A Continuum Three-Zone Model for Swarms

We present a progression of three distinct three-zone, continuum models for swarm behavior based on social interactions with neighbors in order to explain simple coherent structures in popular biological models of aggregations. In continuum models, individuals are replaced with density and velocity functions. Individual behavior is modeled with convolutions acting within three interaction zones corresponding to repulsion, orientation, and attraction, respectively. We begin with a variable-speed first-order model in which the velocity depends directly on the interactions. Next, we present a variable-speed second-order model. Finally, we present a constant-speed second-order model that is coordinated with popular individual-based models. For all three models, linear stability analysis shows that the growth or decay of perturbations in an infinite, uniform swarm depends on the strength of attraction relative to repulsion and orientation. We verify that the continuum models predict the behavior of a swarm of individuals by comparing the linear stability results with an individual-based model that uses the same social interaction kernels. In some unstable regimes, we observe that the uniform state will evolve toward a radially symmetric attractor with a variable density. In other unstable regimes, we observe an incoherent swarming state.     

Modeling Vortex Swarming In Daphnia

Based on experimental observations in Daphnia, we introduce an agent-based model for the motion of single and swarms of animals. Each agent is described by a stochastic equation that also considers the conditions for active biological motion. An environmental potential further reflects local conditions for Daphnia, such as attraction to light sources. This model is sufficient to describe the observed cycling behavior of single Daphnia. To simulate vortex swarming of many Daphnia, i.e. the collective rotation of the swarm in one direction, we extend the model by considering avoidance of collisions. Two different ansatzes to model such a behavior are developed and compared. By means of computer simulations of a multi-agent system we show that local avoidance—as a special form of asymmetric repulsion between animals—leads to the emergence of a vortex swarm. The transition from uncorrelated rotation of single agents to the vortex swarming as a function of the swarm size is investigated. Eventually, some evidence of avoidance behavior in Daphnia is provided by comparing experimental and simulation results for two animals.     

A non-local model for a swarm

 This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles. Received: 17 September 1997 / Revised version: 17 March 1998     

Smart swarms of bacteria-inspired agents with performance adaptable interactions

Collective navigation and swarming have been studied in animal groups, such as fish schools, bird flocks, bacteria, and slime molds. Computer modeling has shown that collective behavior of simple agents can result from simple interactions between the agents, which include short range repulsion, intermediate range alignment, and long range attraction. Here we study collective navigation of bacteria-inspired smart agents in complex terrains, with adaptive interactions that depend on performance. More specifically, each agent adjusts its interactions with the other agents according to its local environment--by decreasing the peers' influence while navigating in a beneficial direction, and increasing it otherwise. We show that inclusion of such performance dependent adaptable interactions significantly improves the collective swarming performance, leading to highly efficient navigation, especially in complex terrains. Notably, to afford such adaptable interactions, each modeled agent requires only simple computational capabilities with short-term memory, which can easily be implemented in simple swarming robots.     

The impact of individual perceptual and cognitive factors on collective states in a data-driven fish school model

In moving animal groups, social interactions play a key role in the ability of individuals to achieve coordinated motion. However, a large number of environmental and cognitive factors are able to modulate the expression of these interactions and the characteristics of the collective movements that result from these interactions. Here, we use a data-driven fish school model to quantitatively investigate the impact of perceptual and cognitive factors on coordination and collective swimming patterns. The model describes the interactions involved in the coordination of burst-and-coast swimming in groups of Hemigrammus rhodostomus. We perform a comprehensive investigation of the respective impacts of two interactions strategies between fish based on the selection of the most or the two most influential neighbors, of the range and intensity of social interactions, of the intensity of individual random behavioral fluctuations, and of the group size, on the ability of groups of fish to coordinate their movements. We find that fish are able to coordinate their movements when they interact with their most or two most influential neighbors, provided that a minimal level of attraction between fish exist to maintain group cohesion. A minimal level of alignment is also required to allow the formation of schooling and milling. However, increasing the strength of social interactions does not necessarily enhance group cohesion and coordination. When attraction and alignment strengths are too high, or when the heading random fluctuations are too large, schooling and milling can no longer be maintained and the school switches to a swarming phase. Increasing the interaction range between fish has a similar impact on collective dynamics as increasing the strengths of attraction and alignment. Finally, we find that coordination and schooling occurs for a wider range of attraction and alignment strength in small group sizes.     

