Self-organized Spatial Patterns Research Articles

Self-organization in spatial ecology

Biologists have long known that populations of organisms - microbes, plants, animals - can self-organize into emergent patterns. Yet, the fact that such patterns can arise with remarkable symmetry at the scale of entire ecosystems remains astonishing, even as aerial imagery has documented their existence across all continents. As the enormous scale of landscape patterns makes them experimentally intractable, ecologists have relied on theoretical modelling - typically rooted in physics - to investigate the underlying pattern-forming mechanisms. Such models have succeeded in generating mechanistic hypotheses and indicate that self-organized spatial patterns can reflect the health of an ecosystem. However, most of these hypotheses remain untested. This essay reflects on our current understanding of the causes and consequences of ecosystem-scale pattern formation.

Self-organization as a mechanism of resilience in dryland ecosystems

Self-organized spatial patterns are a common feature of complex systems, ranging from microbial communities to mussel beds and drylands. While the theoretical implications of these patterns for ecosystem-level processes, such as functioning and resilience, have been extensively studied, empirical evidence remains scarce. To address this gap, we analyzed global drylands along an aridity gradient using remote sensing, field data, and modeling. We found that the spatial structure of the vegetation strengthens as aridity increases, which is associated with the maintenance of a high level of soil multifunctionality, even as aridity levels rise up to a certain threshold. The combination of these results with those of two individual-based models indicate that self-organized vegetation patterns not only form in response to stressful environmental conditions but also provide drylands with the ability to adapt to changing conditions while maintaining their functioning, an adaptive capacity which is lost in degraded ecosystems. Self-organization thereby plays a vital role in enhancing the resilience of drylands. Overall, our findings contribute to a deeper understanding of the relationship between spatial vegetation patterns and dryland resilience. They also represent a significant step forward in the development of indicators for ecosystem resilience, which are critical tools for managing and preserving these valuable ecosystems in a warmer and more arid world.

Microplastic leachates inhibit small-scale self-organization in mussel beds

Self-organized spatial patterns are increasingly recognized for their contribution to ecosystem functioning. They can improve the ecosystem's ability to respond to perturbation and thus increase its resilience to environmental stress. Plastic pollution has now emerged as major threat to aquatic and terrestrial biota. Under laboratory conditions, we tested whether plastic leachates from pellets collected in the intertidal can impair small-scale, spatial self-organization and byssal threads production of intertidal mussels and whether the effect varied depending on where the pellets come from. Specifically, leachates originating from plastic pellets collected from relatively pristine and polluted areas respectively impaired and inhibited the ability of mussels to self-organize at small-scale and to produce byssal threads compared to control conditions (i.e., seawater without leaching solution). Limitations to natural self-organizing processes and threads formation may translate to a declined capacity of natural ecosystems to avoid tipping points and to a reduced restoration success of disturbed ecosystems.

Jamming and flocking in the restricted active Potts model.

We study the active Potts model with either site occupancy restriction or on-site repulsion to explore jamming and kinetic arrest in a flocking model. The incorporation of such volume exclusion features leads to a surprisingly rich variety of self-organized spatial patterns. While bands and lanes of moving particles commonly occur without or under weak volume exclusion, strong volume exclusion along with low temperature, high activity, and large particle density facilitates jams due to motility-induced phase separation. Through several phase diagrams, we identify the phase boundaries separating the jammed and free-flowing phases and study the transition between these phases which provide us with both qualitative and quantitative predictions of how jamming might be delayed or dissolved. We further formulate and analyze a hydrodynamic theory for the restricted APM which predicts various features of the microscopic model.

Phase-separation physics underlies new theory for the resilience of patchy ecosystems.

