A Continuum Three-Zone Model for Swarms

Abstract

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.

Patterned dynamics of delay-coupled swarms with random communication graphs

“…6 is understood as follows. The radius of the center of mass computed from (23) and (24) assumes that agent positions' deviate only slightly from the center of mass. However, as the mean coupling coefficient decreases the agents become spread out over an extended arc (as seen in Fig.…”

“…Existing literature on the subject provides a great selection of both individualbased [5], [6], [20], [4] and continuum models [21], [22], [2] of group motion. Many biologically-inspired models are based on "zones" of attraction, repulsion, and/or alignment interactions between agents [23], [24], [25]. Setting different ranges for attraction/repulsion interactions between all agents in the swarm or introducing heterogeneous dynamics can lead to interesting behaviors, such as splitting of a large swarm into smaller groups [26], [27], [28].…”

Measuring Information Dynamics in Swarms

“…This indicates that communications within the system are less informative, thus leading to less well-formed cascades. Finally, we note that the final configurations of the models are very different (see Miller et al (2011) for details), configurations driven by the differences in the collective memory and communication within the swarms.…”

First-order aggregation models with alignment

“…Models of the latter type are typically referred to as "three-zone" models, as each particular interaction type acts at different ranges (repulsion acts at short distances, while alignment and attraction are present at intermediate and long ranges, respectively). This class of models has had many successful applications in biological and sociological modelling [CKJ + 02, Rey87,MKJNR12].…”

“…Addressing this challenge is one of the major goals of the present paper. The non-uniqueness of solutions to models of type (1.1) has been noted in [MKJNR12], but no resolution was offered. To our best knowledge, the present paper is the first systematic study of a first-order continuum model for aggregation that includes both attractive/repulsive and alignment interactions.…”