Phase separation driven by density-dependent movement: A novel mechanism for ecological patterns

Density-dependent within-patch movement behavior of two competing species

Movement behavior is central to understanding species distributions, population dynamics and coexistence with other species. Although the relationship between conspecific density and emigration has been well studied, little attention has been paid to how interspecific competitor density affects another species' movement behavior. We conducted releases of two species of competing Tribolium flour beetles at different densities, alone and together in homogeneous microcosms, and tested whether their recaptures-with-distance were well described by a random-diffusion model. We also determined whether mean displacement distances varied with the release density of conspecific and heterospecific beetles. A diffusion model provided a good fit to the redistribution of T. castaneum and T. confusum at all release densities, explaining an average of >60% of the variation in recaptures. For both species, mean displacement (directly proportional to the diffusion rate) exhibited a humped-shaped relationship with conspecific density. Finally, we found that both species of beetle impacted the within-patch movement rates of the other species, but the effect depended on density. For T. castaneum in the highest density treatment, the addition of equal numbers of T. castaneum or T. confusum had the same effect, with mean displacements reduced by approximately one half. The same result occurred for T. confusum released at an intermediate density. In both cases, it was total beetle abundance, not species identity that mattered to mean displacement. We suggest that displacement or diffusion rates that exhibit a nonlinear relationship with density or depend on the presence or abundance of interacting species should be considered when attempting to predict the spatial spread of populations or scaling up to heterogeneous landscapes.

Eco-evolutionary dynamics in finite network-structured populations with migration

We consider the effect of network structure on the evolution of a population. Models of this kind typically consider a population of fixed size and distribution. Here we consider eco-evolutionary dynamics where population size and distribution can change through birth, death and migration, all of which are separate processes. This allows complex interaction and migration behaviours that are dependent on competition. For migration, we assume that the response of individuals to competition is governed by tolerance to their group members, such that less tolerant individuals are more likely to move away due to competition. We look at the success of a mutant in the rare mutation limit for the complete, cycle and star networks. Unlike models with fixed population size and distribution, the distribution of the individuals per site is explicitly modelled by considering the dynamics of the population. This in turn determines the mutant appearance distribution for each network. Where a mutant appears impacts its success as it determines the competition it faces. For low and high migration rates the complete and cycle networks have similar mutant appearance distributions resulting in similar success levels for an invading mutant. A higher migration rate in the star network is detrimental for mutant success because migration results in a crowded central site where a mutant is more likely to appear.

Patch-level processes of vegetation underlying site-level restoration patterns in a megatidal salt marsh

Vegetation patterns during salt marsh restoration reflect underlying processes related to colonization, reproduction, and interactions of halotolerant plants. Examining both pattern and process during recovery is valuable for understanding and managing salt marsh restoration projects. We present a decade of vegetation dynamics during salt marsh restoration (2011–2020) at a study site in the Bay of Fundy with megatidal amplitudes, strong currents, cold winter temperatures, and ice. We mainly investigated reproduction (asexual and sexual) and associated spread rates of Spartina grasses, and their health-related states (stem density, canopy height, and percent flowering) which help inform the probability of processes occurring. We also estimated modes of colonization and began quantifying the effects of interspecific interactions and environmental conditions on plant state. Spartina pectinata was the only pastureland plant to survive dike-breaching and saltwater intrusion in 2010; however, it was stunted compared to reference plants. Spartina pectinata patches remained consistent initially, before decreasing in size, and disappearing by the fifth year (2015). This early dynamic may provide initial protection to a developing salt marsh before Spartina alterniflora becomes established. Spartina alterniflora first colonized the sites in year 2 (2012), likely via deposition of rhizomal material, and then spread asexually before seedlings (sexual reproduction) appeared in year 4 (2014). Vegetation cover subsequently increased greatly until near-complete in year 9 (2019). The early successional dynamics of S. pectinata and S. alterniflora occurred spatially independently of each other, and likely contributed to sediment retention, creating an improved environment for S. patens, the dominant high marsh species in our region. Spartina patens have been slowly spreading into restoration sites from high elevation areas since year 6 (2016). We expect that competition between S. alterniflora and S. patens will result in the typical distinct zonation between high and low marsh zones. A next study will use the quantified processes for spatial-explicit modeling to simulate patterns of vegetation recovery, and to evaluate different salt marsh restoration strategies for the Bay of Fundy and elsewhere. Thus, proper identification and quantification of pattern-building processes in salt marsh vegetation recovery, the focus of our present study, was an essential step.

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.

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.

