Alignment in a fish school: a mixed Lagrangian–Eulerian approach

Abstract

Social organization is one of the fundamental aspects of animal behavior, and has received attention both from experimental and theoretical perspectives. Examples of social groups appear at every size scale from the microscopic aggregates of mammalian cells (such as fibroblasts) to macroscopic herds of wildbeast, flocks of birds, and fish schools. There are two general frameworks when modeling such problems: the Lagrangian viewpoint and the Eulerian one. In this paper, we use both the approaches in the study of fish alignment. An individual-based model (IBM) (Lagrangian) provides a virtual world where fish forming a fish school try to adopt a common angular position. Fish are assumed to lie in horizontal planes, an individual angular position is the angle made by the oriented axis associated with the individual (tail to head) with a fixed direction. Two main forces are acting, a force of alignment, whose strength is assumed to be fixed in a given experiment but may be modified, and a force of dispersion, accounting for all disturbances. A transition from dispersion-dominant to alignment-dominant can be observed in the IBM experiments. A related PDE model (Eulerian) is used to determine the transition with sufficient accuracy.

De l'équation mathématique au programme informatique, dialogue entre mathématiciens et informaticiens

“…La dynamique d'alignement d'individus entrant dans un réseau d'interactions est illustrée dans le cas d'un banc de poissons (Adioui et al, 2003). Il y a compétition entre deux mécanismes locaux : la tendance des poissons à s'aligner sur les autres et l'agitation globale.…”

“…Certaines modélisations font intervenir un nombre fixe d'agents de nature identique, comme les automates cellulaires, des modélisations dites lagrangiennes de phénomènes d'écoulements (Servat et al, 1999), de formation de bancs de poissons (Adioui et al, 2003). En ce cas, la formalisation de ce qu'est l'espace des états possibles du système est simple.…”

A framework for models of movement in geographic space

“…In contrast, the Lagrangian approach usually quantifies the movement of individual macro-organisms over larger spatio-temporal scales (Nathan et al 2008, Turchin 1998, Benhamou 2004. A mixed Eulerian-Lagrangian approach has been proposed by Adioui et al (2003) for the alignment of fish schooling A u t h o r ' s m a n u s c r i p t July 28, 2015 15:25 International Journal of Geographical Information Science "movement special issue" 4…”

Effect of vision angle on the phase transition in flocking behavior of animal groups

“…Well-known examples are found in populations such as large schools of fish, gathering of birds, swarming of ants and herding of sheep [2][3][4][5][6][7]. Biologically, it has been known that the flocking behavior is advantageous for survival of a population [8][9][10]: reducing the risk of capture by predator, higher mating efficiency, easier search for food, efficient learning of external stimuli, and reducing overall aggression [11][12][13][14].…”