ArticleLiterature Review

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

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract and Figures

Many ecosystems develop strikingly regular spatial patterns because of small-scale interactions between organisms, a process generally referred to as spatial self-organization. Self-organized spatial patterns are important determinants of the functioning of ecosystems, promoting the growth and survival of the involved organisms, and affecting the capacity of the organisms to cope with changing environmental conditions. The predominant explanation for self-organized pattern formation is spatial heterogeneity in establishment, growth and mortality, resulting from the self-organization processes. A number of recent studies, however, have revealed that movement of organisms can be an important driving process creating extensive spatial patterning in many ecosystems. Here, we review studies that detail movement-based pattern formation in contrasting ecological settings. Our review highlights that a common principle, where movement of organisms is density-dependent, explains observed spatial regular patterns in all of these studies. This principle, well known to physics as the Cahn-Hilliard principle of phase separation, has so-far remained unrecognized as a general mechanism for self-organized complexity in ecology. Using the examples presented in this paper, we explain how this movement principle can be discerned in ecological settings, and clarify how to test this mechanism experimentally. Our study highlights that animal movement, both in isolation and in unison with other processes, is an important mechanism for regular pattern formation in ecosystems.
Content may be subject to copyright.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... Hence, density-dependent aggregation is characterized by spatial patterns consisting of coexisting phases (resembling alternative stable states), as shown in the model simulation of Fig. 3 C and D. In physics, the emergence of such alternating phases out of uniformity is referred to as phase separation (75,76). Phase separation finds its origin in the unmoving of two immiscible fluids like oil and water as put forward by Cahn and Hilliard (75,76), but has also been identified as a mechanism behind cell polarization (51,(77)(78)(79) and has been suggested to be the principle mechanism behind the aggregation of animals (24,42,(80)(81)(82) in recent years. Non-regular and scale-free patterns have been put forward by a wide range of modeling approaches (see refs. 6, 7, 27, 83, and 84 for examples). ...
... This results in a patch coarsening process, where larger patches absorb more and more resources, while smaller patches wither away (Figs. 3 and 4). Such coarsening dynamics are a defining characteristic for phase-separating dynamics in ecosystems (24,42,80,90). ...
... In this paper, we have extended the use of phase separation as a pattern-generating process from single-species examples [e.g., aggregation of mussels or ungulate herbivores (42,80)] to universal two-species ecosystems, potentially expanding the applicability of phase separation within ecology. Our study demonstrates that within-patch bistability (e.g., patch collapse following a disturbance), between-patch competition, and hysteresis cause intrinsic vulnerability of patchy ecosystems governed by density-dependent aggregation. ...
Article
Full-text available
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.
... The present prevailing ecological self-organization theoretical framework is based on Turing principle, which well describes the formation of persistent patterns. Turingtype models couple the processes of dispersal, growth and mortality, so that they describe non-equilibrium patterns based on a demographic process (Liu et al., 2016;Meinhardt et al., 2009). Vegetation patterns of arid ecosystems tardily arise from the scale-dependent feedbacks between plant and soil water within decades or even centuries (Rietkerk et al., 2002). ...
... Vegetation patterns of arid ecosystems tardily arise from the scale-dependent feedbacks between plant and soil water within decades or even centuries (Rietkerk et al., 2002). Different from scale-dependent feedbacks, motility-induced phase separation (MIPS) that is driven by density-dependent movement behaviours can rapidly produce arrested or transient spatial patterns in the timescale of days or years (Table 1, Liu et al., 2016). ...
... In ecosystems, MIPS has been theoretically and experimentally confirmed to perform in a single trophic level (Liu et al., 2013;de Paoli et al., 2017). The densitydependent movement broadly appears in many ecosystems like bacteria, elegans, ants, birds and elk (Demir et al., 2020;Liu et al., 2016). Thus, phase separation has the potential to become a novel theoretical framework of behaviour-driven ecological self-organization. ...
Article
Full-text available
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. Biological behavior‐driven self‐organized patterns have recently been confirmed to play a key role in ecosystem functioning. Here we develop a theoretical phase‐separation model describing spatiotemporal self‐similar dynamics, which is a consequence of behavior‐driven trophic interactions in short‐time scales. Our findings potentially provide new insights into self‐organized patterns as an indicator of ecosystem transitions under a stressful environment.
... Movement in herd animals -including in human crowds -appears to be initiated by only a few individuals, and whether or not movement can be initiated by a particular individual is correlated with the degree of social interaction that individual has had with others in the group ( Krueger et al. 2014). Yet group aggregation led by individuals does not explain everything: all species clump together in the face of danger and sometimes do so in a highly complex and unexplained manner ( Liu et al. 2016). Across species, attachment behaviours are elicited at times of stress and result in members of social groups seeking close physical proximity with one another in order to achieve strength and protection in numbers (Bowlby 1982). ...
... Across species, attachment behaviours are elicited at times of stress and result in members of social groups seeking close physical proximity with one another in order to achieve strength and protection in numbers (Bowlby 1982). The formation of herds, flocks or shoals, through aggregation of individuals of the same species, is a universal anti-predator mechanism that occurs across all species and taxa -from mammals to microbes ( Liu et al. 2016). The process of aggregation therefore does not depend on language or sight, nor on the availability of a leader -although sensory information (Kimbell and Morrell 2015) and leadership ( Andrieu et al. 2016) can enhance these processes. ...
... The emerging field of "social signal processing" (a new cross-disciplinary research domain that aims at understanding and modelling social interactions) has begun to identify the many subtle ways in which human beings communicate non-verbally such as with gesture, interpersonal distance, posture and mutual gaze ( Vinciarelli et al. 2012), but there are still many unanswered questions about how group propagative phenomena operate. A recent review suggests that a common, but poorly understood, principle of "density-dependent movement" (where clumping of individuals depends on the density of the group) might be involved in group aggregation across species ( Liu et al. 2016). We suggest that, in higher animals, the role of the highly folded brain surface in facilitating such processes is worthy of further investigation. ...
Article
Full-text available
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.
... Formation of these biological patterns is particularly initiated by aggregation. At high population density, animals tend to aggregate as animal to animal interactions cause instability in uniform distribution [9][10][11][12] . Different factors such as motility of animals and chemical cues can trigger these instabilities by altering the behavior of animals. ...
... When thousands of worms are forced to feed together, aggregation-induced bacterial accumulation and oxygen depletion create unstable conditions and further trigger phase separations. The principle of phase separation is mainly based on sudden change in the animal's motility 11,34 . We also found that the dynamics of the entire process is controlled by the sensitivity of the oxygen-sensing neurons, which gives rise to strong variations in animals' collective response. ...
... These are the Keller-Segel like equations 45 aggregates, stripes, and holes (Fig. 2d). These are the basic patterns that are frequently seen in many biological systems 11,47 . ...
Preprint
Full-text available
Many animals in their natural habitat exhibit collective motion and form complex patterns to tackle environmental difficulties. Several physical and biological 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 all these factors and controls the dynamics of collective response and pattern formation. Here, we use a model organism, Caenorhabditis elegans, to study the direct relation between oxygen sensing, pattern formation, and the emergence of swarming in active worm aggregates. We find that when thousands of animals gather on food, bacteria-mediated decrease in oxygen levels slowed down the animals and triggers motility-induced phase separation. Three coupled factors bacterial accumulation, aerotaxis, and population density act together and control the dynamics of pattern formation. Through several intermediate stages, aggregates converge to a large scale swarming phase and collectively move across the bacterial lawn. Additionally, our theoretical model captures behavioral differences resulting from the genetic variations and oxygen sensitivity. Altogether, our study provides many physical insights and a new platform for investigating the complex relationship between neural sensitivity, collective dynamics, and pattern formation.
... Van de Koppel's [50] model includes a positive feedback to describe these facilitated effects at higher mussel densities. Hitherto, many ecologists and mathematicians have focused on mussel bed development with mussel-algae models [3,12,[17][18][19][20][21]52], in which the theoretical studies suggest that self-organized patterns would affect the emergent properties of ecosystems in large-scale space [50]. Based on nonlinear numerical continue approaches, Wang et al. [52] have found that spatial patterns would exist at a remarkably lower food concentration compared with the classic linear stability, which explores the validity of predicting the pattern existence near the tipping point. ...
... Although advection and diffusion are two different ecological processes, in real mussel bed ecosystems, these processes normally coexist and share the same activator-inhibitor mechanism [21]. For the emergent properties of spatial self-organization patterns, the advection and diffusion are equivalent. ...