Foraging behaviours lead to spatiotemporal self-similar dynamics in grazing ecosystems

Biological behaviour‐driven self‐organized patterns have recently been confirmed to play a key role in ecosystem functioning. Here, we develop a theoretical phase‐separation model to describe spatiotemporal self‐similar dynamics, which is a consequence of behaviour‐driven trophic interactions in short‐time scales. Our framework integrates scale‐dependent feedback and density‐dependent movement into grazing ecosystems. This model derives six types of selective foraging behaviours that trigger pattern formation for top‐down grazing ecosystems, and one of which is consistent with existing foraging theories. Self‐organized patterns nucleate under moderate grazing intensity and are destroyed by overgrazing, which suggests ecosystem degradation. Theoretical results qualitatively agree with observed grazing ecosystems that display spatial heterogeneities under variable grazing intensity. Our findings potentially provide new insights into self‐organized patterns as an indicator of ecosystem transitions under a stressful environment.

Evasion of tipping in complex systems through spatial pattern formation

[Figure: see text].

Ice needles weave patterns of stones in freezing landscapes

Patterned ground, defined by the segregation of stones in soil according to size, is one of the most strikingly self-organized characteristics of polar and high-alpine landscapes. The presence of such patterns on Mars has been proposed as evidence for the past presence of surface liquid water. Despite their ubiquity, the dearth of quantitative field data on the patterns and their slow dynamics have hindered fundamental understanding of the pattern formation mechanisms. Here, we use laboratory experiments to show that stone transport is strongly dependent on local stone concentration and the height of ice needles, leading effectively to pattern formation driven by needle ice activity. Through numerical simulations, theory, and experiments, we show that the nonlinear amplification of long wavelength instabilities leads to self-similar dynamics that resemble phase separation patterns in binary alloys, characterized by scaling laws and spatial structure formation. Our results illustrate insights to be gained into patterns in landscapes by viewing the pattern formation through the lens of phase separation. Moreover, they may help interpret spatial structures that arise on diverse planetary landscapes, including ground patterns recently examined using the rover Curiosity on Mars.

Field-theoretic model for chemotaxis in run and tumble particles

In this paper, we develop a field-theoretic description for run and tumble chemotaxis, based on a density-functional description of crystalline materials modified to capture orientational ordering. We show that this framework, with its in-built multiparticle interactions, soft-core repulsion, and elasticity, is ideal for describing continuum collective phases with particle resolution, but on diffusive timescales. We show that our model exhibits particle aggregation in an externally imposed constant attractant field, as is observed for phototactic or thermotactic agents. We also show that this model captures particle aggregation through self-chemotaxis, an important mechanism that aids quorum-dependent cellular interactions.

Growth, death, and resource competition in sessile organisms

Population-level scaling in ecological systems arises from individual growth and death with competitive constraints. We build on a minimal dynamical model of metabolic growth where the tension between individual growth and mortality determines population size distribution. We then separately include resource competition based on shared capture area. By varying rates of growth, death, and competitive attrition, we connect regular and random spatial patterns across sessile organisms from forests to ants, termites, and fairy circles. Then, we consider transient temporal dynamics in the context of asymmetric competition, such as canopy shading or large colony dominance, whose effects primarily weaken the smaller of two competitors. When such competition couples slow timescales of growth to fast competitive death, it generates population shocks and demographic oscillations similar to those observed in forest data. Our minimal quantitative theory unifies spatiotemporal patterns across sessile organisms through local competition mediated by the laws of metabolic growth, which in turn, are the result of long-term evolutionary dynamics.

Nonlinear self-organized population dynamics induced by external selective nonlocal processes

Self-organization evolution of a population is studied considering generalized reaction-diffusion equations. We proposed a model based on non-local operators that has several of the equations traditionally used in research on population dynamics as particular cases. Then, employing a relatively simple functional form of the non-local kernel, we determined the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we established a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our proposal reproduces some of the experimental results that other approaches considered previously had not been able to obtain.

Nonlinear self-organized population dynamics induced by external selective nonlocal processes

Self-organization evolution of a population is studied considering generalized reaction-diffusion equations. We proposed a model based on non-local operators that has several of the equations traditionally used in research on population dynamics as particular cases. Then, employing a relatively simple functional form of the non-local kernel, we determined the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we established a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our proposal reproduces some of the experimental results that other approaches considered previously had not been able to obtain.

A computational study of density-dependent individual movement and the formation of population clusters in two-dimensional spatial domains

The patterns of collective behaviour in a population emerging from individual animal movement have long been of interest to ecologists, as has the emergence of heterogeneous patterns among a population. In this paper we will consider these phenomena by using an individual-based modelling approach to simulate a population whose individuals undergo density-dependent movement in 2D spatial domains. We first show that the introduction of density-dependent movement in the form of two parameters, a perception radius and a probability of directed movement, leads to the formation of clusters. We then show that the properties of the clusters and their stability over time are different between populations of Brownian and non-Brownian walkers and are also dependent on the choice of parameters. Finally, we consider the effect of the probability of directed movement on the temporal stability of clusters and show that while clusters formed by Brownian and non-Brownian walkers may have similar properties with certain parameter sets, the spatio-temporal dynamics remain different.

