In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.

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Coordination of groups of mobile autonomous agents using nearest neighbor rules

In this paper, we present a theoretical framework for design and analysis of distributed flocking algorithms. Two cases of flocking in free-space and presence of multiple obstacles are considered. We present three flocking algorithms: two for free-flocking and one for constrained flocking. A comprehensive analysis of the first two algorithms is provided. We demonstrate the first algorithm embodies all three rules of Reynolds. This is a formal approach to extraction of interaction rules that lead to the emergence of collective behavior. We show that the first algorithm generically leads to regular fragmentation, whereas the second and third algorithms both lead to flocking. A systematic method is provided for construction of cost functions (or collective potentials) for flocking. These collective potentials penalize deviation from a class of lattice-shape objects called /spl alpha/-lattices. We use a multi-species framework for construction of collective potentials that consist of flock-members, or /spl alpha/-agents, and virtual agents associated with /spl alpha/-agents called /spl beta/- and /spl gamma/-agents. We show that migration of flocks can be performed using a peer-to-peer network of agents, i.e., "flocks need no leaders." A "universal" definition of flocking for particle systems with similarities to Lyapunov stability is given. Several simulation results are provided that demonstrate performing 2-D and 3-D flocking, split/rejoin maneuver, and squeezing maneuver for hundreds of agents using the proposed algorithms.

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Flocking for multi-agent dynamic systems: algorithms and theory

This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed. Introduction The immersed boundary (IB) method was introduced to study flow patterns around heart valves and has evolved into a generally useful method for problems of fluid–structure interaction. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all.

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The immersed boundary method

For animals that forage or travel in groups, making movement decisions often depends on social interactions among group members. However, in many cases, few individuals have pertinent information, such as knowledge about the location of a food source, or of a migration route. Using a simple model we show how information can be transferred within groups both without signalling and when group members do not know which individuals, if any, have information. We reveal that the larger the group the smaller the proportion of informed individuals needed to guide the group, and that only a very small proportion of informed individuals is required to achieve great accuracy. We also demonstrate how groups can make consensus decisions, even though informed individuals do not know whether they are in a majority or minority, how the quality of their information compares with that of others, or even whether there are any other informed individuals. Our model provides new insights into the mechanisms of effective leadership and decision-making in biological systems.

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Effective leadership and decision-making in animal groups on the move

We review the observations and the basic laws describing the essential aspects of collective motion -- being one of the most common and spectacular manifestation of coordinated behavior Our aim is to provide a balanced discussion of the various facets of this highly multidisciplinary field, including experiments, mathematical methods and models for simulations, so that readers with a variety of background could get both the basics and a broader, more detailed picture of the field The observations we report on include systems consisting of units ranging from macromolecules through metallic rods and robots to groups of animals and people Some emphasis is put on models that are simple and realistic enough to reproduce the numerous related observations and are useful for developing concepts for a better understanding of the complexity of systems consisting of many simultaneously moving entities As such, these models allow the establishing of a few fundamental principles of flocking In particular, it is demonstrated, that in spite of considerable differences, a number of deep analogies exist between equilibrium statistical physics systems and those made of self-propelled (in most cases living) units In both cases only a few well defined macroscopic/collective states occur and the transitions between these states follow a similar scenario, involving discontinuity and algebraic divergences

Collective Motion

Heterogeneous, "aggregated" patterns in the spatial distributions of individuals are almost universal across living organisms, from bacteria to higher vertebrates. Whereas specific features of aggregations are often visually striking to human eyes, a heuristic analysis based on human vision is usually not sufficient to answer fundamental questions about how and why organisms aggregate. What are the individual-level behavioral traits that give rise to these features? When qualitatively similar spatial patterns arise from purely physical mechanisms, are these patterns in organisms biologically significant, or are they simply epiphenomena that are likely characteristics of any set of interacting autonomous individuals? If specific features of spatial aggregations do confer advantages or disadvantages in the fitness of group members, how has evolution operated to shape individual behavior in balancing costs and benefits at the individual and group levels? Mathematical models of social behaviors such as schooling in fishes provide a promising avenue to address some of these questions. However, the literature on schooling models has lacked a common framework to objectively and quantitatively characterize relationships between individual-level behaviors and group-level patterns. In this paper, we briefly survey similarities and differences in behavioral algorithms and aggregation statistics among existing schooling models. We present preliminary results of our efforts to develop a modeling framework that synthesizes much of this previous work, and to identify relationships between behavioral parameters and group-level statistics.

Self-organized fish schools: an examination of emergent properties.