How the Spatial Position of Individuals Affects Their Influence on Swarms: A Numerical Comparison of Two Popular Swarm Dynamics Models

Schools of fish and flocks of birds are examples of self-organized animal groups that arise through social interactions among individuals. We numerically study two individual-based models, which recent empirical studies have suggested to explain self-organized group animal behavior: (i) a zone-based model where the group communication topology is determined by finite interacting zones of repulsion, attraction, and orientation among individuals; and (ii) a model where the communication topology is described by Delaunay triangulation, which is defined by each individual''s Voronoi neighbors. The models include a tunable parameter that controls an individual''s relative weighting of attraction and alignment. We perform computational experiments to investigate how effectively simulated groups transfer information in the form of velocity when an individual is perturbed. A cross-correlation function is used to measure the sensitivity of groups to sudden perturbations in the heading of individual members. The results show how relative weighting of attraction and alignment, location of the perturbed individual, population size, and the communication topology affect group structure and response to perturbation. We find that in the Delaunay-based model an individual who is perturbed is capable of triggering a cascade of responses, ultimately leading to the group changing direction. This phenomenon has been seen in self-organized animal groups in both experiments and nature.     

Social interactions in myxobacterial swarming

Swarming, a collective motion of many thousands of cells, produces colonies that rapidly spread over surfaces. In this paper, we introduce a cell-based model to study how interactions between neighboring cells facilitate swarming. We chose to study Myxococcus xanthus, a species of myxobacteria, because it swarms rapidly and has well-defined cell–cell interactions mediated by type IV pili and by slime trails. The aim of this paper is to test whether the cell contact interactions, which are inherent in pili-based S motility and slime-based A motility, are sufficient to explain the observed expansion of wild-type swarms. The simulations yield a constant rate of swarm expansion, which has been observed experimentally. Also, the model is able to quantify the contributions of S motility and A motility to swarming. Some pathogenic bacteria spread over infected tissue by swarming. The model described here may shed some light on their colonization process.     

Swarms of particle agents with harmonic interactions

Summary Agent-based modeling is a powerful methodology to describe the occurence of complex behavior in biological systems. The interaction of a large number of individuals (agents) may for example lead to the emergence of new forms of collective motion. In this paper, we investigate a particle-based approach to the coherent motion of a swarm with parabolic (i. e. harmonic) interactions between the agents. It is based on generalized Langevin equations for the particle agents, which take into account (i) energetic conditions for active motion, (ii) linear attractive forces between each two agents. The complex collective motion observed can be explained as the result of these different influences: the active motion of the agents, which is driven by the energy-take up, would eventually lead to a spatial dispersion of the swarm, while the mutual interaction of the agents results in a tendency of spatial concentration. In addition to particle-based computer simulations, we also provide a mathematical framework for investigating the collective dynamics. Dedicated to the memory of Michael Conrad     

Dynamics of Bacterial Swarming

When vegetative bacteria that can swim are grown in a rich medium on an agar surface, they become multinucleate, elongate, synthesize large numbers of flagella, produce wetting agents, and move across the surface in coordinated packs: they swarm. We examined the motion of swarming Escherichia coli, comparing the motion of individual cells to their motion during swimming. Swarming cells' speeds are comparable to bulk swimming speeds, but very broadly distributed. Their speeds and orientations are correlated over a short distance (several cell lengths), but this correlation is not isotropic. We observe the swirling that is conspicuous in many swarming systems, probably due to increasingly long-lived correlations among cells that associate into groups. The normal run-tumble behavior seen in swimming chemotaxis is largely suppressed, instead, cells are continually reoriented by random jostling by their neighbors, randomizing their directions in a few tenths of a second. At the edge of the swarm, cells often pause, then swim back toward the center of the swarm or along its edge. Local alignment among cells, a necessary condition of many flocking theories, is accomplished by cell body collisions and/or short-range hydrodynamic interactions.     

