A Conceptual Model for Milling Formations in Biological Aggregates

Abstract

Collective behavior of swarms and flocks has been studied from several perspectives, including continuous (Eulerian) and individual-based (Lagrangian) models. Here, we use the latter approach to examine a minimal model for the formation and maintenance of group structure, with specific emphasis on a simple milling pattern in which particles follow one another around a closed circular path.We explore how rules and interactions at the level of the individuals lead to this pattern at the level of the group. In contrast to many studies based on simulation results, our model is sufficiently simple that we can obtain analytical predictions. We consider a Newtonian framework with distance-dependent pairwise interaction-force. Unlike some other studies, our mill formations do not depend on domain boundaries, nor on centrally attracting force-fields or rotor chemotaxis.By focusing on a simple geometry and simple distant-dependent interactions, we characterize mill formations and derive existence conditions in terms of model parameters. An eigenvalue equation specifies stability regions based on properties of the interaction function. We explore this equation numerically, and validate the stability conclusions via simulation, showing distinct behavior inside, outside, and on the boundary of stability regions. Moving mill formations are then investigated, showing the effect of individual autonomous self-propulsion on group-level motion. The simplified framework allows us to clearly relate individual properties (via model parameters) to group-level structure. These relationships provide insight into the more complicated milling formations observed in nature, and suggest design properties of artificial schools where such rotational motion is desired.

Swarming, schooling, milling: phase diagram of a data-driven fish school model

“…For larger values of the alignment parameter, despite having higher overall values for the milling parameter, the transition does not occur. This indicates that in the model studied here, like in many of those proposed before [25], milling dynamics does not emerge in the arbitrarily large infinite-size limit.…”

Social behaviour and collective motion in plant-animal worms

“…In processionary caterpillars and army ants, circular milling is underpinned by individuals laying trails that others follow and reinforce [15,16]. In fish, it occurs because of rules of attraction and alignment [11,[20][21][22]. A recent sophisticated analysis of collective motion in glass prawns demonstrates that a weak form of circular milling can occur in an annular arena because these supposedly non-social Crustacea influence one another's movements even after a substantial delay following an encounter [23].…”

Collective motion from local attraction

“…A number of models of flocking, often referred to as self-propelled particle (SPP) models, have been constructed and analysed in recent years (Aoki, 1982;Huth et al, 1991;Vicsek et al, 1995;Grégorie et al, 2003;Czirók et al, 1997;Couzin et al, 2002;Czirók et al, 1999;D'Orsogna et al, 2006;Wood et al, 2007;Romanczuk et al, 2008;Lukeman et al, 2009). The main difference between the different models is the form of the local interaction rule and how neighbouring particles affect each other.…”