Determining individual-level interactions that govern highly coordinated motion in animal groups or cellular aggregates has been a long-standing challenge, central to understanding the mechanisms and evolution of collective behavior. Numerous models have been proposed, many of which display realistic-looking dynamics, but nonetheless rely on untested assumptions about how individuals integrate information to guide movement. Here we infer behavioral rules directly from experimental data. We begin by analyzing trajectories of golden shiners (Notemigonus crysoleucas) swimming in two-fish and three-fish shoals to map the mean effective forces as a function of fish positions and velocities. Speeding and turning responses are dynamically modulated and clearly delineated. Speed regulation is a dominant component of how fish interact, and changes in speed are transmitted to those both behind and ahead. Alignment emerges from attraction and repulsion, and fish tend to copy directional changes made by those ahead. We find no evidence for explicit matching of body orientation. By comparing data from two-fish and three-fish shoals, we challenge the standard assumption, ubiquitous in physics-inspired models of collective behavior, that individual motion results from averaging responses to each neighbor considered separately; three-body interactions make a substantial contribution to fish dynamics. However, pairwise interactions qualitatively capture the correct spatial interaction structure in small groups, and this structure persists in larger groups of 10 and 30 fish. The interactions revealed here may help account for the rapid changes in speed and direction that enable real animal groups to stay cohesive and amplify important social information.A fundamental problem in a wide range of biological disciplines is understanding how functional complexity at a macroscopic scale (such as the functioning of a biological tissue) results from the actions and interactions among the individual components (such as the cells forming the tissue). Animal groups such as bird flocks, fish schools, and insect swarms frequently exhibit complex and coordinated collective behaviors and present unrivaled opportunities to link the behavior of individuals with dynamic group-level properties. With the advent of tracking technologies such as computer vision and global positioning systems, group behavior can be reduced to a set of trajectories in space and time. Consequently, in principle, it is possible to deduce the individual interaction rules starting from the observed kinematics. However, calculating interindividual interactions from trajectories means solving a fundamental inverse problem that appears universally in many-body systems. In general, such problems are very hard to solve and, even if they can be solved, their solution is often not unique.To avoid solving these inverse problems (and because detailed kinematic data were not available until recently), many attempts have been made to replicate the patterns observed in animal groups by...

The spontaneous emergence of pattern formation is ubiquitous in nature, often arising as a collective phenomenon from interactions among a large number of individual constituents or sub-systems. Understanding, and controlling, collective behavior is dependent on determining the low-level dynamical principles from which spatial and temporal patterns emerge; a key question is whether different group-level patterns result from all components of a system responding to the same external factor, individual components changing behavior but in a distributed self-organized way, or whether multiple collective states co-exist for the same individual behaviors. Using schooling fish (golden shiners, in groups of 30 to 300 fish) as a model system, we demonstrate that collective motion can be effectively mapped onto a set of order parameters describing the macroscopic group structure, revealing the existence of at least three dynamically-stable collective states; swarm, milling and polarized groups. Swarms are characterized by slow individual motion and a relatively dense, disordered structure. Increasing swim speed is associated with a transition to one of two locally-ordered states, milling or highly-mobile polarized groups. The stability of the discrete collective behaviors exhibited by a group depends on the number of group members. Transitions between states are influenced by both external (boundary-driven) and internal (changing motion of group members) factors. Whereas transitions between locally-disordered and locally-ordered group states are speed dependent, analysis of local and global properties of groups suggests that, congruent with theory, milling and polarized states co-exist in a bistable regime with transitions largely driven by perturbations. Our study allows us to relate theoretical and empirical understanding of animal group behavior and emphasizes dynamic changes in the structure of such groups.