Movement patterns of the grey field slug (Deroceras reticulatum) in an arable field

We report the results of an experiment on radio-tracking of individual grey field slugs in an arable field and associated data modelling designed to investigate the effect of slug population density in their movement. Slugs were collected in a commercial winter wheat field in which a 5x6 trapping grid had been established with 2m distance between traps. The slugs were taken to the laboratory, radio-tagged using a recently developed procedure, and following a recovery period released into the same field. Seventeen tagged slugs were released singly (sparse release) on the same grid node on which they had been caught. Eleven tagged slugs were released as a group (dense release). Each of the slugs was radio-tracked for approximately 10 h during which their position was recorded ten times. The tracking data were analysed using the Correlated Random Walk framework. The analysis revealed that all components of slug movement (mean speed, turning angles and movement/resting times) were significantly different between the two treatments. On average, the slugs released as a group disperse more slowly than slugs released individually and their turning angle has a clear anticlockwise bias. The results clearly suggest that population density is a factor regulating slug movement.

Dynamics of pattern formation and emergence of swarming in Caenorhabditis elegans

Many animals collectively form complex patterns to tackle environmental difficulties. Several biological and physical factors, such as animal motility, population densities, and chemical cues, play significant roles in this process. However, very little is known about how sensory information interplays with these factors and controls the dynamics of pattern formation. Here, we study the direct relation between oxygen sensing, pattern formation, and emergence of swarming in active Caenorhabditis elegans aggregates. We find that when thousands of animals gather on food, bacteria-mediated decrease in oxygen level slows down the animals and triggers motility-induced phase separation. Three coupled factors—bacterial accumulation, aerotaxis, and population density—act together and control the entire dynamics. Furthermore, we find that biofilm-forming bacterial lawns including Bacillus subtilis and Pseudomonas aeruginosa strongly alter the collective dynamics due to the limited diffusibility of bacteria. Additionally, our theoretical model captures behavioral differences resulting from genetic variations and oxygen sensitivity.

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.

Hypothesis: The highly folded brain surface might be structured and located so as to facilitate inter-brain synchronization

We integrate recent findings from neuro-anatomy, electroencephalography, quantum biology and social/neurodevelopment to propose that the brain surface might be specialised for communication with other brains.

Ground breaking, but still small-scale, research has demonstrated that human brains can act in synchrony and detect the brain activity of other human brains. Group aggregation, in all species, maximises community support and safety but does not depend on verbal or visual interaction. The morphology of the brain’s outermost layers, across a wide range of species, exhibits a highly folded fractal structure that is likely to maximise exchange at the surface: in humans, a reduced brain surface area is associated with disorders of social communication. The brain sits in a vulnerable exposed location where it is prone to damage, rather than being housed in a central location such as within the ribcage.

These observations have led us to the hypothesis that the brain surface might be specialised for interacting with other brains at its surface, allowing synchronous non-verbal interaction. To our knowledge, this has not previously been proposed or investigated.

Local adaptation, dispersal evolution, and the spatial eco-evolutionary dynamics of invasion

Local adaptation and dispersal evolution are key evolutionary processes shaping the invasion dynamics of populations colonizing new environments. Yet their interaction is largely unresolved. Using a single-species population model along a one-dimensional environmental gradient, we show how local competition and dispersal jointly shape the eco-evolutionary dynamics and speed of invasion. From a focal introduction site, the generic pattern predicted by our model features a temporal transition from wave-like to pulsed invasion. Each regime is driven primarily by local adaptation, while the transition is caused by eco-evolutionary feedbacks mediated by dispersal. The interaction range and cost of dispersal arise as key factors of the duration and speed of each phase. Our results demonstrate that spatial eco-evolutionary feedbacks along environmental gradients can drive strong temporal variation in the rate and structure of population spread, and must be considered to better understand and forecast invasion rates and range dynamics.