Spatial self-organization of ecosystems into large-scale (from micron to meters) patterns is an important phenomenon in ecology, enabling organisms to cope with harsh environmental conditions and buffering ecosystem degradation. Scale-dependent feedbacks provide the predominant conceptual framework for self-organized spatial patterns, explaining regular patterns observed in, e.g., arid ecosystems or mussel beds. Here, we highlight an alternative mechanism for self-organized patterns, based on the aggregation of a biotic or abiotic species, such as herbivores, sediment, or nutrients. Using a generalized mathematical model, we demonstrate that ecosystems with aggregation-driven patterns have fundamentally different dynamics and resilience properties than ecosystems with patterns that formed through scale-dependent feedbacks. Building on the physics theory for phase-separation dynamics, we show that patchy ecosystems with aggregation patterns are more vulnerable than systems with patterns formed through scale-dependent feedbacks, especially at small spatial scales. This is because local disturbances can trigger large-scale redistribution of resources, amplifying local degradation. Finally, we show that insights from physics, by providing mechanistic understanding of the initiation of aggregation patterns and their tendency to coarsen, provide a new indicator framework to signal proximity to ecological tipping points and subsequent ecosystem degradation for this class of patchy ecosystems.

Integrating theory and experiments to link local mechanisms and ecosystem-level consequences of vegetation patterns in drylands

Self-organized spatial patterns of vegetation are frequent in drylands and, because pattern shape correlates with water availability, they have been suggested as important indicators of ecosystem health. However, the mechanisms underlying pattern emergence remain unclear. Some theories hypothesize that patterns could result from a water-mediated scale-dependent feedback (SDF) whereby interactions favoring plant growth dominate at short distances and growth–inhibitory interactions dominate in the long range. However, we know little about how the presence of a focal plant affects the fitness of its neighbors as a function of the inter-individual distance, which is expected to be highly ecosystem-dependent. This lack of empirical knowledge and system dependency challenge the relevance of SDF as a unifying theory for vegetation pattern formation. Assuming that plant interactions are always inhibitory and only their intensity is scale-dependent, alternative theories also recover the typical vegetation patterns observed in nature. Importantly, although these alternative hypotheses lead to visually indistinguishable patterns, they predict contrasting desertification dynamics, which questions the potential use of vegetation patterns as ecosystem-state indicators. To help resolve this issue, we first review existing theories for vegetation self-organization and their conflicting predictions about desertification dynamics. Second, we discuss potential empirical tests via manipulative experiments to identify pattern-forming mechanisms and link them to specific desertification dynamics. A comprehensive view of models, the mechanisms they intend to capture, and experiments to test them in the field will help to better understand both how patterns emerge and improve predictions on the fate of the ecosystems where they form.

Distributed information encoding and decoding using self-organized spatial patterns

SummaryDynamical systems often generate distinct outputs according to different initial conditions, and one can infer the corresponding input configuration given an output. This property captures the essence of information encoding and decoding. Here, we demonstrate the use of self-organized patterns that generate high-dimensional outputs, combined with machine learning, to achieve distributed information encoding and decoding. Our approach exploits a critical property of many natural pattern-formation systems: in repeated realizations, each initial configuration generates similar but not identical output patterns due to randomness in the patterning process. However, for sufficiently small randomness, different groups of patterns that arise from different initial configurations can be distinguished from one another. Modulating the pattern-generation and machine learning model training can tune the tradeoff between encoding capacity and security. We further show that this strategy is scalable by implementing the encoding and decoding of all characters of the standard English keyboard.

Spatial patterns in ecological systems: from microbial colonies to landscapes.

Self-organized spatial patterns are ubiquitous in ecological systems and allow populations to adopt non-trivial spatial distributions starting from disordered configurations. These patterns form due to diverse nonlinear interactions among organisms and between organisms and their environment, and lead to the emergence of new (eco)system-level properties unique to self-organized systems. Such pattern consequences include higher resilience and resistance to environmental changes, abrupt ecosystem collapse, hysteresis loops, and reversal of competitive exclusion. Here, we review ecological systems exhibiting self-organized patterns. We establish two broad pattern categories depending on whether the self-organizing process is primarily driven by nonlinear density-dependent demographic rates or by nonlinear density-dependent movement. Using this organization, we examine a wide range of observational scales, from microbial colonies to whole ecosystems, and discuss the mechanisms hypothesized to underlie observed patterns and their system-level consequences. For each example, we review both the empirical evidence and the existing theoretical frameworks developed to identify the causes and consequences of patterning. Finally, we trace qualitative similarities across systems and propose possible ways of developing a more quantitative understanding of how self-organization operates across systems and observational scales in ecology.