Extracellular proteolytic activity plays a central role in swarming motility in Bacillus subtilis

Natural isolates of Bacillus subtilis exhibit a robust multicellular behavior known as swarming. A form of motility, swarming is characterized by a rapid, coordinated progression of a bacterial population across a surface. As a collective bacterial process, swarming is often associated with biofilm formation and has been linked to virulence factor expression in pathogenic bacteria. While the swarming phenotype has been well documented for Bacillus species, an understanding of the molecular mechanisms responsible remains largely isolated to gram-negative bacteria. To better understand how swarming is controlled in members of the genus Bacillus, we investigated the effect of a series of gene deletions on swarm motility. Our analysis revealed that a strain deficient for the production of surfactin and extracellular proteolytic activity did not swarm or form biofilm. While it is known that surfactin, a lipoprotein surfactant, functions in swarming motility by reducing surface tension, this is the first report demonstrating that general extracellular protease activity also has an important function. These results not only help to define the factors involved in eliciting swarm migration but support the idea that swarming and biofilm formation may have overlapping control mechanisms.     

Locust Collective Motion and Its Modeling

Over the past decade, technological advances in experimental and animal tracking techniques have motivated a renewed theoretical interest in animal collective motion and, in particular, locust swarming. This review offers a comprehensive biological background followed by comparative analysis of recent models of locust collective motion, in particular locust marching, their settings, and underlying assumptions. We describe a wide range of recent modeling and simulation approaches, from discrete agent-based models of self-propelled particles to continuous models of integro-differential equations, aimed at describing and analyzing the fascinating phenomenon of locust collective motion. These modeling efforts have a dual role: The first views locusts as a quintessential example of animal collective motion. As such, they aim at abstraction and coarse-graining, often utilizing the tools of statistical physics. The second, which originates from a more biological perspective, views locust swarming as a scientific problem of its own exceptional merit. The main goal should, thus, be the analysis and prediction of natural swarm dynamics. We discuss the properties of swarm dynamics using the tools of statistical physics, as well as the implications for laboratory experiments and natural swarms. Finally, we stress the importance of a combined-interdisciplinary, biological-theoretical effort in successfully confronting the challenges that locusts pose at both the theoretical and practical levels.     

A one-dimensional model of trail propagation by army ants

We develop and analyze a model for the swarming behaviour observed in army ants. The model assumes that the ants coordinate their movements by using chemical pheromones as trail markers. The markers continuously evaporate, and are reinforced by new markers laid down by the ants as they move. The motion of the swarm is modelled by a system of partial differential equations (PDEs). The equations are derived from the motions of the individuals, but represent the collective motion of the group, and the formation and decay of the trail network. The PDEs have travelling wave solutions which correspond to the propagation of the leading edge of the swarm. We describe these solutions qualitatively, and use them to determine how both the shape and the speed of the swarm depend on the parameters describing the motion of the individual ants.     

Fluid Forces Enhance the Performance of an Aspirant Leader in Self-Organized Living Groups

In this paper, the performance of an individual aiming at guiding a self-organized group is numerically investigated. A collective behavioural model is adopted, accounting for the mutual repulsion, attraction and orientation experienced by the individuals. Moreover, these represent a set of solid particles which are supposed to be immersed in a fictitious viscous fluid. In particular, the lattice Boltzmann and Immersed boundary methods are used to predict the fluid dynamics, whereas the effect of the hydrodynamic forces on particles is accounted for by solving the equation of the solid motion through the time discontinuous Galerkin scheme. Numerical simulations are carried out by involving the individuals in a dichotomous process. On the one hand, an aspirant leader (AL) additional individual is added to the system. AL is forced to move along a prescribed direction which intersects the group. On the other hand, these tend to depart from an obstacle represented by a rotating lamina which is placed in the fluid domain. A numerical campaign is carried out by varying the fluid viscosity and, as a consequence, the hydrodynamic field. Moreover, scenarios characterized by different values of the size of the group are investigated. In order to estimate the AL''s performance, a proper parameter is introduced, depending on the number of individuals following AL. Present findings show that the sole collective behavioural equations are insufficient to predict the AL''s performance, since the motion is drastically affected by the presence of the surrounding fluid. With respect to the existing literature, the proposed numerical model is enriched by accounting for the presence of the encompassing fluid, thus computing the hydrodynamic forces arising when the individuals move.     

