Pattern formation – A missing link in the study of ecosystem response to environmental changes

Delayed loss of stability of periodic travelling waves: Insights from the analysis of essential spectra

Periodic travelling waves (PTW) are a common solution type of partial differential equations. Such models exhibit multistability of PTWs, typically visualised through the Busse balloon, and parameter changes typically lead to a cascade of wavelength changes through the Busse balloon. In the past, the stability boundaries of the Busse balloon have been used to predict such wavelength changes. Here, motivated by anecdotal evidence from previous work, we provide compelling evidence that the Busse balloon provides insufficient information to predict wavelength changes due to a delayed loss of stability phenomenon. Using two different reaction-advection-diffusion systems, we relate the delay that occurs between the crossing of a stability boundary in the Busse balloon and the occurrence of a wavelength change to features of the essential spectrum of the destabilised PTW. This leads to a predictive framework that can estimate the order of magnitude of such a time delay, which provides a novel "early warning sign" for pattern destabilisation. We illustrate the implementation of the predictive framework to predict under what conditions a wavelength change of a PTW occurs.

Vegetation restoration strategies in arid or semi-arid regions-From the perspective of optimal control

Inappropriate human activities contribute to the degradation of ecosystems in arid or semi-arid regions. Therefore, emphasizing the importance of strategies for restoring vegetation in these areas cannot be overstated. However, there has been insufficient research on how to develop effective restoration strategies at minimal cost. This paper addresses this gap by studying how optimizing the spatiotemporal distribution of human activities through local and boundary controls can reduce the level of desertification in vegetation pattern structures, thereby facilitating the recovery of arid land vegetation. The results indicate that vegetation restoration depends on the proportion and number of human activity areas, with a trade-off between them. Furthermore, consistent conclusions were obtained on circular regions, demonstrating the robustness of the approach to boundary shapes. This paper aims to offer new insights into the restoration of arid land vegetation and the prevention of catastrophic ecosystem changes from the perspective of optimal control.

Ecogeographic Study of Ipomoea Species in Mauritius, Indian Ocean

The wild relatives of crops play a critical role in enhancing agricultural resilience and sustainability by contributing valuable traits for crop improvement. Shifts in climatic conditions and human activities threaten plant genetic resources for food and agriculture (PGRFA), jeopardizing contributions to future food production and security. Studies and inventories of the extant agrobiodiversity, in terms of numbers and distribution patterns of species and their genetic diversity, are primordial for developing effective and comprehensive conservation strategies. We conducted an ecogeographic study on Ipomoea species and assessed their diversity, distribution, and ecological preferences across different topographic, altitudinal, geographical, and climatic gradients, at a total of 450 sites across Mauritius. Species distribution maps overlaid with climatic data highlighted specific ecological distribution. Principal Component Analysis (PCA) revealed species distribution was influenced by geographical factors. Regional richness analyses indicated varying densities, with some species exhibiting localized distributions and specific ecological preferences while the other species showed diverse distribution patterns. Field surveys identified 14 species and 2 subspecies out of 21 species and 2 subspecies of Ipomoea reported in Mauritius. A gap in ex situ germplasm collections was observed and several species were identified as threatened. Further investigations and a more long-term monitoring effort to better guide conservation decisions are proposed.

The relationship between structure and ecosystem services of forest and grassland based on pattern analysis method: A case study of the Mongolian Plateau

Global warming has led to severe land desertification on the Mongolian plateau. It puts great environmental pressure on vegetation communities. This pressure leads to fragmentation of land use and landscape patterns, thus triggering changes in the spatial distribution patterns of vegetation. The spatial distribution pattern of vegetation is crucial for the performance of its ecosystem services. However, there is not enough research on the relationship between large-scale spatial distribution patterns of vegetation and ecosystem services. Therefore, this study is to construct an ecological spatial network on the Mongolian Plateau based on landscape ecology and complex network theory. Combining pattern analysis methods to analyze the network, we obtained the spatial and temporal trends of forest and grass spatial distribution patterns from 2000 to 2100, and explored the relationship between the topological properties of source patches and ecosystem services in different patterns. It was found that there are four basic patterns of spatial distribution of forest and grass in the Mongolian Plateau. The Core-Linked Ring pattern accounts for 40.74 % and exhibits the highest stability. Under the SSP5-RCP8.5 scenario, source patches are reduced by 22.76 % in 2100. Topological indicators of source patches showed significant correlations with ecosystem services. For example, the CUE of grassland patches in the Centralized Star pattern was positively correlated with betweeness centrality. The most significant improvement in WUE after optimization is 19.90 % compared to pre-optimization. The conclusion of the study shows that the spatial distribution pattern of vegetation can be used to enhance the stability of ecological spatial network and improve ecosystem services at a larger scale. It can provide a certain reference for the study of spatial patterns of vegetation distribution in arid and semi-arid areas.