... In modeling the mussel bed ecosystems, the advection term is usually used to depict the tidal fluid direction where the water come from the sea to coast, and the diffusion term depicts the isotropic dispersion on horizontal planes. Although they are involved in two different ecological processes, advection and diffusion terms have the same mechanism-an activeinhibitor principle [21]. For the emergent properties of spatial self-organization patterns, the advection and diffusion are equivalent. ...
Article
Intertidal mussels can self-organize into periodic spot, stripe, labyrinth, and gap patterns ranging from centimeter to meter scales. The leading mathematical explanations for these phenomena are the reaction-diffusion-advection model and the phase separation model. This paper continues the series studies on analytically understanding the existence of pattern solutions in the reaction-diffusion mussel-algae model. The stability of the positive constant steady state and the existence of Hopf and steady-state bifurcations are studied by analyzing the corresponding characteristic equation. Furthermore, we focus on the Turing-Hopf (TH) bifurcation and obtain the explicit dynamical classification in its neighborhood by calculating and investigating the normal form on the center manifold. Using theoretical and numerical simulations, we demonstrates that this TH interaction would significantly enhance the diversity of spatial patterns and trigger the alternative paths for the pattern development.
... Inter-individual interactions may establish various positive and negative feedbacks on the organisms and the environment, which might locally enhance or diminish demographic rates and movement. If positive and negative feedbacks act at different spatial scales (scaledependent feedbacks, SDF), they can drive the emergence of a regular self-organized spatial distribution in population density [6,8,9]. Although density-dependent feedbacks often act simultaneously on demographic rates and movement [10], depending on which of these two processes is more strongly shaped by nonlinear feedbacks, it is possible to classify emergent patterns in two broad categories: demography-driven patterns in which patterns form due to SDFs modulating birth/death dynamics, and movement-driven patterns, in which the total population size is mostly conserved and individuals rearrange in space and time due to active movement with density-dependent velocity. ...
... Colonies of motile microbes (e.g. low and intermediate agar concentrations for bacterial inoculations), show a diversity of movement behaviors, including gradient-sensing navigation (such as chemotaxis) [61] or densitydependent speed [9,62]. These behaviors can lead to different colony-level spatial patterns, such as regular or elongated spots, stripes, concentric rings, branching patterns, or vortices [21]. ...
Article
Self-organized spatial patterns are ubiquitous in ecological systems and allow popula- tions 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, hyster- esis 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 eco- systems, and discuss the mechanisms hypothesized to underlie observed patterns and their system-level consequences. For each example, we review both the empirical evi- dence and the existing theoretical frameworks developed to identify the causes and con- sequences 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.
... Phase separation has in the past decades become a central physical principle for self-organized patterning in cell structure (21,22), gravitational fluid (23,24), active matter (18,(25)(26)(27)(28), and ecological systems (29)(30)(31). Herein, we suggest that the principle leads to insights into pattern formation in geomorphic systems. It is important to note that it is distinct from the principle at the core of sophisticated phase-field models, which arise from regularized partial differential equations designed to solve moving boundary problems (32,33), such as solidification fronts (34,35), fracture (36,37), and so on. ...
... The concentration of stones in the neighborhood of a tracked stone was estimated by measuring the fractional cover of the stones within the distances of 1.2-, 2.0-, 3.0-, 4.0-, 5.0-, 6.0-, 7.0-, 8.0-, and 9.0-fold of the diameter (see Fig. 2F for an example). These specific setups correspond to the spatial scales of about 12,18,24,30,42,48, and 54 mm for stones with an ∼6-mm stone diameter. Following the methods proposed by van de Koppel and coauthors (63), images were converted to binary bitmaps indicating the presence or absence of stones using a custom-made MATLAB program. ...
Article
Full-text available
Significance Self-organization is increasingly recognized as fundamental to pattern formation in geomorphology. Relative to other fields, however, underlying mechanisms have received little attention from theoreticians. Here, we introduce phase separation theory to study the formation of sorted patterned ground in cold regions; “sorted” refers to the segregation of soil and stones due to feedbacks between stone concentration and recurring ice growth. Using detailed measurements of the concentration of stones in soil and their displacements, we demonstrate that phase separation accounts for the observed sorting and patterns. Our study highlights phase separation theory as a source of important insight into studying ground patterns in cold regions and their potential value in signaling important changes in ground conditions with the warming climate.
... Numerous laboratory results show that mussels can actively move both within and between clusters, the influence of advection with tidal flow at a small-scale space on the mussel bed is very small. These imply that advection and diffusion are two different ecological processes; in real mussel bed ecosystems, these processes normally coexist and share the same activator-inhibitor mechanism [13,16]. Advection and diffusion are equivalent due to the emergence of spatial self-organizing patterns. ...
... Wang et al. [17] derived the conditions for differential-flow instability that causes the formation of spatial patterns, and then systematically investigated the influence of parameters on pattern formation. Liu et al. [16] verified that the dimensionless model is an extension of the original model and in exact accordance with their laboratory experiment. Ghazaryan and Manukian [15] used the geometric singular perturbation theory to study the nonlinear mechanisms of pattern and wave formation. ...
Article
Full-text available
In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of the corresponding kinetic system, the existence and stability conditions of the equilibrium and the type of Hopf bifurcation are studied. Via the center manifold and Hopf bifurcation theorem, sufficient conditions for Turing instability in equilibrium and limit cycles are obtained, respectively. In addition, we find that the strip patterns are mainly induced by Turing instability in equilibrium and spot patterns are mainly induced by Turing instability in limit cycles by numerical simulations. These provide a comprehension on the complex pattern formation of a mussel–algae system.
... Some populations aggregate as a consequence of animal 'sociality' (Grünbaum and Okubo, 1994;Gueron et al., 1996) and modify their movement based on movement characteristics of their close neighbours as seen in flocks, herds or swarms (Gueron et al., 1996;Kolokolnikov et al., 2013;Reynolds et al., 2017). Patterns of high population density have also been shown to be formed by a change in the speed of movement with density whereby an animal moving into a higher density area will slow down causing clusters (Liu et al., 2013;Liu et al., 2016). Other animals will simply be more inclined to move towards their conspecifics and aggregate as a consequence of this densitydependent movement (Keller and Segel, 1970;Liu et al., 2016;Mogilner et al., 2003;Tyutyunov et al., 2004;Tyutyunov et al., 2009). ...
... Patterns of high population density have also been shown to be formed by a change in the speed of movement with density whereby an animal moving into a higher density area will slow down causing clusters (Liu et al., 2013;Liu et al., 2016). Other animals will simply be more inclined to move towards their conspecifics and aggregate as a consequence of this densitydependent movement (Keller and Segel, 1970;Liu et al., 2016;Mogilner et al., 2003;Tyutyunov et al., 2004;Tyutyunov et al., 2009). ...
Article
The patterns of collective behaviour in a population emerging from individual animal movement has 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.
... On the other hand, low tolerance to group members means individuals are sensitive and more likely to move away from non-beneficial interactions. Markov movement is a way of capturing density-dependent movement that explains a wide variety of ecological aggregations Liu et al. (2016). In this paper we implement a version of this Markov movement where tolerance to group members plays a key role in determining whether individuals move or stay. ...
Preprint
Full-text available
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 looked 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.
... Generally speaking, the moving speed of a herbivore (V ) depends on the local resource concentration P (X, T ), where X is the spatial position and T is the time [79][80][81]. And the moving speed is in a parabolic relationship with the local density ( Fig. 5(b)) [82], which can be mathematically expressed as: ...
Article
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 modelling 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.
... The experiments had shown that mussels could move actively within and between clusters, meaning that advection and tidal currents in small scales had little effect on mussel beds. Although advection and diffusion were two different ecological processes, in real mussel bed ecosystems, the two processes were usually coexisting and had the same activation-inhibition mechanism [8]. ...
Article
Full-text available
In this paper, the kinetics of a class of delayed reaction-diffusion mussel-algae system under Neumann boundary conditions with the half-saturation constant is studied. The global existence and priori bounds as well as the existence conditions of positive equilibrium are obtained. The half-saturation constant affects the stability of the system and may result in Turing instability. When the half-saturation constant exceeds a certain critical value, the boundary equilibrium is globally asymptotically stable which means that the larger half-saturation constant forces the mussel population toward extinction. By analyzing the distribution of the roots of the characteristic equation with two delays, the stability conditions of the positive equilibrium in the parameter space are obtained. The stability of the positive equilibrium can be changed by steady-state bifurcation, Hopf bifurcation, Hopf-Hopf bifurcation or Hopf-steady state bifurcation, which can be verified by some numerical simulations. Among the parameters, the half-saturation constant and two delays drive the complexity of system dynamics.