Diffusion sustains cooperation via forming diverse spatial patterns in prisoner's dilemma game

How cooperation arouses and maintains in social dilemmas has long been the core problem in evolutionary biology. Assortative interaction has been identified as a fundamental mechanism for cooperation in the social dilemma games. Many studies have shown that spatial limitation can lead to cooperation evolution, but how spatial pattern shape the cooperation is studied rarely. In this paper, using a spatial model of eco-evolutionary prisoner's dilemma game, where the player's diffusion is incorporated definitely, we show that the diffusion can trigger out rich spatial patterns. These self-organized spatial patterns could profoundly affect the cooperation level and system stability. Totally, six types of patterns can be identified according to their influences on cooperation evolution. Cooperation is affected by the spatial configuration of population distributions: spot- or stripe- type pattern promotes the cooperation, but mixed form of them or spatial synchrony suppresses cooperation. In contrast, uniform pattern has no effect on cooperation. Meanwhile, the analysis of spatial autocorrelation about population distributions via local Moran index reveals that the difference of spatial correlations among adjacent sites may be responsible for the functional differentiation of such self-organized structures. Spatial diffusion arouses vastly different spatial patterns, which renders the spatial extension system greatly expands the survival region of cooperators, i.e., the game dynamics continues after the homogeneous system has collapsed. Those results are helpful to understand the function of spatial patterns in supporting cooperation and fostering behavioral diversity.

The shaping role of self-organization: linking vegetation patterning, plant traits and ecosystem functioning.

Self-organized spatial patterns are increasingly recognized for their contribution to ecosystem functioning, in terms of enhanced productivity, ecosystem stability, and species diversity in terrestrial as well as marine ecosystems. Most studies on the impact of spatial self-organization have focused on systems that exhibit regular patterns. However, there is an abundance of patterns in many ecosystems which are not strictly regular. Understanding of how these patterns are formed and how they affect ecosystem function is crucial for the broad acceptance of self-organization as a keystone process in ecological theory. Here, using transplantation experiments in salt marsh ecosystems dominated by Scirpus mariqueter, we demonstrate that scale-dependent feedback is driving irregular spatial pattern formation of vegetation. Field observations and experiments have revealed that this self-organization process affects a range of plant traits, including shoot-to-root ratio, rhizome orientation, rhizome node number, and rhizome length, and enhances vegetation productivity. Moreover, patchiness in self-organized salt marsh vegetation can support a better microhabitat for macrobenthos, promoting their total abundance and spatial heterogeneity of species richness. Our results extend existing concepts of self-organization and its effects on productivity and biodiversity to the spatial irregular patterns that are observed in many systems. Our work also helps to link between the so-far largely unconnected fields of self-organization theory and trait-based, functional ecology.

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.

Local interactions and self-organized spatial patterns stabilize microbial cross-feeding against cheaters.

Mutualisms are ubiquitous, but models predict they should be susceptible to cheating. Resolving this paradox has become relevant to synthetic ecology: cooperative cross-feeding, a nutrient-exchange mutualism, has been proposed to stabilize microbial consortia. Previous attempts to understand how cross-feeders remain robust to non-producing cheaters have relied on complex behaviour (e.g. cheater punishment) or group selection. Using a stochastic spatial model, we demonstrate two novel mechanisms that can allow cross-feeders to outcompete cheaters, rather than just escape from them. Both mechanisms work through the spatial segregation of the resources, which prevents individual cheaters from acquiring the resources they need to reproduce. First, if microbe dispersal is low but resources are shared widely, then the cross-feeders self-organize into stable spatial patterns. Here the cross-feeders can build up where the resource they need is abundant, and send their resource to where their partner is, separating resources at regular intervals in space. Second, if dispersal is high but resource sharing is local, then random variation in population density creates small-scale variation in resource density, separating the resources from each other by chance. These results suggest that cross-feeding may be more robust than previously expected and offer strategies to engineer stable consortia.