Impact of climate change on vegetation patterns in Altay Prefecture, China

Altay Prefecture, a typical arid region in northwestern China, has experienced the climate transition from warming-drying to warming-wetting since 1980s and has attracted widespread attention. Nonetheless, it is still unclear how climate change has influenced the distribution of vegetation in this region. In this paper, a reaction-diffusion model of the climate-vegetation system is proposed to study the impact of climate change (precipitation, temperature and carbon dioxide concentration) on vegetation patterns in Altay Prefecture. Our results indicate that the tendency of vegetation growth in Altay Prefecture improved gradually from 1985 to 2010. Under the current climate conditions, the increase of precipitation results in the change of vegetation pattern structures, and eventually vegetation coverage tends to be uniform. Moreover, we found that there exists an optimal temperature where the spot vegetation pattern structure remains stable. Furthermore, the increase in carbon dioxide concentration induces vegetation pattern transition. Based on four climate change scenarios of the Coupled Model Intercomparison Project Phase 6 (CMIP6), we used the power law range (PLR) to predict the optimal scenario for the sustainable development of the vegetation ecosystem in Altay Prefecture.

Introducing desirable patches to initiate ecosystem transitions and accelerate ecosystem restoration

Meeting restoration targets may require active strategies to accelerate natural regeneration rates or overcome the resilience associated with degraded ecosystem states. Introducing desired ecosystem patches in degraded landscapes constitutes a promising active restoration strategy, with various mechanisms potentially causing these patches to become foci from which desired species can re-establish throughout the landscape. This study considers three mechanisms previously identified as potential drivers of introduced patch dynamics: autocatalytic nucleation, directed dispersal, and resource concentration. These mechanisms reflect qualitatively different positive feedbacks. We developed an ecological model framework that compared how the occurrence of each mechanism was reflected in spatio-temporal patch dynamics. We then analyzed the implications of these relationships for optimal restoration design. We found that patch expansion accelerated over time when driven by the autocatalytic nucleation mechanism, while patch expansion driven by the directed dispersal or resource concentration mechanisms decelerated over time. Additionally, when driven by autocatalytic nucleation, patch expansion was independent of patch position in the landscape. However, the proximity of other patches affected patch expansion either positively or negatively when driven by directed dispersal or resource concentration. For autocatalytic nucleation, introducing many small patches was a favorable strategy, provided that each individual patch exceeded a critical patch size. Introducing a single patch or a few large patches was the most effective restoration strategy to initiate the directed dispersal mechanism. Introducing many small patches was the most effective strategy for reaching restored ecosystem states driven by a resource concentration mechanism. Our model results suggest that introducing desirable patches can substantially accelerate ecosystem restoration, or even induce a critical transition from an otherwise stable degraded state toward a desired ecosystem state. However, the potential of this type of restoration strategy for a particular ecosystem may strongly depend on the mechanism driving patch dynamics. In turn, which mechanism drives patch dynamics may affect the optimal spatial design of an active restoration strategy. Each of the three mechanisms considered reflects distinct spatio-temporal patch dynamics, providing novel opportunities for empirically identifying key mechanisms, and restoration designs that introduce desired patches in degraded landscapes according to these patch dynamics.

