternative stable states. Cascades of such transitions, possible in various pattern forming systems, result in gradual rather than abrupt regime shifts.
dynamics of desertification, restoration of degraded landscapes, biodiversity changes along environmental gradients, and shrubland-grassland transitions.
on the dynamics of the ecosystem, for the view of fairy circles as a self-organization phenomenon driven by water–vegetation interactions. The study further suggests that fairy circle birth and death processes correspond to spatially confined transitions between alternative stable states. Cascades of such transitions, possible in various pattern forming systems, result in gradual rather than abrupt regime shifts.
dynamics of desertification, restoration of degraded landscapes, biodiversity changes along environmental gradients, and shrubland-grassland transitions.