Effect of directional migration on Lotka-Volterra system with desert

Migration is observed across many species. Several authors have studied ecological migration by applying cellular automaton (CA). In this paper, we present a directional migration model with desert on a one-dimensional lattice where a traffic CA model and a lattice Lotka-Volterra system are connected. Here predators correspond to locomotive animals while prey is immobile plants. Predators migrate between deserts and fertile lands repeatedly. Computer simulations reveal the two types of phase transition: coexistence of both species and prey dominance, which is caused by both benefit and cost of migration. In the coexistence phase, the steady-state density of predators usually increases by migration as long as the desert size is small and their mortality rate is low. In contrast, the prey density increases, even if the desert size becomes large. Such a paradox comes from the indirect effect: predators go extinct by the increase of desert size, so that the plant density can increase. Moreover, we find several self-organized spatial patterns: 1) predators form a stripe pattern; namely swarms. 2) The velocity of predators is high on deserts, but very low on fertile land. 3) Predators give birth only on fertile lands.

Reduction of species coexistence through mixing in a spatial competition model

Many ecological systems exhibit self-organized spatial patterns due to local interactions. Such patterns can promote species diversity and therefore serve as an important mechanism for biodiversity maintenance. Previous work has shown that when species interactions occurred at local spatial scales, species diversity was greatest when robust mosaic spatial patterns formed. Also, intransitive interactions led to the emergence of spiral patterns, frequently resulting in multispecies coexistence. In some instances, intransitive interactions reduced species diversity as the consequence of competitive hierarchies. Here, we extend and broaden this line of investigation and examine the role of global competition along a continuum ranging from spatial mosaics to spiral patterns. While previous models have predicted that species diversity is reduced when interactions occur over larger spatial scales, our model considers the effects of various levels of mixing on species diversity, in the context of various network structures as measured by the covariance of row and column sums of the competition matrix. First, we compare local competition (unmixed system) versus global competition (mixed systems) and show that greater species diversity is maintained under a positive covariance. Second, we show that under various levels of mixing, species diversity declines more rapidly under a negative covariance. Lastly, we demonstrate that time to extinction in our model occurs much more rapidly under a negative covariance.

Behavioral self-organization underlies the resilience of a coastal ecosystem

Self-organized spatial patterns occur in many terrestrial, aquatic, and marine ecosystems. Theoretical models and observational studies suggest self-organization, the formation of patterns due to ecological interactions, is critical for enhanced ecosystem resilience. However, experimental tests of this cross-ecosystem theory are lacking. In this study, we experimentally test the hypothesis that self-organized pattern formation improves the persistence of mussel beds (Mytilus edulis) on intertidal flats. In natural beds, mussels generate self-organized patterns at two different spatial scales: regularly spaced clusters of mussels at centimeter scale driven by behavioral aggregation and large-scale, regularly spaced bands at meter scale driven by ecological feedback mechanisms. To test for the relative importance of these two spatial scales of self-organization on mussel bed persistence, we conducted field manipulations in which we factorially constructed small-scale and/or large-scale patterns. Our results revealed that both forms of self-organization enhanced the persistence of the constructed mussel beds in comparison to nonorganized beds. Small-scale, behaviorally driven cluster patterns were found to be crucial for persistence, and thus resistance to wave disturbance, whereas large-scale, self-organized patterns facilitated reformation of small-scale patterns if mussels were dislodged. This study provides experimental evidence that self-organization can be paramount to enhancing ecosystem persistence. We conclude that ecosystems with self-organized spatial patterns are likely to benefit greatly from conservation and restoration actions that use the emergent effects of self-organization to increase ecosystem resistance to disturbance.

Intradomain Textures in Block Copolymers: Multizone Alignment and Biaxiality.

Block copolymer (BCP) melt assembly has been studied for decades, focusing largely on self-organized spatial patterns of periodically ordered segment density. Here, we demonstrate that underlying the well-known composition profiles (i.e., ordered lamella, cylinders, spheres, and networks) are generic and heterogeneous patterns of segment orientation that couple strongly to morphology, even in the absence of specific factors that promote intra or interchain segment alignment. We employ both self-consistent field theory and coarse-grained simulation methods to measure polar and nematic order parameters of segments in a freely jointed chain model of diblock melts. We show that BCP morphologies have a multizone texture, with segments predominantly aligned normal and parallel to interdomain interfaces in the respective brush and interfacial regions of the microdomain. Further, morphologies with anisotropically curved interfaces (i.e., cylinders and networks) exhibit biaxial order that is aligned to the principal curvature axes of the interface.