An important characteristic of flocks of birds, school of fish, and many similar assemblies of selfpropelled particles is the emergence of states of collective order in which the particles move in the same direction. When noise is added into the system, the onset of such collective order occurs through a dynamical phase transition controlled by the noise intensity. While originally thought to be continuous, the phase transition has been claimed to be discontinuous on the basis of recently reported numerical evidence. We address this issue by analyzing two representative network models closely related to systems of self-propelled particles. We present analytical as well as numerical results showing that the nature of the phase transition depends crucially on the way in which noise is introduced into the system. PACS numbers: 05.70. Fh, 87.17.Jj, The collective motion of a group of autonomous particles is a subject of intense research that has potential applications in biology, physics and engineering [1,2,3]. One of the most remarkable characteristics of systems such as a flock of birds, a school of fish or a swarm of locusts, is the emergence of ordered states in which the particles move in the same direction, in spite of the fact that the interactions between the particles are (presumably) of short range. Given that these systems are generally out of equilibrium, the emergence of ordered states cannot be accounted for by the standard theorems in statistical mechanics that explain the existence of ordered states in equilibrium systems typified by ferromagnets.A particularly simple model to describe the collective motion of a group of self-propelled particles was proposed by Vicsek et al. [4]. In this model each particle tends to move in the average direction of motion of its neighbors while being simultaneously subjected to noise. As the amplitude of the noise increases the system undergoes a phase transition from an ordered state in which the particles move collectively in the same direction, to a disordered state in which the particles move independently in random directions. This phase transition was originally thought to be of second order. However, due to a lack of a general formalism to analyze the collective dynamics of the Vicsek model, the nature of the phase transition (i.e. whether it is second or first order) has been brought into question [5].In this letter we show that the nature of the phase transition can depend strongly on the way in which the noise is introduced into these systems. We illustrate this by presenting analytical results on two different network systems that are closely related to the self-propelled particle models. We show that in these two network models the phase transition switches from second to first order when the way in which the noise is introduced changes from the one presented in [4] to the one described in [5].The first network model, which we will refer to as the vectorial network model, consists of a network of N 2D-vectors (represented as complex numbers), {σ 1 = e iθ...

Attractive Bose-Einstein condensates are investigated with numerical continuation methods capturing stationary solutions of the Gross-Pitaevskii equation. The branches of stable (elliptic) and unstable (hyperbolic) solutions are found to meet at a critical particle number through a generic Hamiltonian saddle node bifurcation. The condensate decay rates corresponding to macroscopic quantum tunneling, two and three body inelastic collisions, and thermally induced collapse are computed from the exact numerical solutions. These rates show experimentally significant differences with previously published rates. Universal scaling laws stemming from the bifurcation are derived. [S0031-9007(99)08550-6] PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 47.20.Ky Experimental Bose-Einstein condensation (BEC) in ultracold vapors of 7 Li atoms [1] opened a new field in the study of macroscopic quantum phenomena. Condensates with attractive interactions are known to be metastable in spatially localized systems, provided that the number of condensed particles is below a critical value N c [2]. Various physical processes compete to determine the lifetime of attractive condensates. Among them one can distinguish macroscopic quantum tunneling (MQT) [3,4], inelastic two and three body collisions (ICO) [5][6][7], and thermally induced collapse (TIC) [4,8]. The MQT and TIC contributions were evaluated in the literature using a variational Gaussian approximation to the condensate density. However, this approximation is known to be in substantial quantitative error-e.g., as high as 17% on N c [3,9]-when compared to the exact solution of the Gross-Pitaevskii (GP) equation. Experimentally, the recent observations of Feshbach resonances in BEC of sodium atoms offer new possibilities to investigate the dynamics of condensates with negative scattering lengths close to zero temperature (in the nK range) [10]. Reliable theoretical evaluations of the lifetime of metastable condensates are thus needed for quantitative comparisons with experiments.The basic goal of the present Letter is to numerically compute the bifurcation diagram of the stationary solutions of the GP equation. Both the stable (elliptic) and unstable (hyperbolic) branches of solutions will then be used to obtain decay rates and compare them to the known (Gaussian approximation) ones. At low enough temperature, neglecting the thermal and quantum fluctuations, a Bose condensate can be represented by a complex wave function c͑x, t͒ that obeys the dynamics of the GP equation [11,12]. Specifically, we consider a condensate of N particles of mass m and (negative) effective scattering length a in a radial confining harmonic potential V ͑r͒ mv 2 r 2 ͞2. Using variables rescaled by the natural quantum harmonic oscillator units of time t 0 1͞v and length L 0 ph ͞mv:t t͞t 0 ,x x͞L 0 , andã 4pa͞L 0 , the condensate is described by the action