Swarm robotics: a review from the swarm engineering perspective

Swarm robotics is an approach to collective robotics that takes inspiration from the self-organized behaviors of social animals. Through simple rules and local interactions, swarm robotics aims at designing robust, scalable, and flexible collective behaviors for the coordination of large numbers of robots. In this paper, we analyze the literature from the point of view of swarm engineering: we focus mainly on ideas and concepts that contribute to the advancement of swarm robotics as an engineering field and that could be relevant to tackle real-world applications. Swarm engineering is an emerging discipline that aims at defining systematic and well founded procedures for modeling, designing, realizing, verifying, validating, operating, and maintaining a swarm robotics system. We propose two taxonomies: in the first taxonomy, we classify works that deal with design and analysis methods; in the second taxonomy, we classify works according to the collective behavior studied. We conclude with a discussion of the current limits of swarm robotics as an engineering discipline and with suggestions for future research directions.     

Collective motion of bacteria in two dimensions

Collective motion can be observed in biological systems over a wide range of length scales, from large animals to bacteria. Collective motion is thought to confer an advantage for defense and adaptation. A central question in the study of biological collective motion is how the traits of individuals give rise to the emergent behavior at population level. This question is relevant to the dynamics of general self-propelled particle systems, biological self-organization, and active fluids. Bacteria provide a tractable system to address this question, because bacteria are simple and their behavior is relatively easy to control. In this mini review we will focus on a special form of bacterial collective motion, i.e., bacterial swarming in two dimensions. We will introduce some organization principles known in bacterial swarming and discuss potential means of controlling its dynamics. The simplicity and controllability of 2D bacterial behavior during swarming would allow experimental examination of theory predictions on general collective motion.     

Statistical tests for analysing directed movement of self-organising animal groups

We discuss some theory concerning directional data and introduce a suite of statistical tools that researchers interested in the directional movement of animal groups can use to analyse results from their models. We illustrate these tools by analysing the results of a model of groups moving under the duress of certain informed indistinguishable individuals, that arises in the context of honeybee (Apis mellifera) swarming behaviour. We modify an existing model of collective motion, based on inter-individual social interactions, allowing knowledgeable individuals to guide group members to the goal by travelling through the group in a direct line aligned with the goal direction.     

The biological principles of swarm intelligence

The roots of swarm intelligence are deeply embedded in the biological study of self-organized behaviors in social insects. From the routing of traffic in telecommunication networks to the design of control algorithms for groups of autonomous robots, the collective behaviors of these animals have inspired many of the foundational works in this emerging research field. For the first issue of this journal dedicated to swarm intelligence, we review the main biological principles that underlie the organization of insects’ colonies. We begin with some reminders about the decentralized nature of such systems and we describe the underlying mechanisms of complex collective behaviors of social insects, from the concept of stigmergy to the theory of self-organization in biological systems. We emphasize in particular the role of interactions and the importance of bifurcations that appear in the collective output of the colony when some of the system’s parameters change. We then propose to categorize the collective behaviors displayed by insect colonies according to four functions that emerge at the level of the colony and that organize its global behavior. Finally, we address the role of modulations of individual behaviors by disturbances (either environmental or internal to the colony) in the overall flexibility of insect colonies. We conclude that future studies about self-organized biological behaviors should investigate such modulations to better understand how insect colonies adapt to uncertain worlds.     