The hidden order of Turing patterns in arid and semi-arid vegetation ecosystems

Vegetation Turing patterns play a critical role in the ecological functioning of arid and semi-arid ecosystems. However, the long-range spatial features of these patterns have been neglected compared to short-range features like patch shape and spatial wavelength. Drawing inspiration from hyperuniform structures in material science, we find that the arid and semi-arid vegetation Turing pattern exhibits long-range dispersion similar to hyperuniformity. As the degree of hyperuniformity of the vegetation Turing pattern increases, so does the water-use efficiency of the vegetation. This finding supports previous studies that suggest that Turing patterns represent a spatially optimized self-organization of ecosystems for water acquisition. The degree of hyperuniformity of Turing-type ecosystems exhibits significant critical slowing down near the tipping point, indicating that these ecosystems have non-negligible transient dynamical behavior. Reduced rainfall not only decreases the resilience of the steady state of the ecosystem but also slows down the rate of spatial optimization of water-use efficiency in long transient regimes. We propose that the degree of hyperuniformity indicates the spatial resilience of Turing-type ecosystems after strong, short-term disturbances. Spatially heterogeneous disturbances that reduce hyperuniformity lead to longer recovery times than spatially homogeneous disturbances that maintain hyperuniformity.

Transitions between dissipative localized structures in the simplified Gilad-Meron model for dryland plant ecology

Spatially extended patterns and multistability of possible different states are common in many ecosystems, and their combination has an important impact on their dynamical behaviors. One potential combination involves tristability between a patterned state and two different uniform states. Using a simplified version of the Gilad-Meron model for dryland ecosystems, we study the organization, in bifurcation terms, of the localized structures arising in tristable regimes. These states are generally related to the concept of wave front locking and appear in the form of spots and gaps of vegetation. We find that the coexistence of localized spots and gaps, within tristable configurations, yields the appearance of hybrid states. We also study the emergence of spatiotemporal localized states consisting of a portion of a periodic pattern embedded in a uniform Hopf-like oscillatory background in a subcritical Turing-Hopf dynamical regime.

Landscapes of Biochemical Warfare: Spatial Self-Organization Woven from Allelopathic Interactions

Evidence shows that diversity and spatial distributions of biological communities are largely driven by the race of living organisms in their adaptation to chemicals synthesized by their neighbors. In this report, the emergence of mathematical models on pure spatial self-organization induced by biochemical suppression (allelopathy) and competition between species were investigated through numerical analysis. For both random and patched initial spatial distributions of species, we demonstrate that warfare survivors are self-organized on the landscape in Turing-like patterns driven by diffusive instabilities of allelochemicals. These patterns are simple; either all species coexist at low diffusion rates or are massively extinct, except for a few at high diffusivities, but they are complex and biodiversity-sustained at intermediate diffusion rates. "Defensive alliances" and ecotones seem to be basic mechanisms that sustain great biodiversity in our hybrid cellular automata model. Moreover, species coexistence and extinction exhibit multi-stationarity.

Impacts of climate change on vegetation pattern: Mathematical modeling and data analysis

Climate change has become increasingly severe, threatening ecosystem stability and, in particular, biodiversity. As a typical indicator of ecosystem evolution, vegetation growth is inevitably affected by climate change, and therefore has a great potential to provide valuable information for addressing such ecosystem problems. However, the impacts of climate change on vegetation growth, especially the spatial and temporal distribution of vegetation, are still lacking of comprehensive exposition. To this end, this review systematically reveals the influences of climate change on vegetation dynamics in both time and space by dynamical modeling the interactions of meteorological elements and vegetation growth. Moreover, we characterize the long-term evolution trend of vegetation growth under climate change in some typical regions based on data analysis. This work is expected to lay a necessary foundation for systematically revealing the coupling effect of climate change on the ecosystem.

Positive impacts of livestock and wild ungulate routes on functioning of dryland ecosystems

Livestock grazing is often perceived as being detrimental to the quality and functioning of dryland ecosystems. For example, a study in a semiarid Kenyan savanna proposed that cattle form bare spaces throughout the landscape, which indicate ecosystem degradation. Other studies, conducted in north-eastern Spain, where climatic conditions range between semiarid and Mediterranean subhumid, reported that sheep and goat trails have increased the emergence of rill erosion processes. Sometimes, this negative perception is extended to include wild, large ungulate herbivores as well. Here, we challenge this perception by highlighting the generally nonadverse and even ameliorative impacts of moderate animal rate on geoecosystem functioning of hilly drylands. Specifically, trampling routes (also known as treading paths, livestock terracettes, cattle trails, migration tracks, cowtours, etc.) formed across hillslopes by grazing animals-being either domesticated livestock or native large herbivores-transform the original two-phase vegetation mosaic of shrubby patches and interpatch spaces into a three-phase mosaic. The animal routes increase the complexity of ecosystem, by strengthening the spatial redistribution of water and soil resources at the patch scale and decreasing hydrological connectivity at the hillslope scale. As a consequence, the animal routes improve functioning of hilly drylands and increase their resilience to long-term droughts and climatic change. Therefore, instead of viewing the animal routes as degraded spots, they should be perceived at a wider perspective that allows to properly understand their overall role in sustaining dryland geoecosystems.