... in ecosystems and Earth system components as well (58). For instance, some of the spatial aggregation of organisms and resources can be interpreted as such. ...
Article
Resilience to tipping points in ecosystems Spatial pattern formation has been proposed as an early warning signal for dangerous tipping points and imminent critical transitions in complex systems, including ecosystems. Rietkerk et al . review how ecosystems and Earth system components can actually evade catastrophic tipping through various pathways of spatial pattern formation. With mathematical and real-world examples, they argue that evading tipping and enhancing resilience could be relevant for many ecosystems and Earth system components that until now were known as tipping prone. Many of these complex systems may be more resilient than currently thought because of overlooked spatial dynamics and multiple stable states, and may thus not undergo critical or catastrophic transitions with global change. —AMS
... clusters, meaning that advection and tidal currents in small scales had little effect on mussel beds. Although advection and diffusion were two different ecological processes, in real mussel bed ecosystems, the two processes were usually coexisting and had the same activation-inhibition mechanism [8]. ...
Preprint
Full-text available
In this paper, the kinetics of a class of delayed reaction-diffusion musselalgae system under Neumann boundary conditions with the half- saturation constant is studied. The global existence and priori bounds as well as the existence conditions of positive equilibrium are obtained. The half-saturation constant affect the stability of the system and may result in Turing instability. When the half-saturation constant exceeds a certain critical value, the boundary equilibrium is globally asymptotically stable which means that the larger half-saturation constant forces the mussel population toward extinction. By analyzing the distribution of the roots of the characteristic equation with two delays, the stability conditions of the positive equilibrium in the parameter space are obtained. The stability of the positive equilibrium can be changed by steady-state bifurcation, Hopf bifurcation, Hopf-Hopf bifurcation or Hopf-steady state bifurcation, which can be verified by some numerical simulations. Among parameters, the half-saturation constant and two delays drive the complexity of the system dynamics.
... In the mussel bed ecosystems, although advection term and diffusion term involved in two different ecological processes, they play an important role in detecting the mechanism of pattern formation [47] . For the emergent properties of spatial selforganization patterns, the advection and diffusion are equivalent. ...
Article
Full-text available
The spatiotemporal dynamics of a semi-discrete Mussel-Algae system with advection are investigated in this paper. The stability of equilibrium, the direction of Andronov-Hopf bifurcation and Bautin bifurcation to kinetic system are obtained via linear stability analysis, the Lyapunov coefficients and regularity. In order to analyze the Turing instability, the characteristic equations of the advection operator ∇ is considered. Combined with linear analysis, the critical condition of Turing instability conditions for advection term is obtained. Simulations are performed to illustrate the above theoretical results, such as bifurcation diagram, phase orbits and pattern formations. In addition, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. As a result, the theoretical results for Turing instability of some model with advection term may trigger some significance results for further research.
... The mechanisms underlying such pattern formation have been a source of robust debate, especially in the context of vegetation (8,9). Following Turing's seminal work on scale-dependent feedback, namely local activation and long-range inhibition, similar principles of pattern formation with local density dependence have been considered (10)(11)(12)(13), touching on the more general question of how multiple scales of time and space emerge (14)(15)(16). More recent work has connected these principles with mechanisms of biological interaction and environmental feedback (17)(18)(19). ...
Article
Full-text available
Significance Although termite mounds stand out as an example of remarkably regular patterns emerging over long times from local interactions, ecological spatial patterns range from regular to random, and temporal patterns range from transient to stable. We propose a minimal quantitative framework to unify this variety by accounting for how quickly sessile organisms grow and die mediated by competition for fluctuating resources. Building on metabolic scaling theory for forests, we reproduce a wide range of spatial patterns and predict transient features such as population shock waves that align with observations. By connecting diverse ecological dynamics, our work will help apply lessons from model systems more broadly (e.g., by leveraging remote mapping to infer local ecological conditions).
... As such, an extension of this equation describes the movement-driven, spatial self-organization of animals, where species tend to disperse at low and very high densities but aggregate at intermediate density. Such patterned clumping has been observed in a number of animals, including mussels and ants [8]. ...
... Hence, pattern formation in mussels follows a process similar to phase separation in physics. Liu et al. mentioned the findings in a review [55]: "Phase separation driven by density-dependent movement: A novel mechanism for ecological patterns". In biological systems, ant corpses within ant cemeteries show regular patterns [52]; this is because the rate at which ants pick up and discard corpses varies with the local density of corpses. ...
Article
Full-text available
How populations distribute in both space and time is one of the key issues in ecological systems, which can characterize the relationship between populations, space–time structure and evolution law. Consequently, pattern dynamics in ecosystems has been widely investigated including their causes and ecological functions. In order to systematically understand the interactions in ecosystems, we summarize the related results in pattern formation of ecological systems. Based on mathematical modeling and analysis, we show the mechanisms of different patterns including feedback, scale-dependent, phase separation, nonlocal effects, time delay and spatial heterogeneity. This work offers assistance for better understanding the complexity of ecosystems and provides new insights for self-organizations evolution and ecosystem protection. We hope that our results may be applied in other related fields such as epidemiology, medical science, atmospheric science and so on.
... A combination of attractive and repulsive forces acting on different scales is, for instance, believed to be responsible for the formation of regular stripes in mussel beds [11]. Other models that investigate the formation of different structures in animal groupings also rely on similar attraction-repulsion or activation-inhibition principles [104][105][106][107][108][109]. On the other hand, several biological systems also self-organize only as a consequence of repulsive or growth-inhibitory interactions alone. ...
Preprint
Full-text available
Self-organized spatial patterns of vegetation are frequent in water-limited regions and have been suggested as important indicators of ecosystem health. However, the mechanisms underlying their 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 net plant-to-plant interactions may shift from positive to negative as a function of inter-individual distance, and in the absence of strong empirical support, the relevance of this SDF for vegetation pattern formation remains disputed. These theories predict a sequential change in pattern shape from gapped to labyrinthine to spotted spatial patterns as precipitation declines. Nonetheless, alternative theories show that the same sequence of patterns could emerge even if net interactions between plants were always inhibitory (purely competitive feedbacks, PCF). Importantly, although these alternative hypotheses lead to visually indistinguishable patterns they predict very different desertification dynamics following the spotted pattern, limiting their potential use as an ecosystem-state indicator. Moreover, vegetation interaction with other ecosystem components can introduce additional spatio-temporal scales that reshape both the patterns and the desertification dynamics. Therefore, to make reliable ecological predictions for a focal ecosystem, it is crucial that models accurately capture the mechanisms at play in the system of interest. Here, we review existing theories for vegetation self-organization and their conflicting predictions about desertification dynamics. We further discuss possible ways for reconciling these predictions and potential empirical tests via manipulative experiments to improve our understanding of how vegetation self-organizes and better predict the fate of the ecosystems where they form.
... We aim to mention an alternative procedure due to P. Podio-Guidugli [38] that leads to another viscous Cahn-Hilliard system of nonstandard type [11,10]. In addition, we report that in recent years Cahn-Hilliard and viscous Cahn-Hilliard systems have been employed successfully in many other branches of Science and Engineering, fields in which the segregation of a diffusant leads to pattern formation, such as population dynamics [26], image processing [4], dynamics for mixtures of fluids [18] and tumor modelling [9]. In the case of the variable ϕ understood as concentration, the recent paper [5] faces with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of ϕ and the potential f for ϕ are introduced, thus leading to an equation more general than (1.3). ...
Article
In the present contribution we study a viscous Cahn–Hilliard system where a further leading term in the expression for the chemical potential \begin{document}$ \mu $\end{document} is present. This term consists of a subdifferential operator \begin{document}$ S $\end{document} in \begin{document}$ L^2(\Omega) $\end{document} (where \begin{document}$ \Omega $\end{document} is the domain where the evolution takes place) acting on the difference of the phase variable \begin{document}$ \varphi $\end{document} and a given state \begin{document}$ {\varphi^*} $\end{document}, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential \begin{document}$ \mu $\end{document}. Next, by assuming that \begin{document}$ S = \rho\;{\rm{sign}} $\end{document}, a multiple of the \begin{document}$ \;{\rm{sign}} $\end{document} operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for \begin{document}$ \mu $\end{document} and under suitable conditions on \begin{document}$ \rho $\end{document} and \begin{document}$ \Omega $\end{document}, we also prove the sliding mode property, that is, that \begin{document}$ \varphi $\end{document} is forced to join the evolution of \begin{document}$ {\varphi^*} $\end{document} in some time \begin{document}$ T^* $\end{document} lower than the given final time \begin{document}$ T $\end{document}. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on \begin{document}$ \varphi $\end{document}.