Cooperation in carbon source degradation shapes spatial self-organization of microbial consortia on hydrated surfaces

Mounting evidence suggests that natural microbial communities exhibit a high level of spatial organization at the micrometric scale that facilitate ecological interactions and support biogeochemical cycles. Microbial patterns are difficult to study definitively in natural environments due to complex biodiversity, observability and variable physicochemical factors. Here, we examine how trophic dependencies give rise to self-organized spatial patterns of a well-defined bacterial consortium grown on hydrated surfaces. The model consortium consisted of two Pseudomonas putida mutant strains that can fully degrade the aromatic hydrocarbon toluene. We demonstrated that obligate cooperation in toluene degradation (cooperative mutualism) favored convergence of 1:1 partner ratio and strong intermixing at the microscale (10–100 μm). In contrast, competition for benzoate, a compound degraded independently by both strains, led to distinct segregation patterns. Emergence of a persistent spatial pattern has been predicted for surface attached microbial activity in liquid films that mediate diffusive exchanges while permitting limited cell movement (colony expansion). This study of a simple microbial consortium offers mechanistic glimpses into the rules governing the assembly and functioning of complex sessile communities, and points to general principles of spatial organization with potential applications for natural and engineered microbial systems.

A new mechanism for spatial pattern formation via lateral and protrusion-mediated lateral signalling.

Tissue organization and patterning are critical during development when genetically identical cells take on different fates. Lateral signalling plays an important role in this process by helping to generate self-organized spatial patterns in an otherwise uniform collection of cells. Recent data suggest that lateral signalling can be mediated both by junctional contacts between neighbouring cells and via cellular protrusions that allow non-neighbouring cells to interact with one another at a distance. However, it remains unclear precisely how signalling mediated by these distinct types of cell–cell contact can physically contribute to the generation of complex patterns without the assistance of diffusible morphogens or pre-patterns. To explore this question, in this work we develop a model of lateral signalling based on a single receptor/ligand pair as exemplified by Notch and Delta. We show that allowing the signalling kinetics to differ at junctional versus protrusion-mediated contacts, an assumption inspired by recent data which show that the cleavage of Notch in several systems requires both Delta binding and the application of mechanical force, permits individual cells to act to promote both lateral activation and lateral inhibition. Strikingly, under this model, in which Delta can sequester Notch, a variety of patterns resembling those typical of reaction–diffusion systems is observed, together with more unusual patterns that arise when we consider changes in signalling kinetics, and in the length and distribution of protrusions. Importantly, these patterns are self-organizing—so that local interactions drive tissue-scale patterning. Together, these data show that protrusions can, in principle, generate different types of patterns in addition to contributing to long-range signalling and to pattern refinement.

A Five Species Cyclically Dominant Evolutionary Game with Fixed Direction: A New Way to Produce Self-Organized Spatial Patterns

Cyclically dominant systems are hot issues in academia, and they play an important role in explaining biodiversity in Nature. In this paper, we construct a five-strategy cyclically dominant system. Each individual in our system changes its strategy along a fixed direction. The dominant strategy can promote a change in the dominated strategy, and the dominated strategy can block a change in the dominant strategy. We use mean-field theory and cellular automaton simulation to discuss the evolving characters of the system. In the cellular automaton simulation, we find the emergence of spiral waves on spatial patterns without a migration rate, which suggests a new way to produce self-organized spatial patterns.

Advancing microbial sciences by individual-based modelling.

Remarkable technological advances have revealed ever more properties and behaviours of individual microorganisms, but the novel data generated by these techniques have not yet been fully exploited. In this Opinion article, we explain how individual-based models (IBMs) can be constructed based on the findings of such techniques and how they help to explore competitive and cooperative microbial interactions. Furthermore, we describe how IBMs have provided insights into self-organized spatial patterns from biofilms to the oceans of the world, phage-CRISPR dynamics and other emergent phenomena. Finally, we discuss how combining individual-based observations with IBMs can advance our understanding at both the individual and population levels, leading to the new approach of microbial individual-based ecology (μIBE).