High-integrity human intervention in ecosystems: Tracking self-organization modes

Humans play major roles in shaping and transforming the ecology of Earth. Unlike natural drivers of ecosystem change, which are erratic and unpredictable, human intervention in ecosystems generally involves planning and management, but often results in detrimental outcomes. Using model studies and aerial-image analysis, we argue that the design of a successful human intervention form calls for the identification of the self-organization modes that drive ecosystem change, and for studying their dynamics. We demonstrate this approach with two examples: grazing management in drought-prone ecosystems, and rehabilitation of degraded vegetation by water harvesting. We show that grazing can increase the resilience to droughts, rather than imposing an additional stress, if managed in a spatially non-uniform manner, and that fragmental restoration along contour bunds is more resilient than the common practice of continuous restoration in vegetation stripes. We conclude by discussing the need for additional studies of self-organization modes and their dynamics.

Human intervention in ecosystems is motivated by various functional needs, such as provisioning ecosystem services, but often has unexpected detrimental outcomes. A major question in ecology is how to manage human intervention so as to achieve its goal without impairing ecosystem function. The main idea pursued here is the need to identify the inherent response ways of ecosystems to disturbances, and use them as road maps for conducting interventions. This approach is demonstrated mathematically using two contexts, grazing management and vegetation restoration, and compared to remote sensing data for the latter. Among the surprising insights obtained is the beneficial effect of grazing, in terms of resilience to droughts, that can be achieved by managing it non-uniformly in space.

Linking spatial self-organization to community assembly and biodiversity

Temporal shifts to drier climates impose environmental stresses on plant communities that may result in community reassembly and threatened ecosystem services, but also may trigger self-organization in spatial patterns of biota and resources, which act to relax these stresses. The complex relationships between these counteracting processes - community reassembly and spatial self-organization - have hardly been studied. Using a spatio-temporal model of dryland plant communities and a trait-based approach, we study the response of such communities to increasing water-deficit stress. We first show that spatial patterning acts to reverse shifts from fast-growing species to stress-tolerant species, as well as to reverse functional-diversity loss. We then show that spatial self-organization buffers the impact of further stress on community structure. Finally, we identify multistability ranges of uniform and patterned community states and use them to propose forms of non-uniform ecosystem management that integrate the need for provisioning ecosystem services with the need to preserve community structure.

Belowground feedbacks as drivers of spatial self-organization and community assembly

Vegetation patterning in water-limited and other resource-limited ecosystems highlights spatial self-organization processes as potentially key drivers of community assembly. These processes provide insight into predictable landscape-level relationships between organisms and their abiotic environment in the form of regular and irregular patterns of biota and resources. However, two aspects have largely been overlooked; the roles played by plant - soil-biota feedbacks and allelopathy in spatial self-organization, and their potential contribution, along with plant-resource feedbacks, to community assembly through spatial self-organization. Here, we expand the drivers of spatial self-organization from a focus on plant-resource feedbacks to include plant - soil-biota feedbacks and allelopathy, and integrate concepts of nonlinear physics and community ecology to generate a new hypothesis. According to this hypothesis, below-ground processes can affect community assemblages through two types of spatial self-organization, global and local. The former occurs simultaneously across whole ecosystems, leading to self-organized patterns of biota, allelochemicals and resources, and niche partitioning. The latter occurs locally in ecotones, and determines ecotone structure and motion, invasion dynamics, and species coexistence. Studies of the two forms of spatial self-organization are important for understanding the organization of plant communities in drier climates which are likely to involve spatial patterning or re-patterning. Such studies are also important for developing new practices of ecosystem management, based on local manipulations at ecotones, to slow invasion dynamics or induce transitions from transitive to intransitive networks of interspecific interactions which increase species diversity.