... There is considerable field evidence that patches of high slug density are stable in time, at least within a given season 28,48 . In addition, there are many theoretical results showing that density dependent individual movement is a factor that can increase patch stability and even lead to patch (cluster) formation 19,21,49,50 . Correspondingly, in this section we analyse the field data on slug movement in the context of patch dynamics. ...
Article
Full-text available
Abstract 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.
... Phase separation in these systems typically occurs due to differences in mobility as a function of density; constituents move slowly through crowded regions, and quickly through low density regions. Mobility-induced phase separation has been observed (or predicted) in systems as varied as swimming bacteria 60 , self-propelled colloids 61,62 , mussels 51 , granular rods 63 , active filaments 64,65 , rotating particles 66 , among other systems 52 . In the current system, activity is derived from reproduction and killing events at high density rather than constituent mobility 67 , leading to a 'Model A' transition. ...
Preprint
By nature of their small size, dense growth and frequent need for extracellular metabolism, microbes face persistent public goods dilemmas 1–5 . Spatial assortment can act as a general solution to social conflict by allowing extracellular goods to be utilized preferentially by productive genotypes 1,6,7 . Established mechanisms that generate microbial assortment depend on the availability of free space 8–14 ; however, microbes often live in densely-packed environments, wherein these mechanisms are ineffective. Here, we describe a novel class of self-organized pattern formation that facilitates the development of spatial structure within densely-packed bacterial colonies. Contact-mediated killing through the Type VI secretion system (T6SS) drives high levels of assortment by precipitating phase separation, even in initially well-mixed populations that do not necessarily exhibit net growth. We examine these dynamics using three different classes of mathematical models and experiments with mutually antagonistic strains of Vibrio cholerae growing on solid media, and find that all appear to de-mix via the same ‘Model A’ universality class of order-disorder transition. We mathematically demonstrate that contact killing should favour the evolution of public goods cooperation, and empirically examine the relationship between T6SSs and potential cooperation through phylogenetic analysis. Across 26 genera of Proteobacteria and Bacteroidetes, the proportion of a strain’s genome that codes for potentially-exploitable secreted proteins increases significantly with boththe number of Type 6 secretion systems and the number of T6SS effectors that it possesses. This work demonstrates how antagonistic traits—likely evolved for the purpose of killing competitors—can indirectlylead to the evolution of cooperation by driving genetic phase separation.
... The mechanisms underlying such pattern formation have been a source of robust debate especially in the context of vegetation (8,9). Following Turing's seminal work on scaledependent feedback, namely local activation and long-range inhibition, similar principles of pattern formation with local density dependence have been considered (10)(11)(12)(13), touching on the more general question of how multiple scales of time and space emerge (14)(15)(16). More recent work has connected these principles with mechanisms of biological interaction and environmental feedback (17)(18)(19). ...
Preprint
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 include resource competition based on shared capture area separately. By varying relative 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 that primarily weakens the smaller of two competitors such as canopy shading or large colony dominance. When such competition couples slow timescales of growth with fast competitive death, it generates population shock waves similar to those observed in forest demographic data. Our minimal quantitative theory unifies spatiotemporal patterns across sessile organisms through local competition mediated by the laws of metabolic growth which in turn result from long-term evolutionary dynamics.
... As mussels beds in nature are a self-organised system, density at the larger scale can drive self-organisation processes (e.g. Liu et al. 2016) and modify small scale density effects via the resulting patterning. However, also J o u r n a l P r e -p r o o f 'within-pattern' effects are observed in mussel patches, more commonly thought of as 'edgeeffects', resulting, for example, in larger mussels at the edges compared to middle of patches (Okamura, 1986). ...
Article
Shellfish aquaculture is considered a sustainable way to help meet rising protein demands worldwide. In shallow coastal dynamic ecosystems mussels can be cultivated directly on the seabed, however this method has a low return as mussels exposed to natural environments risk dislodgment, high predation rates, sedimentation and competition. The formation of spatial patterns in natural mussel beds, that result in ‘organized patchiness’, is thought to be an adaptive mechanism to reduce population losses. The driver and effects of this patterning need to be disentangled at multiple spatial scales in which patterns are observed. With a field experiment we aimed to understand how small-scale density (actual cover) and patterning (perimeter: area ratio of clumps and number of mussel layers) can be altered by manipulating large scale density (re-laying biomass), that farmers could control during seeding activity. Within this study we considered the interplay between environmental conditions (manipulating flow rate with the use of large mesh cages) and density for pattern development and persistence, and the repercussions of this on mussel productivity (growth and condition). We further investigated local scale processes, such as the role of within-clump biological activity (biodeposition), that may be a predictor for the larger scale observations of losses and persistence relative to density. We found that manipulating density by controlling seeding biomass from boats is not an accurate predictor of actual seabed density and resulting patterning. The growth and condition of the mussels was only influenced by local scale effects, resulting in high ‘within clumps’ variation. Aiming for an intermediate density to avoid both excessive fragmentation and excessive layering may be viewed as an optimal strategy to maximise returns, but we encourage the incorporation of the hierarchy of multiple scales of density in future studies of patterning that will allow the inclusion of these effects in a model of growth and productivity.
... Formation of these biological patterns is particularly initiated by aggregation. At high population density, animals tend to aggregate as animal-to-animal interactions cause instability in uniform distribution (Gregor et al., 2010;Buhl et al., 2006;Liu et al., 2016;Vicsek et al., 1995). Different factors such as motility of animals and chemical cues can trigger these instabilities by altering the behavior of animals. ...
Article
Full-text available
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.
... Formation of these biological patterns is particularly initiated by aggregation. At high population density, animals tend to aggregate as animal-to-animal interactions cause instability in uniform distribution (Gregor et al., 2010;Buhl et al., 2006;Liu et al., 2016;Vicsek et al., 1995). Different factors such as motility of animals and chemical cues can trigger these instabilities by altering the behavior of animals. ...
Article
Full-text available
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.
... Besides being a fundamental contribution to Materials Science, the C-H system has had considerable success in many other branches of Science and Engineering where segregation of a diffusant leads to pattern formation, such as population dynamics [20], image processing [6], dynamics for mixtures of fluids [16], tumor modelling [1,12,13], to name a few. ...
Article
Full-text available
In this paper we deal with a doubly nonlinear Cahn–Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential includes two regularizations: a viscous and a diffusive term. First of all, we prove existence and uniqueness of a bounded solution to the system using a nonstandard maximum-principle argument for time-discretizations of doubly nonlinear equations. Possibly including singular potentials, this novel result brings improvements over previous approaches to this problem. Secondly, under suitable assumptions on the data, we show the convergence of solutions to the respective limit problems once either of the two regularization parameters vanishes.
... We aim to mention an alternative procedure due to P. Podio-Guidugli [38] that leads to another viscous Cahn-Hilliard system of nonstandard type [10,11]. In addition, we report that in recent years Cahn-Hilliard and viscous Cahn-Hilliard systems have been employed successfully in many other branches of Science and Engineering, fields in which the segregation of a diffusant leads to pattern formation, such as population dynamics [26], image processing [4], dynamics for mixtures of fluids [18] and tumor modelling [9]. In the case of the variable ϕ understood as concentration, the recent paper [5] faces with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of ϕ and the potential f for ϕ are introduced, thus leading to an equation more general than (1.3). ...
Preprint
In the present contribution we study a viscous Cahn-Hilliard system where a further leading term in the expression for the chemical potential $ \mu$ is present. This term consists of a subdifferential operator $S$ in $L^2(\Omega)$ (where $\Omega$ is the domain where the evolution takes place) acting on the difference of the phase variable $\varphi$ and a given state $\varphi^* $, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential $\mu$. Next, by assuming that $S=\rho\,$sign, a multiple of the sign operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for $\mu$ and under suitable conditions on $\rho$ and $\Omega$, we also prove the sliding mode property, that is, that $\varphi$ is forced to join the evolution of $\varphi^* $ in some time $T^*$ lower than the given final time $T$. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on $\varphi$.
... It describes the change and the stability of spatial structure with time. There are still many other potential mechanisms to generate spatially self-organized patterns in the ecosystems [47,48] such as animal aggregation due to taxis and density-dependence [49], but they are beyond the scope of the discussion here. It is interesting for further comparison in the future research. ...