From Individuals to Aggregations: the Interplay between Behavior and Physics

Many aquatic animals face a fundamental problem during foraging and migratory movements: while their resources commonly vary at large spatial scales, they can only sample and assess their environment at relatively small, local spatial scales. Thus, they are unable to choose movement directions by directly sampling distant parts of their environment. A common strategy to overcome this problem is taxis, a behaviour in which an animal performs a biased random walk by changing direction more rapidly when local conditions are getting worse. Such an animal spends more time moving in right directions than wrong ones, and eventually gets to a favourable area. Taxis is inefficient, however, when environmental gradients are weak or overlain by ‘noisy’ small-scale fluctuations. In this paper, I show that schooling behaviour can improve the ability of animals performing taxis to climb gradients, even under conditions when asocial taxis would be ineffective. Schooling is a social behaviour incorporating tendencies to remain close to and align with fellow members of a group. It enhances taxis because the alignment tendency produces tight angular distributions within groups, and dampens the stochastic effects of individual sampling errors. As a result, more school members orient up-gradient than in the comparable asocial case. However, overly strong schooling behaviour makes the school slow in responding to changing gradient directions. This trade-off suggests an optimal level of schooling behaviour for given spatio-temporal scales of environmental variations. Social taxis may enhance the selective value of schooling in pelagic grazers such as herrings, anchovies and Antarctic krill. Furthermore, the degree of aggregation in a population of schooling animals may affect directly the rate and direction of migration and foraging movements.

Schooling as a strategy for taxis in a noisy environment

It is hard to find animals in nature that do not aggregate for one reason or another. The details of such aggregations are important because they influence numerous fundamental processes like mate-finding, prey-detection, predator avoidance, and disease transmission. Yet, despite the near universality of aggregation and its profound consequences, biologists have only recently begun to probe its underlying mechanisms. In this chapter we review theoretical approaches to animal aggregation, concentrating on aggregations which are caused by social interactions. We emphasize methods and limitations, and suggest what we think are the most promising avenues for future research. For an earlier review article on dynamical aspects of animal grouping consult Okubo (1986).

Modelling social animal aggregations

The success of most foragers is constrained by limits vidual-level movements and physiology while preying on to their sensory perception, memory, and locomotion. However, a the goldenrod aphid (Uroleucon nigrotuberculatum) were general and quantitative understanding of how these constraints affect foraging benefits, and the trade-offs they imply for foraging quantified in detail by Kareiva and Odell (1987). Finally, strategies, is difficult to achieve. This article develops foraging per- I argue that because a large class of sophisticated foraging formance statistics to assess constraints and define trade-offs for behaviors share this form of population-level movement foragers using biased random walk behaviors, a widespread class model, the approach used here could find wide applica- of foraging strategies that includes area-restricted searches, kineses, bility in a broad array of taxa. and taxes. The statistics are expected payoff and expected travel The foraging behaviors addressed in this article are time and assess two components of foraging performance: how ef- known in the mathematical literature as biased random fectively foragers distinguish between resource-poor and resource- rich parts of their environments and how quickly foragers in poor walks (Okubo 1980, 1986; Othmer et al. 1988), but more parts of the environment locate resource concentrations. These commonly to biologists as area-restricted search, kinesis, statistics provide a link between mechanistic models of individuals' and taxis (Fraenkel and Gunn 1961; DeAngelis and Yeh movement and functional responses, population-level models of 1982; Tranquillo and Alt 1990). A biased random walk is forager distributions in space and time, and foraging theory pre- a behavior in which a forager's movement decisions are dictions of optimal forager distributions and criteria for abandon- stochastic but have biases that make some outcomes ing resource patches. Application of the analysis to area-restricted more likely than others and thus lead to long-term di- search in coccinellid beetles suggests that the most essential aspect of these predators' foraging strategy is the ''turning threshold,'' the rected movement. In the case of foraging or search be- prey density at which ladybirds switch from slow to rapid turning. haviors, the statistical biases in a forager's behavior are This threshold effectively determines whether a forager exploits or determined at least in part by its sensory perception of abandons a resource concentration. Foraging is most effective local resources and changes in internal states (e.g., physi- when the threshold is tuned to match physiological or energetic ology or cognition) that result from recent foraging his- requirements. These performance statistics also help anticipate and tory. For example, a forager might respond primarily to interpret the dynamics of complex spatially and temporally varying local gradients in the availability of resources (taxis), to forager-resource systems. the absolute level of resource availability (area-restricted

https://www.jstor.org/stable/10.1086/286105

Daniel Grünbaum

Papers

Self-organized fish schools: an examination of emergent properties.

From Individuals to Aggregations: the Interplay between Behavior and Physics

Schooling as a strategy for taxis in a noisy environment

Modelling social animal aggregations

Using spatially explicit models to characterize foraging performance in heterogeneous landscapes.