A structure-function based approach to floc hierarchy and evidence for the non-fractal nature of natural sediment flocs

Natural sediment flocs are fragile, highly irregular, loosely bound aggregates comprising minerogenic and organic material. They contribute a major component of suspended sediment load and are critical for the fate and flux of sediment, carbon and pollutants in aquatic environments. Understanding their behaviour is essential to the sustainable management of waterways, fisheries and marine industries. For several decades, modelling approaches have utilised fractal mathematics and observations of two dimensional (2D) floc size distributions to infer levels of aggregation and predict their behaviour. Whilst this is a computationally simple solution, it is highly unlikely to reflect the complexity of natural sediment flocs and current models predicting fine sediment hydrodynamics are not efficient. Here, we show how new observations of fragile floc structures in three dimensions (3D) demonstrate unequivocally that natural flocs are non-fractal. We propose that floc hierarchy is based on observations of 3D structure and function rather than 2D size distribution. In contrast to fractal theory, our data indicate that flocs possess characteristics of emergent systems including non-linearity and scale-dependent feedbacks. These concepts and new data to quantify floc structures offer the opportunity to explore new emergence-based floc frameworks which better represent natural floc behaviour and could advance our predictive capacity.

Effect of vegetation patchiness on the subsurface water distribution in abandoned farmland of the Loess Plateau, China

Patchiness of grassland results in important effects on ecohydrological processes in arid and semiarid areas; however, the influences on subsurface water flow and soil water distribution remain poorly understood, particularly during vegetation succession on slopes. This study examined these effects by comparing the water flow behaviors and preferential infiltration between vegetation patches (VP) and interspace patches (IP) in three sites at different states of vegetation succession (grass, subshrub and shrub) in abandoned farmland of the Loess Plateau, China. Dye tracer infiltration showed that patchiness of vegetation increased spatial variations of soil water and preferential infiltration by increasing the densities of fine root length and fine root volume in the soil profile. Moreover, the more abundant and intricate roots following a lateral direction beneath VP likely contributed to lateral flow and infiltration variability. However, differences between VP and IP were not significant because considerable living fine roots and decayed roots of IP also provided preferential flow pathways. Our finding indicated that IP could compete with VP for access to soil water resources, which potentially increased hillslope-scale infiltration and reduced surface runoff and erosion risk. Under the different states of vegetation succession, subshrub patches showed significantly greater preferential infiltration volume (28.53 mm) and contribution of preferential infiltration to total infiltration (60.58%) than grass and shrub patches. Vegetation patch size made positive effects on improving preferential flow and water movement. Greater preferential flow in subshrub patches played a positive role in soil water storage and replenishment. Therefore, natural restoration of a slope area with small heterogeneity in preferential flow can be successfully applied in the Loess Plateau, particularly during the subshrub succession state.

The role of spatial self-organization in the design of agroforestry systems

The development of sustainable agricultural systems in drylands is currently a crucial issue in the context of mitigating the outcomes of population growth under the conditions of climatic changes. The need to meet the growing demand for food, fodder, and fuel, together with the hazards due to climate change, requires cross-disciplinary studies of ways to increase livelihood while minimizing the impact on the environment. Practices of agroforestry systems, in which herbaceous species are intercropped between rows of woody species plantations, have been shown to mitigate several of the predicaments of climatic changes. Focusing on agroforestry in drylands, we address the question of how we can improve the performance of agroforestry systems in those areas. As vegetation in drylands tends to self-organize in various patterns, it seems essential to explore the various patterns that agroforestry systems tend to form and their impact on the performance of these systems in terms of biomass production, resilience to droughts, and water use efficiency. We use a two-soil-layers vegetation model to study the relationship between deep-rooted woody vegetation and shallow herbaceous vegetation, and explore how self-organization in different spatial patterns influences the performance of agroforestry systems. We focus on three generic classes of patterns, spots, gaps, and stripes, assess these patterns using common metrics for agroforestry systems, and examine their resilience to droughts. We show that in contrast to the widespread practice of planting the woody and herbaceous species in alternating rows, that is, in a stripe pattern, planting the woody species in hexagonal spot patterns may increase the system's resilience to droughts. Furthermore, hexagonal spot patterns reduce the suppression of herbs growth by the woody vegetation, therefore maintaining higher crop yields. We conclude by discussing some limitations of this study and their significance.