Article
Full-text available
Although the diversity of spatial patterns has gained extensive attention on ecosystems, it is still a challenge to discern the underlying ecological processes and mechanisms. Dynamical system models, such partial differential equations (PDEs), are some of the most widely used frameworks to unravel the spatial pattern formation, and to explore the potential ecological processes and mechanisms. Here, comparing the similarity of patterned dynamics among Allen–Cahn (AC) model, Cahn–Hilliard (CH) model, and Cahn–Hilliard with population demographics (CHPD) model, we show that integrated spatiotemporal behaviors of the structure factors, the density-fluctuation scaling, the Lifshitz–Slyozov (LS) scaling, and the saturation status are useful indicators to infer the underlying ecological processes, even though they display the indistinguishable spatial patterns. First, there is a remarkable peak of structure factors of the CH model and CHPD model, but absent in AC model. Second, both CH and CHPD models reveal a hyperuniform behavior with scaling of −2.90 and −2.60, respectively, but AC model displays a random distribution with scaling of −1.91. Third, both AC and CH display uniform LS behaviors with slightly different scaling of 0.37 and 0.32, respectively, but CHPD model has scaling of 0.19 at short-time scales and saturation at long-time scales. In sum, we provide insights into the dynamical indicators/behaviors of spatial patterns, obtained from pure spatial data and spatiotemporal related data, and a potential application to infer ecological processes.
... Density is a key driver of spatial patterning , Capelle et al. 2014, Liu et al. 2016. Density-dependent mechanisms, such as growth and mortality, are common features in resource-limited (by food, space) environments (Brook and Bradshaw 2006). ...
Article
Spatial patterns formed through the process of self‐organization are found in nature across a variety of ecosystems. Pattern formation may reduce the costs of competition while maximizing the benefits of group living, and thus promote ecosystem persistence. This leads to the prediction that self‐organizing to obtain locally intermediate densities will be the optimal solution to balance costs and benefits. However, despite much evidence documenting pattern formation in natural ecosystems, there is limited empirical evidence of how these patterns both influence and are influenced by tradeoffs between costs and benefits. Using mussels as a model system, we coupled field observations in mussel‐culture plots with manipulative laboratory experiments to address the following hypotheses: 1) labyrinthine spatial patterns, characteristically found at intermediate to high patch densities, are the most persistent over time; this is because labyrinthine patterns 2) result in adequately heavy patches that can maximize resistance to dislodgement while 3) increasing water turbulence with spacing, which will maximize food delivery processes. In the field, we observed that labyrinthine ‘stripes’ patterns are indeed the most persistent over time, confirming our first hypothesis. Furthermore, with laboratory experiments, we found the ‘stripes’ pattern to be highly resistant to dislodgement, confirming the second hypothesis. Finally, with regards to the third hypothesis, we found positive effects of this pattern on local turbulence. These results suggest that the mechanisms of intraspecific facilitation not only depend on initial organism densities, but may also be influenced by spatial patterning. We hence recommend taking into account spatial patterns to maximize productivity and persistence in shellfish‐cultivation practices and to increase the restoration success of ecosystems with self‐organizing properties. This article is protected by copyright. All rights reserved.
... Besides being a fundamental contribution to Materials Science, the C-H system has had considerable success in many other branches of Science and Engineering where segregation of a diffusant leads to pattern formation, such as population dynamics [20], image processing [6], dynamics for mixtures of fluids [16], tumor modelling [1,12,13], to name a few. ...
Preprint
Full-text available
In this paper we deal with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential includes two regularizations: a viscosity and a diffusive term. First of all, we prove existence and uniqueness of a bounded solution to the system using a nonstandard maximum-principle argument for time-discretizations of doubly nonlinear equations. Possibly including singular potentials, this novel result brings improvements over previous approaches to this problem. Secondly, under suitable assumptions on the data, we show the convergence of solutions to the respective limit problems once either of the two regularization parameters vanishes.
... While these ecosystems are distinct in many respects, the formation of their distinctive Turing-like patterns could be driven by universal mechanisms, including scale-dependent feedback (SDF, referred to as coupled & 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution short-range positive feedbacks and long-range negative feedbacks; see [14,15]) and behavioural-driven phase separation [16,17]. These mechanisms are essentially at the core of selforganization processes [14,15,18 -20], the important role of which has been increasingly demonstrated at all levels of organisms, from molecules to ecosystems [12,19,21 -23]. ...
Article
Full-text available
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.
... The diffusion effects in ecological models are quite common due to the movement of species from higher to lower concentration areas as a result of good living environment, food, etc. and thus reaction-diffusion terms associated with preypredator model should be taken into account [Murray, 2001;Banerjee, 2010;Sivakumar et al., 2015;Sivakumar et al., 2017;Choudhury & Nasipuri, 2015;Sun et al., 2016]. The applications of spatial diffusion can be found in other models also, such as Gierer-Meinhardt model [Iron et al., 2001], Brusselator model [Golovin et al., 2008], chemotaxis model [Kuto et al., 2012], ecosystem models [Liu et al., 2014;Liu et al., 2016] and herbivore-plant model [Sun et al., 2015;Li et al., 2016]. Specifically pattern formation (also knowns as diffusion-driven instability or Turing instability) is an essential mechanism in reaction-diffusion systems which was initiated by Turing [1952] and since then this topic became more popular among researchers. ...
Article
This study focuses on the spatial-temporal dynamics of predator–prey model with cross-diffusion where the intake rate of prey is per capita predator according to ratio-dependent functional response and the prey is harvested through nonlinear harvesting strategy. The permanence analysis and local stability analysis of the proposed model without cross-diffusion are analyzed. We derive the conditions for the appearance of diffusion-driven instability and global stability of the considered model. Also the parameter space for Turing region is specified by keeping the cross-diffusion coefficient as one of the crucial parameters. Numerical simulations are given to justify the proposed theoretical results and to show that the cross-diffusion term plays a significant role in the pattern formation.
... To test whether adding dissipation to the model would induce aggregated reefscape patterns in the attractor, the model was modified to include CO diffusion using a range of diffusion constants (figure 6; electronic supplementary material, ESM5). The results indicate that diffusion enables self-organization and clumping in the model, which is in line with previous studies that have linked diffusive or differential flow processes with regular pattern formation [47,49]. However, diffusion also acts to depress the steadystate CO fractional cover. ...
Article
Full-text available
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.
... Density-dependent movement resulting in striking spatial patterns is a well-known phenomena in physics known as the Cahn -Hilliard principle of phase separation. Studies have highlighted its underuse in ecology even though examples satisfying the principle abound in nature from sperm cells to mussel beds [61,62]. The relationship between the movement of mussels and their density is similar to the human migration pattern proposed herein. ...
Article
Full-text available
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.
... Unless it can be attributed to a distinct environmental heterogeneity (e.g. animal grouping at a better feeding ground), this phenomenon is thought to be either a consequence of animal 'sociality' ( Grüunbaum and Okubo, 1994;Gueron et al., 1996 ) or the effect of the density-dependence of the movement ( Liu, 2016;Tyutyunov et al., 2004 ). The specific animal's reaction can be of somewhat different type. ...
Article
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.
... To our knowledge, despite several well-established routes available for stabilizing droplets in non-active systems [63], potential strategies to suppress Ostwald ripening in the presence of self-propelled particles remain largely unexplored. In fact, in one-component suspensions of (non-aligning) self-propelled particles, finite domains (formed as a result of the interplay between motility-induced aggregation and intrinsic active fluctuations in the system) are found to coarsen to a state of complete phase separation [68][69][70][71][72][73][74][75]. The same applies to binary mixtures of active and non-active particles, where it is shown that even suitably prepared circular droplets of non-active particles surrounded by active selfpropellers are unstable to macroscopic phase separation [76,77]. ...
Article
Full-text available
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 as models representing 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 on 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 and suspended in a normal bulk behave in accordance with standard capillary paradigms, 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. As a result, we find a regime of anomalous capillarity for a single droplet, where the inside droplet pressure increases upon increasing the droplet size. In the case of two interconnected droplets, we show that mechanical equilibrium can occur also when they have different sizes. We further identify a regime of anomalous ripening, where two unequal-sized droplets can reach a final state of equal sizes upon interconnection, in stark contrast with the standard Ostwald ripening phenomenon, implying shrinkage of the smaller droplet in favor of the larger one.
... High mussel densities increase the competition for nutrients over long distances but they facilitate mussel-attachment to the sediment on the shorter range (van de Koppel et al. 2005;Rietkerk & van de Koppel 2008). Other models to study the formation of different structures in animal grouping also rely on similar attraction-repulsion principles (Couzin et al. 2002;Couzin 2003;Martínez-García et al. 2015;Liu et al. 2013;Liu et al. 2016;Vicsek & Zafeiris 2012). On the other hand, even though the fact that only competitive interactions may lead to clustering and pattern formation seems counterintuitive, it has been observed in several scenarios as well. ...