Influence of aggregation on benthic coral reef spatio-temporal dynamics

Spatial patterning of coral reef sessile benthic organisms can constrain competitive and demographic rates, with implications for dynamics over a range of time scales. However, techniques for quantifying and analysing reefscape behaviour, particularly at short to intermediate time scales (weeks to decades), are lacking. An analysis of the dynamics of coral reefscapes simulated with a lattice model shows consistent trends that can be categorized into four stages: a repelling stage that moves rapidly away from an unstable initial condition, a transient stage where spatial rearrangements bring key competitors into contact, an attracting stage where the reefscape decays to a steady-state attractor, and an attractor stage. The transient stage exhibits nonlinear dynamics, whereas the other stages are linear. The relative durations of the stages are affected by the initial spatial configuration as characterized by coral aggregation-a measure of spatial clumpiness, which together with coral and macroalgae fractional cover, more completely describe modelled reefscape dynamics. Incorporating diffusional processes results in aggregated patterns persisting in the attractor. Our quantitative characterization of reefscape dynamics has possible applications to other spatio-temporal systems and implications for reef restoration: high initial aggregation patterns slow losses in herbivore-limited systems and low initial aggregation configurations accelerate growth in herbivore-dominated systems.

Ecological feedback on diffusion dynamics

Spatial patterns are ubiquitous across different scales of organization in ecological systems. Animal coat pattern, spatial organization of insect colonies and vegetation in arid areas are prominent examples from such diverse ecologies. Typically, pattern formation has been described by reaction-diffusion equations, which consider individuals dispersing between subpopulations of a global pool. This framework applied to public goods game nicely showed the endurance of populations via diffusion and generation of spatial patterns. However, how the spatial characteristics, such as diffusion, are related to the eco-evolutionary process as well as the nature of the feedback from evolution to ecology and vice versa, has been so far neglected. We present a thorough analysis of the ecologically driven evolutionary dynamics in a spatially extended version of ecological public goods games. Furthermore, we show how these evolutionary dynamics feed back into shaping the ecology, thus together determining the fate of the system.

Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs Levy flights

Individual animal movement has been a focus of intense research and considerable controversy over the last two decades, however the understanding of wider ecological implications of various movement behaviours is lacking. In this paper, we consider this issue in the context of pattern formation. Using an individual-based modelling approach and computer simulations, we first show that density dependence ("auto-taxis") of the individual movement in a population of random walkers typically results in the formation of a strongly heterogeneous population distribution consisting of clearly defined animal clusters or patches. We then show that, when the movement takes place in a large spatial domain, the properties of the clusters are significantly different in the populations of Brownian and non-Brownian walkers. Whilst clusters tend to be stable in the case of Brownian motion, in the population of Levy walkers clusters are dynamical so that the number of clusters fluctuates in the course of time. We also show that the population dynamics of non-Brownian walkers exhibits two different time scales: a short time scale of the relaxation of the initial condition and a long time scale when one type of dynamics is replaced by another. Finally, we show that the distribution of sample values in the populations of Brownian and non-Brownian walkers is significantly different.

Active fluids at circular boundaries: swim pressure and anomalous droplet ripening

We investigate the swim pressure exerted by non-chiral and chiral active particles on convex or concave circular boundaries. Active particles are modeled as non-interacting and non-aligning self-propelled Brownian particles. The convex and concave circular boundaries are used to model a fixed inclusion immersed in an active bath and a cavity (or container) enclosing the active particles, respectively. We first present a detailed analysis of the role of convex versus concave boundary curvature and of the chirality of active particles in their spatial distribution, chirality-induced currents, and the swim pressure they exert on the bounding surfaces. The results will then be used to predict the mechanical equilibria of suspended fluid enclosures (generically referred to as 'droplets') in a bulk with active particles being present either inside the bulk fluid or within the suspended droplets. We show that, while droplets containing active particles behave in accordance with standard capillary paradigms when suspended in a normal bulk, those containing a normal fluid exhibit anomalous behaviors when suspended in an active bulk. In the latter case, the excess swim pressure results in non-monotonic dependence of the inside droplet pressure on the droplet radius; hence, revealing an anomalous regime of behavior beyond a threshold radius, in which the inside droplet pressure increases upon increasing the droplet size. Furthermore, for two interconnected droplets, mechanical equilibrium can occur also when the droplets have different sizes. We thus identify a regime of anomalous droplet ripening, where two unequal-sized droplets can reach a final state of equal size upon interconnection, in stark contrast with the standard Ostwald ripening phenomenon, implying shrinkage of the smaller droplet in favor of the larger one.

Waves of seed propagation induced by delayed animal dispersion

We study a model of seed dispersal that considers the inclusion of an animal disperser moving diffusively, feeding on fruits and transporting the seeds, which are later deposited and capable of germination. The dynamics depends on several population parameters of growth, decay, harvesting, transport, digestion and germination. In particular, the deposition of transported seeds at places away from their collection sites produces a delay in the dynamics, whose effects are the focus of this work. Analytical and numerical solutions of different simplified scenarios show the existence of travelling waves. The effect of zoochory is apparent in the increase of the velocity of these waves. The results support the hypothesis of the relevance of animal mediated seed dispersion when trying to understand the origin of the high rates of vegetable invasion observed in real systems.

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.