Organizing principles for vegetation dynamics

Plants and vegetation play a critical-but largely unpredictable-role in global environmental changes due to the multitude of contributing processes at widely different spatial and temporal scales. In this Perspective, we explore approaches to master this complexity and improve our ability to predict vegetation dynamics by explicitly taking account of principles that constrain plant and ecosystem behaviour: natural selection, self-organization and entropy maximization. These ideas are increasingly being used in vegetation models, but we argue that their full potential has yet to be realized. We demonstrate the power of natural selection-based optimality principles to predict photosynthetic and carbon allocation responses to multiple environmental drivers, as well as how individual plasticity leads to the predictable self-organization of forest canopies. We show how models of natural selection acting on a few key traits can generate realistic plant communities and how entropy maximization can identify the most probable outcomes of community dynamics in space- and time-varying environments. Finally, we present a roadmap indicating how these principles could be combined in a new generation of models with stronger theoretical foundations and an improved capacity to predict complex vegetation responses to environmental change.

A model study of terraced riverbeds as novel ecosystems

Riverbed terracing has been introduced in ancient times to retain water and soil, to reduce hydrological connectivity and erosion and to increase primary and secondary productivity of agro-ecological systems. These presently abandoned human-made landscapes have become novel ecosystems and a potential source of ecosystem services to humans in drylands. We use a mathematical-modeling approach to study factors that regulate terraced riverbeds and affect community and ecosystem attributes such as productivity, functional diversity and resilience to droughts. We introduce a model that captures the relationships between rainfall pattern, runoff coupling between adjacent terraces, and vegetation growth, taking into account competition for water and light. We found that a large number of weak rainfall events results in lower total biomass and functional diversity across the terraced riverbed compared with a few strong rainfall events. We further analyzed the filtering of species traits from pools of functional groups that make different tradeoffs between investment in above-ground biomass to capture canopy resources and investment in below-ground biomass to capture soil resources. Pools characterized by concave tradeoffs give rise to higher functional diversity, lower biomass production and lower resilience to droughts, as compared with convex pools. New empirical studies are needed to test these model predictions.

Period doubling as an indicator for ecosystem sensitivity to climate extremes

The predictions for a warmer and drier climate and for increased likelihood of climate extremes raise high concerns about the possible collapse of dryland ecosystems, and about the formation of new drylands where native species are less tolerant to water stress. Using a dryland-vegetation model for plant species that display different tradeoffs between fast growth and tolerance to droughts, we find that ecosystems subjected to strong seasonal variability, typical for drylands, exhibit a temporal period-doubling route to chaos that results in early collapse to bare soil. We further find that fast-growing plants go through period doubling sooner and span wider chaotic ranges than stress-tolerant plants. We propose the detection of period-doubling signatures in power spectra as early indicators of ecosystem collapse that outperform existing indicators in their ability to warn against climate extremes and capture the heightened vulnerability of newly-formed drylands. The proposed indicator is expected to apply to other types of ecosystems, such as consumer–resource and predator–prey systems. We conclude by delineating the conditions ecosystems should meet in order for the proposed indicator to apply.

Continuum Modeling of Discrete Plant Communities: Why Does It Work and Why Is It Advantageous?

Understanding ecosystem response to drier climates calls for modeling the dynamics of dryland plant populations, which are crucial determinants of ecosystem function, as they constitute the basal level of whole food webs. Two modeling approaches are widely used in population dynamics, individual (agent)-based models and continuum partial-differential-equation (PDE) models. The latter are advantageous in lending themselves to powerful methodologies of mathematical analysis, but the question of whether they are suitable to describe small discrete plant populations, as is often found in dryland ecosystems, has remained largely unaddressed. In this paper, we first draw attention to two aspects of plants that distinguish them from most other organisms—high phenotypic plasticity and dispersal of stress-tolerant seeds—and argue in favor of PDE modeling, where the state variables that describe population sizes are not discrete number densities, but rather continuous biomass densities. We then discuss a few examples that demonstrate the utility of PDE models in providing deep insights into landscape-scale behaviors, such as the onset of pattern forming instabilities, multiplicity of stable ecosystem states, regular and irregular, and the possible roles of front instabilities in reversing desertification. We briefly mention a few additional examples, and conclude by outlining the nature of the information we should and should not expect to gain from PDE model studies.