Article
Full-text available
Vegetation patterns are abundant in arid and semiarid ecosystems, but how they form remains unclear. One of the most extended theories lies in the existence of scale-dependent feedbacks (SDF) in plant-to-plant and plant-water interactions. Short distances are dominated by facilitative interactions, whereas competitive interactions dominate at larger scales. These feedbacks shape spatially inhomogeneous distributions of water that ultimately drive the emergence of patterns of vegetation. Even though the presence of facilitative and competitive interactions is clear, they are often hard to disentangle in the field, and therefore their relevance in vegetation pattern formation is still disputable. Here, we review the biological processes that have been proposed to explain pattern formation in arid ecosystems and how they have been implemented in mathematical models. We conclude by discussing the existence of similar structures in different biological and physical systems.
... The spatial patterns in ecosystems originate from spatial processes. Some processes generate spatial patterns (Groen et al., 2008;Liu et al., 2016), so called heterogenising processes. At the same time other processes can eradicate these spatial patterns, so called homogenising processes. ...
Article
Large-scale sudden transitions in ecosystems are expected as result of changing global climate or land use. Current theory predicts such sudden transitions especially to occur in spatially homogeneous ecosystems, whereas transitions in spatially heterogeneous systems will be more gradual. The spatial heterogeneity of ecosystems is determined as result of opposing spatial processes that are either increasing or decreasing heterogeneity. Hence, the relative strength of these opposing processes is expected to determine how sensitive the system is to transitions, which has not been explored to date. In our study, fire, as a spatially heterogenising process, and plant dispersion, as a spatially homogenising process, in tropical savannas were modelled to analyse how these processes affect the occurrence of sudden transitions from grass dominance to tree dominance. Savannas are expected to change due to precipitation or land use changes towards either tree dominance or grass dominance. We found that high rates of grass dispersion can create homogeneous grass patches, but only when the spatial extent of fire is limited to small patches that are spread across the landscape. When fires occur in larger patches, a heterogeneous pattern is generated. In spatially heterogeneous savannas, we found a more gradual responses to increasing grazing pressure compared to the sudden transitions when savannas are spatially homogeneous. The most sudden transitions were found in near-homogeneous grass distributions where the interaction between grazing, grass dispersion and fire led to a few homogeneous patches. Within these homogeneous patches, transitions were complete and sudden. We conclude that when spatially heterogenising processes are stronger than spatially homogenising processes, heterogeneous systems are created. In these systems large-scale sudden transitions are less likely to occur, because transitions at smaller scales are averaged over space. We discuss how this has implications for responses of savannas to climatic and land use change.
... Similar phenomena have been observed in systems using camphor. 12,26,34,35) The density-dependent motion has been recognized recently as an important factor for biological clusters, 36) such as mussel beds 37) and the periodic colonization of bacteria. 38) It is an open question whether our droplet system can serve as a model for the biological cluster dynamics. ...
Article
We review our recent results on a simple droplet system consisting of droplets of a mixture of two liquids. Although the system is composed of simple chemicals, it is capable of exhibiting complex collective behaviors as a cluster dynamics. We propose a simple inhomogeneous force model to analyze the experimental observations and discuss the mechanisms of the dynamic ordering in our system. The relevance of the observations to hierarchical ordering generally observed in systems of active elements is also discussed.
... Other transport mechanisms could be considered once a persistent interaction between the plant and the animal has been established and the topology of the landscape has been shaped. In those cases, we could add chemotactic terms or Cahn-Hilliard like equations ( Liu et al., 2016 ) to take into consideration the feedback interactions between both species. ...
Article
We study a model of seed dispersal that consists of an animal moving diffusively, feeding on fruits and dispersing 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 velocity of these waves depends on the delay, and we show that they are slower than the corresponding Fisher equations of logistic growth with diffusion.
Article
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.
Article
Full-text available
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.
Article
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.
Self-organization evolution of a population is studied considering generalized non-local 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 determine the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we establish a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our model reproduces some of the experimental results that other models employed previously had not been able to obtain.
Article
Full-text available
This work presents new approximate analytical solutions for the Riccati equation (RE) resulting from the application of the method of variation of parameters. The original equation is solved using another RE explicitly dependent on the independent variable. The solutions obtained are easy to implement and highly applicable, which is confirmed by solving several examples corresponding to REs whose solution is known, as well as optimizing the method to determine the density of the members that make up a population. In this way, new perspectives are open in the study of the phenomenon of pattern formation.
Article
Full-text available
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 C. 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.
Article
Full-text available
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.
Article
One of the major challenges in animal ecology is to understand the factors and processes driving movement behaviour. Although density may influence movement patterns, the occurrence and nature of density‐dependence in animal movements are still unclear, particularly whether it may vary among populations of a species, or across time within a population. Here, we evaluate the occurrence and nature of density‐dependence in the movements of a Neotropical marsupial, the Grey four‐eyed opossum Philander frenatus (Didelphidae, Didelphimorphia). We quantified fine‐scale path tortuosity of individuals inhabiting continuous forest areas and forest fragments, in different climatic seasons (humid vs. super‐humid). We also determined the relative importance of population size compared to sex and body mass on movements, using a model‐selection approach. In forest fragments, path tortuosity increased with population size in the super‐humid season, but decreased in the humid season. In the continuous forest, path tortuosity was affected only by sex and body mass, being slightly higher in males and negatively related to body mass. The occurrence of density‐dependence on movements only in forest fragments is likely to reflect the higher overall density of P. frenatus in small forest fragments. The variation in the nature of density‐dependence between climatic seasons is likely to reflect a trade off between foraging over large areas (humid season, low resource availability) versus avoiding agonistic encounters (super‐humid season, high resource availability). Our results show that (i) density‐dependence in movements may be context‐dependent occurring only in areas of relatively high overall population density; and (ii) density may affect movements in different ways at different climatic seasons.
Article
Full-text available
Active matter comprised of many self-driven units can exhibit emergent collective behaviors such as pattern formation and phase separation in both biologica and synthetic systems. While these behaviors are increasingly well understood for ensembles of linearly self-propelled particles, less is known about the collective behaviors of active rotating particles where energy input at the particle level gives rise to rotational particle motion. A recent simulation study revealed that active rotation can induce phase separation in mixtures of counter-rotating particles in 2D. In contrast to that of linearly self-propelled particles, the phase separation of counter-rotating fluids is accompanied by steady convective flows that originate at the fluid-fluid interface. Here, we investigate the influence of these flows on the coarsening dynamics of actively rotating binary liquids using a phenomenological, hydrodynamic model that combines a Cahn-Hilliard equation for the fluid composition with a Navier-Stokes equation for the fluid velocity. The effect of active rotation is introduced though an additional force within the Navier-Stokes equations that arises due to gradients in the concentrations of clockwise and counter-clockwise rotating particles. Depending on the strength of active rotation and that of frictional interactions with the stationary surroundings, we observe and explain new dynamical behaviors such as "active coarsening" via self-generated flows as well as the emergence of self-propelled vortex doublets. We confirm that many of the qualitative behaviors identified by the continuum model can also be found in discrete, particle-based simulations of actively rotating liquids. Our results highlight further opportunities for achieving complex dissipative structures in active materials subject to distributed actuation.
Article
Full-text available
Migratory ungulates outnumber residents by an order of magnitude in several savanna ecosystems in Africa, as was apparently the case in other grasslands around the world before the intervention of modern man. Migrants may be more numerous because 1) they use a much larger area, 2) they make more-efficient use of resources, or 3) they are less vulnerable to regulation by predators. These hypotheses were examined using simulation models of migratory and sedentary wildebeest Connochaetes taurinus in the Serengeti ecosystem. Simulations suggest that realistic numbers of predators could regulate resident herbivores at low population densities, whereas such regulation is probably rare for migratory herds. When residents and migrants have overlapping ranges, migrants should always outcompete residents, reducing them to low numbers. Results suggest that differences in the modes of regulation explain the predominance of migratory herbivores in some grassland ecosystems. -from Authors
Article
Full-text available
Self-organized spatial vegetation patterning is widespread and has been described using models of scale-dependent feedback between plants and water on homogeneous substrates. As rainfall decreases, these models yield a characteristic sequence of patterns with increasingly sparse vegetation, followed by sudden collapse to desert. Thus, the final, spot-like pattern may provide early warning for such catastrophic shifts. In many arid ecosystems, however, termite nests impart substrate heterogeneity by altering soil properties, thereby enhancing plant growth. We show that termite-induced heterogeneity interacts with scale-dependent feedbacks to produce vegetation patterns at different spatial grains. Although the coarse-grained patterning resembles that created by scale-dependent feedback alone, it does not indicate imminent desertification. Rather, mound-field landscapes are more robust to aridity, suggesting that termites may help stabilize ecosystems under global change.
