A quest toward a mathematical theory of the dynamics of swarms

Abstract

This paper presents a unified mathematical theory of swarms where the dynamics of social behaviors interacts with the mechanical dynamics of self-propelled particles. The term behavioral swarms is introduced to characterize the specific object of the theory which is subsequently followed by applications. As concrete examples for our unified approach, we show that several Cucker-Smale type models with internal variables fall down to our framework. Subsequently the modeling goes beyond the Cucker-Smale approach and looks ahead to research perspectives. Plan of the paperThe celebrated paper by Cucker and Smale [20] has boosted a huge literature on the modeling, qualitative analysis, and computational applications of the mathematical theory of swarms. Namely, models which describe, within a pseudo-Newtonian framework, the mechanics of many interacting self-propelled particles. Interactions are generally nonlocal and nonlinearly additive. These produce accelerations, where inertia is hidden in the interaction parameters rather than being explicitly taken into account. The mathematical literature in the field has been reviewed and critically analyzed in [3], specifically treated in Sections 5 and 6 which are devoted to the theory of swarms. This topic has been related to the dynamics of crowds as well as to computational models to treat kinetic and multi-agent systems.After Cucker-Smale's (C-S)'s seminal work, the (C-S) model has been extensively studied in literature, for example, we refer to [18] and references therein for mathematical aspects of the model. Among them, we briefly discuss some recent works incorporating social behaviors together with mechanical aspect of the model. Original C-S model describes the temporal evolution of the mechanical variables (positions and momentum of C-S particles), whereas in the modeling for the collective dynamics of biological and social complex systems, one needs to take into account of internal variables such as temperature, spin and excitation [24,27,29] etc, see Section 3.Recently, in [26], the authors found that the one-dimensional C-S model can be converted into the first-order nonlinear consensus model with monotone coupling function on the real line. Then, in 1 arXiv:2006.12932v1 [nlin.AO]

Numerical Simulation of a Multiscale Cell Motility Model Based on the Kinetic Theory of Active Particles

“…The obtained results confirmed that cells tend to follow paths and to align according to the ECM structure. In this sense, it is worth noticing the analogy with crowd dynamics [33], vehicular traffic [34], or swarming phenomena [37], where individual entities also receive some stimulus from the surrounding environment, interact among themselves, and generate certain emergent behaviors.…”

“…Some specific tools of the KTAP approach have so far been used in many other applications. For instance, competition between tumor and immune cells [30] and social dynamics [31,32], crowd dynamics [17,18,[33][34][35][36], and swarming [37]. Thus, the contents of this paper may be useful for the modeling of other phenomena in life sciences.…”

Singular Cucker–Smale Dynamics

“…Remark 2.2 Proposition 2.1 is based only on the structure of the CS system, and holds for any non-negative communication weight ψ. The symmetry of x → ψ(|x|) allows to swap indexes in (6).…”

“…al. [70] form 1995, extensive studies were carried out in various directions including: asymptotics [24,31], pattern formation [54,55], collision avoidance [1,22], variants of the model with preferences [23,45], leadership [21,59] and additional deterministic or stochastic forces [6,7,15,28]. Further directions include the study of the kinetic [9,31,32] and hydrodynamic [29,30,40] limits of the CS particle system and their coupling with classical equations of hydrodynamics [2,3,17,52].…”

On Entropy Dynamics for Active “Living” Particles

“…The interested reader is referred to the book [3] for theoretical aspects and for the rationale to develop specific applications which have been produced to model, for instance, the dynamics of collective learning [4], the immune competition [5], vehicular traffic [6][7][8], the dynamics of crowds [9][10][11], and the collective motion of swarms [12].…”