Metastability as a Coexistence Mechanism in a Model for Dryland Vegetation Patterns

Vegetation patterns are a ubiquitous feature of water-deprived ecosystems. Despite the competition for the same limiting resource, coexistence of several plant species is commonly observed. We propose a two-species reaction-diffusion model based on the single-species Klausmeier model, to analytically investigate the existence of states in which both species coexist. Ecologically, the study finds that coexistence is supported if there is a small difference in the plant species' average fitness, measured by the ratio of a species' capabilities to convert water into new biomass to its mortality rate. Mathematically, coexistence is not a stable solution of the system, but both spatially uniform and patterned coexistence states occur as metastable states. In this context, a metastable solution in which both species coexist corresponds to a long transient (exceeding [Formula: see text] years in dimensional parameters) to a stable one-species state. This behaviour is characterised by the small size of a positive eigenvalue which has the same order of magnitude as the average fitness difference between the two species. Two mechanisms causing the occurrence of metastable solutions are established: a spatially uniform unstable equilibrium and a stable one-species pattern which is unstable to the introduction of a competitor. We further discuss effects of asymmetric interspecific competition (e.g. shading) on the metastability property.

Front Instabilities Can Reverse Desertification

Degradation processes in living systems often take place gradually by front propagation. An important context of such processes is loss of biological productivity in drylands or desertification. Using a dryland-vegetation model, we analyze the stability and dynamics of desertification fronts, identify linear and nonlinear front instabilities, and highlight the significance of these instabilities in inducing self-recovery. The results are based on the derivation and analysis of a universal amplitude equation for pattern-forming living systems for which nonuniform instabilities cannot emerge from the nonviable (zero) state. The results may therefore be applicable to other contexts of animate matter where degradation processes occur by front propagation.

Interplay between exogenous and endogenous factors in seasonal vegetation oscillations

A fundamental question in ecology is whether vegetation oscillations are merely a result of periodic environmental variability, or rather driven by endogenous factors. We address this question using a mathematical model of dryland vegetation subjected to annual rainfall periodicity. We show that while spontaneous oscillations do not exist in realistic parameter ranges, resonant response to periodic precipitation is still possible due to the existence of damped oscillatory modes. Using multiple time-scale analysis, in a restricted parameter range, we find that these endogenous modes can be pumped by the exogenous precipitation forcing to form sustained oscillations. The oscillations amplitude shows a resonance peak that depends on model parameters representing species traits and mean annual precipitation. Extending the study to bistability ranges of uniform vegetation and bare soil, we investigate numerically the implications of resonant oscillations for ecosystem function. We consider trait parameters that represent species with damped oscillatory modes and species that lack such modes, and compare their behaviors. We find that the former are less resilient to droughts, suffer from larger declines in their biomass production as the precipitation amplitude is increased, and, in the presence of spatial disturbances, are likely to go through abrupt collapse to bare soil, rather than gradual, domino-like collapse.

Molding the asymmetry of localized frequency-locking waves by a generalized forcing and implications to the inner ear

Frequency locking to an external forcing frequency is a well-known phenomenon. In the auditory system, it results in a localized traveling wave, the shape of which is essential for efficient discrimination between incoming frequencies. An amplitude equation approach is used to show that the shape of the localized traveling wave depends crucially on the relative strength of additive versus parametric forcing components; the stronger the parametric forcing, the more asymmetric is the response profile and the sharper is the traveling-wave front. The analysis qualitatively captures the empirically observed regions of linear and nonlinear responses and highlights the potential significance of parametric forcing mechanisms in shaping the resonant response in the inner ear.

Implications of tristability in pattern-forming ecosystems

Many ecosystems show both self-organized spatial patterns and multistability of possible states. The combination of these two phenomena in different forms has a significant impact on the behavior of ecosystems in changing environments. One notable case is connected to tristability of two distinct uniform states together with patterned states, which has recently been found in model studies of dryland ecosystems. Using a simple model, we determine the extent of tristability in parameter space, explore its effects on the system dynamics, and consider its implications for state transitions or regime shifts. We analyze the bifurcation structure of model solutions that describe uniform states, periodic patterns, and hybrid states between the former two. We map out the parameter space where these states exist, and note how the different states interact with each other. We further focus on two special implications with ecological significance, breakdown of the snaking range and complex fronts. We find that the organization of the hybrid states within a homoclinic snaking structure breaks down as it meets a Maxwell point where simple fronts are stationary. We also discover a new series of complex fronts between the uniform states, each with its own velocity. We conclude with a brief discussion of the significance of these findings for the dynamics of regime shifts and their potential control.