Article
Full-text available
Self-organized spatial vegetation patterning is widespread and has been described using models of scale-dependent feedback between plants and water on homogeneous substrates. As rainfall decreases, these models yield a characteristic sequence of patterns with increasingly sparse vegetation, followed by sudden collapse to desert. Thus, the final, spot-like pattern may provide early warning for such catastrophic shifts. In many arid ecosystems, however, termite nests impart substrate heterogeneity by altering soil properties, thereby enhancing plant growth. We show that termite-induced heterogeneity interacts with scale-dependent feedbacks to produce vegetation patterns at different spatial grains. Although the coarse-grained patterning resembles that created by scale-dependent feedback alone, it does not indicate imminent desertification. Rather, mound-field landscapes are more robust to aridity, suggesting that termites may help stabilize ecosystems under global change.
Article
Full-text available
Dryland landscapes self-organize to form various patterns of vege-tation patchiness. Two major classes of patterns can be distinguished: regular patterns with characteristic length scales and scale-free patterns. The latter form under conditions of global competition over the water resource. In this pa-per we show that the asymptotic dynamics of scale-free vegetation patterns in-volve patch coarsening similar to Ostwald ripening in two-phase mixtures. We demonstrate it numerically, using a spatially explicit model for water-limited vegetation, and further study it by drawing an analogy to an activator-inhibitor system that shares many properties with the vegetation system. The ecologi-cal implications of patch coarsening may not be highly significant due to the long time scales involved. The reported results, however, raise an interesting pattern formation question associated with the incompatibility of mechanisms that stabilize vegetation spots and the condition of global competition.
Article
Full-text available
Sperm cooperation has evolved in a variety of taxa and is often considered a response to sperm competition, yet the benefit of this form of collective movement remains unclear. Here we use fine-scale imaging and a minimal mathematical model to study sperm aggregation in the rodent genus $Peromyscus$. We demonstrate that as the number of sperm cells in an aggregate increase, the group moves with more persistent linearity but without increasing speed; this benefit, however, is offset in larger aggregates as the geometry of the group forces sperm to swim against one another. The result is a non-monotonic relationship between aggregate size and average velocity with both a theoretically predicted and empirically observed optimum of 6-7 sperm/aggregate. To understand the role of sexual selection in driving these sperm group dynamics, we compared two sister-species with divergent mating systems and find that sperm of $P.\,maniculatus$ (highly promiscuous), which have evolved under intense competition, form optimal-sized aggregates more often than sperm of $P.\,polionotus$ (strictly monogamous), which lack competition. Our combined mathematical and experimental study of coordinated sperm movement reveals the importance of geometry, motion and group size on sperm velocity and suggests how these physical variables interact with evolutionary selective pressures to regulate cooperation in competitive environments.
Article
Full-text available
Self-propelled particles include both self-phoretic synthetic colloids and various micro-organisms. By continually consuming energy, they bypass the laws of equilibrium thermodynamics. These laws enforce the Boltzmann distribution in thermal equilibrium: the steady state is then independent of kinetic parameters. In contrast, self-propelled particles tend to accumulate where they move more slowly. They may also slow down at high density, for either biochemical or steric reasons. This creates positive feedback which can lead to motility-induced phase separation (MIPS) between dense and dilute fluid phases. At leading order in gradients, a mapping relates variable-speed, self-propelled particles to passive particles with attractions. This deep link to equilibrium phase separation is confirmed by simulations, but generally breaks down at higher order in gradients: new effects, with no equilibrium counterpart, then emerge. We give a selective overview of the fast-developing field of MIPS, focusing on theory and simulation but including a brief speculative survey of its experimental implications.
Article
Full-text available
We investigate collective phenomena with rotationally driven spinners of concave shape. Each spinner experiences a constant internal torque in either a clockwise or counterclockwise direction. Although the spinners are modeled as hard, otherwise noninteracting rigid bodies, their active motion induces an effective interaction that favors rotation in the same direction. With increasing density and activity, phase separation occurs via spinodal decomposition, as well as self-organization into rotating crystals. We observe the emergence of cooperative, superdiffusive motion along interfaces, which can transport inactive test particles. Our results demonstrate novel phase behavior of actively rotated particles that is not possible with linear propulsion or in nondriven, equilibrium systems of identical hard particles.
Article
Full-text available
The kinetic separation of repulsive active Brownian particles into a dense and a dilute phase is analyzed using a systematic coarse-graining strategy. We derive an effective Cahn-Hilliard equation on large length and time scales, which implies that the separation process can be mapped onto that of passive particles. A lower density threshold for clustering is found, and using our approach we demonstrate that clustering first proceeds via a hysteretic nucleation scenario and above a higher threshold changes into a spinodal-like instability. Our results are in agreement with particle-resolved computer simulations and can be verified in experiments of artificial or biological microswimmers.
Article
Full-text available
Ecological theory uses Brownian motion as a default template for describing ecological movement, despite limited mechanistic underpinning. The generality of Brownian motion has recently been challenged by empirical studies that highlight alternative movement patterns of animals, especially when foraging in resource-poor environments. Yet, empirical studies reveal animals moving in a Brownian fashion when resources are abundant. We demonstrate that Einstein's original theory of collision-induced Brownian motion in physics provides a parsimonious, mechanistic explanation for these observations. Here, Brownian motion results from frequent encounters between organisms in dense environments. In density-controlled experiments, movement patterns of mussels shifted from Lévy towards Brownian motion with increasing density. When the analysis was restricted to moves not truncated by encounters, this shift did not occur. Using a theoretical argument, we explain that any movement pattern approximates Brownian motion at high-resource densities, provided that movement is interrupted upon encounters. Hence, the observed shift to Brownian motion does not indicate a density-dependent change in movement strategy but rather results from frequent collisions. Our results emphasize the need for a more mechanistic use of Brownian motion in ecology, highlighting that especially in rich environments, Brownian motion emerges from ecological interactions, rather than being a default movement pattern.
Article
Full-text available
Active Brownian particles (ABPs), when subject to purely repulsive interactions, are known to undergo activity-induced phase separation broadly resembling an equilibrium (attraction-induced) gas-liquid coexistence. Here we present an accurate continuum theory for the dynamics of phase-separating ABPs, derived by direct coarse graining, capturing leading-order density gradient terms alongside an effective bulk free energy. Such gradient terms do not obey detailed balance; yet we find coarsening dynamics closely resembling that of equilibrium phase separation. Our continuum theory is numerically compared to large-scale direct simulations of ABPs and accurately accounts for domain growth kinetics, domain topologies, and coexistence densities.
Article
Full-text available
We present a theoretical explanation for the interfacial zig-zag instability that appears in anisotropic systems. Such an instability has been experimentally highlighted for an Ising wall formed in a nematic liquid crystal cell under homeotropic anchoring conditions. From an envelope equation, relevant close to the Fréedericksz transition, we have derived an asymptotic equation describing the interface dynamics in the vicinity of its bifurcation. The asymptotic limit used accounts for a strong difference between two of the elastic constants. The model is characterized by a conservative order parameter which satisfies a Cahn-Hilliard equation. It provides a good qualitative understanding of the experiments.
Article
Full-text available
On the Atlantic coast of Nova Scotia, transitions between alternative states of the subtidal ecosystem, kelp beds and sea urchin barrens, occur on a decadal scale. To explore the process of urchin aggregation within kelp beds that leads to the shift to barrens, we developed a coupled map lattice model to simulate the spatial dynamics of kelp and green sea urchin Strongylocentrotus droebachiensis abundance over time. Our simulations show that the following factors can cause sea urchins to form grazing aggregations that create gaps in a kelp bed: (1) random movement: by > 60% of sea urchins residing in the bed, (2) moderate to high levels of spatial variability in sea urchin recruitment (30 to 90 [urchins m(-2)](2)), (3) localized aggregation of sea urchins (150 urchins m(-2)) amid a moderate to high background density of sea urchins within the kelp bed (> 10 urchins m(-2)), with or without a chemotactic response of sea urchins to kelp, and (4) removal of kelp from areas > 20 m(2) (to simulate physical or biological disturbance, or harvesting). Gaps formed at random locations within the spatial domain and expanded and coalesced to form barrens in which sea urchins were randomly distributed, Sea urchins formed circular fronts around gaps in the kelp bed. The rate of advance of fronts (and increase in gap size) was linearly related to the density of sea urchins at the front. The duration of the transition to the barrens state decreased with increases in (1) the proportion (1),) of sea urchins moving (from > 6 yr for P-m = 0.8 to < 2 yr for P-m = 1) and (2) the variance of sea urchin recruitment (from > 5 yr for 30 [urchins m(-2)](2) to < 3 yr for 90 [urchins m(-2)](2)). Our findings support observations of gap formation within kelp beds that resulted in widespread destructive grazing on this coast in the late 1960s. Our model provides a predictive tool for the design of monitoring programs and field experiments to explore the underlying mechanisms of an ecosystem phase shift that has major ecological consequences.