Variation in hydration forces between neutral phospholipid bilayers: evidence for hydration attraction

It is now generally recognized that hydration forces dominate close interactions of lipid hydrophilic surfaces. The commonality of their characteristics has been reasonably established. However, differences in measured net repulsion, particularly evident when phosphatidylethanolamine (PE) and phosphatidylcholine (PC) bilayers are compared, suggest there exists a variety of behavior wider than expected from earlier models of hydration and fluctuation repulsion balanced by van der Waals attraction. To find a basis for this diverse behavior, we have looked more closely at measured structural parameters, degrees of hydration, and interbilayer repulsive forces for the lamellar phases of the following lipids: 1-palmitoyl-2-oleoyl-PE (POPE), egg PE, transphosphatidylated egg PE (egg PE-T), mono- and dimethylated egg PE-T (MMPE and DMPE), 1-stearoyl-2-oleoyl-PC (SOPC), and mixtures of POPE and SOPC. POPE and SOPC bilayers differ not only in their maximum degrees of hydration but also in the empirical hydration force coefficients and decay lengths that characterize their interaction. When mixed with POPE, SOPC effects sudden and disproportionate increases in hydration. POPE, egg PE, and egg PE-T differ in their degree of hydration, molecular area, and hydration repulsion. A single methylation of egg PE-T almost completely converts its hydration and bilayer repulsive properties to those of egg PC; little progression of hydration is seen with successive methylations. In order to reconcile these observations with the conventional scheme of balancing interbilayer hydration and fluctuation-enhanced repulsion with van der Waals attraction, it is necessary to relinquish the fundamental idea that the decay of hydration forces is a constant determined by the properties of the aqueous medium. Alternatively, one can retain that fundamental idea if one recognizes the possibility that polar group hydration has an attractive component to it. In the latter view, that attractive component originates from interbilayer hydrogen-bonded water bridges between apposing bilayer surfaces, arising from correlation of zwitterionic or other complementary polar groups or from factors that affect polar group solubility. The same Marcelja and Radic formalism that accounts so well for the repulsive component also leads to an estimate of the attractive one. We suggest that the full range of degrees of hydration and of interbilayer spacings observed for different neutral bilayers results in part from variable contributions of the attractive and repulsive hydration components.(ABSTRACT TRUNCATED AT 400 WORDS)     

Order and flexibility in the motion of fish schools

The coexistence of order and flexibility in the motion of fish schools was studied by using a simple numerical model and a computer simulation. The numerical model is based on behavioral rules for individuals in the school by considering attraction, repulsion, and parallel-orientation behavior. Each individual follows the same rules and makes school movements. The simulation results show that school order and flexibility are affected by the number of neighbors interacting with an individual in the school and by the randomness of individual motion. Increase in the number of interacting neighbors leads to high order, especially when the number increases from a low value (between one and three). An optimal number of interacting neighbors exists that is relatively low (two or three) for high flexibility, indicating that a fish needs only to pay attention to a few neighbors to realize both order and flexibility. The low randomness of individual motion benefits both order and flexibility. These results indicate that schooling fish have evolved specialized ability for establishing both school order and flexibility.     

Collective Cell Movement Promotes Synchronization of Coupled Genetic Oscillators

Collective cell movement is a crucial component of embryonic development. Intercellular interactions regulate collective cell movement by allowing cells to transfer information. A key question is how collective cell movement itself influences information flow produced in tissues by intercellular interactions. Here, we study the effect of collective cell movement on the synchronization of locally coupled genetic oscillators. This study is motivated by the segmentation clock in zebrafish somitogenesis, where short-range correlated movement of cells has been observed. We describe the segmentation clock tissue by a Voronoi diagram, cell movement by the force balance of self-propelled and repulsive forces between cells, the dynamics of the direction of self-propelled motion, and the synchronization of genetic oscillators by locally coupled phase oscillators. We find that movement with a correlation length of about 2 ∼ 3 cell diameters is optimal for the synchronization of coupled oscillators. Quantification of cell mixing reveals that this short-range correlation of cell movement allows cells to exchange neighbors most efficiently. Moreover, short-range correlated movement strongly destabilizes nonuniform spatial phase patterns, further promoting global synchronization. Our theoretical results suggest that collective cell movement may enhance the synchronization of the segmentation clock in zebrafish somitogenesis. More generally, collective cell movement may promote information flow in tissues by enhancing cell mixing and destabilizing spurious patterns.