On a nonlocal system for vegetation in drylands

Several mathematical models are proposed to understand spatial patchy vegetation patterns arising in drylands. In this paper, we consider the system with nonlocal dispersal of plants (through a redistribution kernel for seeds) proposed by Pueyo et al. (Oikos 117:1522-1532, 2008) as a model for vegetation in water-limited ecosystems. It consists in two reaction diffusion equations for surface water and soil water, combined with an integro-differential equation for plants. For this system, under suitable assumptions, we prove well-posedness using the Schauder fixed point theorem. In addition, we consider the stationary problem from the viewpoint of vegetated pattern formation, and show a transition of vegetation patterns when parameter values (rainfall, seed dispersal range, seed germination rate) in the system vary. The influence of the shape of the redistribution kernel is also discussed.

Effects of Hydraulic Soil Properties on Vegetation Pattern Formation in Sloping Landscapes

Current models of vegetation pattern formation rely on a system of weakly nonlinear reaction-diffusion equations that are coupled by their source terms. While these equations, which are used to describe a spatiotemporal planar evolution of biomass and soil water, qualitatively capture the emergence of various types of vegetation patterns in arid environments, they are phenomenological and have a limited predictive power. We ameliorate these limitations by deriving the vertically averaged Richards' equation to describe flow (as opposed to "diffusion") of water in partially saturated soils. This establishes conditions under which this nonlinear equation reduces to its weakly nonlinear reaction-diffusion counterpart used in the previous models, thus relating their unphysical parameters (e.g., diffusion coefficient) to the measurable soil properties (e.g., hydraulic conductivity) used to parameterize the Richards equation. Our model is valid for both flat and sloping landscapes and can handle arbitrary topography and boundary conditions. The result is a model that relates the environmental conditions (e.g., precipitation rate, runoff and soil properties) to formation of multiple patterns observed in nature (such as stripes, labyrinth and spots).

A morphometric analysis of vegetation patterns in dryland ecosystems

Vegetation in dryland ecosystems often forms remarkable spatial patterns. These range from regular bands of vegetation alternating with bare ground, to vegetated spots and labyrinths, to regular gaps of bare ground within an otherwise continuous expanse of vegetation. It has been suggested that spotted vegetation patterns could indicate that collapse into a bare ground state is imminent, and the morphology of spatial vegetation patterns, therefore, represents a potentially valuable source of information on the proximity of regime shifts in dryland ecosystems. In this paper, we have developed quantitative methods to characterize the morphology of spatial patterns in dryland vegetation. Our approach is based on algorithmic techniques that have been used to classify pollen grains on the basis of textural patterning, and involves constructing feature vectors to quantify the shapes formed by vegetation patterns. We have analysed images of patterned vegetation produced by a computational model and a small set of satellite images from South Kordofan (South Sudan), which illustrates that our methods are applicable to both simulated and real-world data. Our approach provides a means of quantifying patterns that are frequently described using qualitative terminology, and could be used to classify vegetation patterns in large-scale satellite surveys of dryland ecosystems.

Spatial Self-Organization of Ecosystems: Integrating Multiple Mechanisms of Regular-Pattern Formation

Large-scale regular vegetation patterns are common in nature, but their causes are disputed. Whereas recent theory focuses on scale-dependent feedbacks as a potentially universal mechanism, earlier studies suggest that many regular spatial patterns result from territorial interference competition between colonies of social-insect ecosystem engineers, leading to hexagonally overdispersed nest sites and associated vegetation. Evidence for this latter mechanism is scattered throughout decades of disparate literature and lacks a unified conceptual framework, fueling skepticism about its generality in debates over the origins of patterned landscapes. We review these mechanisms and debates, finding evidence that spotted and gapped vegetation patterns generated by ants, termites, and other subterranean animals are globally widespread, locally important for ecosystem functioning, and consistent with models of intraspecific territoriality. Because these and other mechanisms of regular-pattern formation are not mutually exclusive and can coexist and interact at different scales, the prevailing theoretical outlook on spatial self-organization in ecology must expand to incorporate the dynamic interplay of multiple processes.