Article
Full-text available
The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with long-range inhibition of growth and survival is an essential prerequisite for pattern formation. Here, we show that the physical principle of phase separation, solely based on density-dependent movement by organisms, represents an alternative class of self-organized pattern formation in ecology. Using experiments with self-organizing mussel beds, we derive an empirical relation between the speed of animal movement and local animal density. By incorporating this relation in a partial differential equation, we demonstrate that this model corresponds mathematically to the well-known Cahn-Hilliard equation for phase separation in physics. Finally, we show that the predicted patterns match those found both in field observations and in our experiments. Our results reveal a principle for ecological self-organization, where phase separation rather than activation and inhibition processes drives spatial pattern formation.
Article
Full-text available
Fundamental biological processes including morphogenesis, tissue repair and tumour metastasis require collective cell motions, and to drive these motions cells exert traction forces on their surroundings. Current understanding emphasizes that these traction forces arise mainly in `leader cells' at the front edge of the advancing cell sheet. Our data are contrary to that assumption and show for the first time by direct measurement that traction forces driving collective cell migration arise predominately many cell rows behind the leading front edge and extend across enormous distances. Traction fluctuations are anomalous, moreover, exhibiting broad non-Gaussian distributions characterized by exponential tails. Taken together, these unexpected findings demonstrate that although the leader cell may have a pivotal role in local cell guidance, physical forces that it generates are but a small part of a global tug-of-war involving cells well back from the leading edge.
Article
Full-text available
We study experimentally and numerically a (quasi) two dimensional colloidal suspension of self-propelled spherical particles. The particles are carbon-coated Janus particles, which are propelled due to diffusiophoresis in a near-critical water-lutidine mixture. At low densities, we find that the driving stabilizes small clusters. At higher densities, the suspension undergoes a phase separation into large clusters and a dilute gas phase. The same qualitative behavior is observed in simulations of a minimal model for repulsive self-propelled particles lacking any alignment interactions. The observed behavior is rationalized in terms of a dynamical instability due to the self-trapping of self-propelled particles.
Article
Full-text available
We examine a minimal model for an active colloidal fluid in the form of self-propelled Brownian hard spheres that interact purely through excluded volume. Despite the absence of an aligning interaction, this system shows the signature behaviors of an active fluid, including anomalous number fluctuations and phase separation behavior. Using simulations and analytic modeling, we quantify the phase diagram and separation kinetics. The dense phase is a unique material that we call an active hexatic, which exhibits the structural signatures of a crystalline solid near the crystal-hexatic transition point, but the rheological and transport properties associated with a viscoelastic fluid.
Article
Full-text available
Recent experiments have shown that spreading epithelial sheets exhibit a long-range coordination of motility forces that leads to a buildup of tension in the tissue, which may enhance cell division and the speed of wound healing. Furthermore, the edges of these epithelial sheets commonly show finger-like protrusions whereas the bulk often displays spontaneous swirls of motile cells. To explain these experimental observations, we propose a simple flocking-type mechanism, in which cells tend to align their motility forces with their velocity. Implementing this idea in a mechanical tissue simulation, the proposed model gives rise to efficient spreading and can explain the experimentally observed long-range alignment of motility forces in highly disordered patterns, as well as the buildup of tensile stress throughout the tissue. Our model also qualitatively reproduces the dependence of swirl size and swirl velocity on cell density reported in experiments and exhibits an undulation instability at the edge of the spreading tissue commonly observed in vivo. Finally, we study the dependence of colony spreading speed on important physical and biological parameters and derive simple scaling relations that show that coordination of motility forces leads to an improvement of the wound healing process for realistic tissue parameters.
Article
Full-text available
Mammalian spermatozoa motility is a subject of growing importance because of rising human infertility and the possibility of improving animal breeding. We highlight opportunities for fluid and continuum dynamics to provide novel insights concerning the mechanics of these specialized cells, especially during their remarkable journey to the egg. The biological structure of the motile sperm appendage, the flagellum, is described and placed in the context of the mechanics underlying the migration of mammalian sperm through the numerous environments of the female reproductive tract. This process demands certain specific changes to flagellar movement and motility for which further mechanical insight would be valuable, although this requires improved modeling capabilities, particularly to increase our understanding of sperm progression in vivo. We summarize current theoretical studies, highlighting the synergistic combination of imaging and theory in exploring sperm motility, and discuss the challenges for futur...
Article
Full-text available
Humans and climate affect ecosystems and their services, which may involve continuous and discontinuous transitions from one stable state to another. Discontinuous transitions are abrupt, irreversible and among the most catastrophic changes of ecosystems identified. For terrestrial ecosystems, it has been hypothesized that vegetation patchiness could be used as a signature of imminent transitions. Here, we analyse how vegetation patchiness changes in arid ecosystems with different grazing pressures, using both field data and a modelling approach. In the modelling approach, we extrapolated our analysis to even higher grazing pressures to investigate the vegetation patchiness when desertification is imminent. In three arid Mediterranean ecosystems in Spain, Greece and Morocco, we found that the patch-size distribution of the vegetation follows a power law. Using a stochastic cellular automaton model, we show that local positive interactions among plants can explain such power-law distributions. Furthermore, with increasing grazing pressure, the field data revealed consistent deviations from power laws. Increased grazing pressure leads to similar deviations in the model. When grazing was further increased in the model, we found that these deviations always and only occurred close to transition to desert, independent of the type of transition, and regardless of the vegetation cover. Therefore, we propose that patch-size distributions may be a warning signal for the onset of desertification.
Article
Full-text available
Simple analytical considerations are applied to recently discovered patterns in a generalized Fisher equation. The generalization consists of the inclusion of nonlocal competition interactions among the constituents of the field exhibiting patterns. We show here how stability arguments yield a necessary condition for pattern formation involving the ratio of the pattern wavelength and the effective diffusion length of the individual constituents. We also remark on how a mode−mode coupling analysis may be developed that might be useful in shedding some light on the amplitude of the patterns.
Chapter
This chapter focuses on density-dependent interaction-diffusion systems and discusses some consequences of nonlinearity in the diffusion process. The equation governing gas flow in a porous medium is a nonlinear diffusion equation without any reaction terms. Gurtin–MacCamy population model combines the porous medium flow mechanism with various nonlinear growth terms. In porous medium flow, there is a finite speed of propagation of disturbances from rest. In the predator–prey interaction system, the tendency of each species to migrate can depend on the local densities of both species. Density-dependent migration behavior can be seen in spatially distributed predator–prey interactions, for example, some insect predators can tend to stay in the vicinity of their last meal, at least for some time.
Article
Can physics be an appropriate framework for the understanding of ecological science? Most ecologists would probably agree that there is little relation between the complexity of natural ecosystems and the simplicity of any example derived from Newtonian physics. Though ecologists have long been interested in concepts originally developed by statistical physicists and later applied to explain everything from why stock markets crash to why rivers develop particular branching patterns, applying such concepts to ecosystems has remained a challenge. Self-Organization in Complex Ecosystems is the first book to clearly synthesize what we have learned about the usefulness of tools from statistical physics in ecology. Ricard Solé and Jordi Bascompte provide a comprehensive introduction to complex systems theory, and ask: do universal laws shape the structure of ecosystems, at least at some scales? They offer the most compelling array of theoretical evidence to date of the potential of nonlinear ecological interactions to generate nonrandom, self-organized patterns at all levels. Tackling classic ecological questions--from population dynamics to biodiversity to macroevolution--the book's novel presentation of theories and data shows the power of statistical physics and complexity in ecology. Self-Organization in Complex Ecosystems will be a staple resource for years to come for ecologists interested in complex systems theory as well as mathematicians and physicists interested in ecology.