Special Issue “Kinetic Theory and Swarming Tools to Modeling Complex Systems—Symmetry problems in the Science of Living Systems”—Editorial and Research Perspectives

This editorial paper presents a special issue devoted to the development of mathematical tools from kinetic and swarms theory to the modeling and simulations of the dynamics of living systems constituted by very many interacting living entities. Applications refer to several fields: collective learning, behavioral economy, multicellular systems, vehicular traffic, and human crowds. A forward look to research perspectives is focused on the conceptual links between swarms methods and the kinetic theory approach

Towards a mathematical theory of behavioral swarms

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

On the mathematical theory of behavioral swarms emerging collective dynamics

This paper considers a system consisting of a number of interacting living entities whose state at the microscopic scale is heterogeneously distributed among the said entities. This state includes, in addition, the classical mechanical variables, such as position and velocity, also a behavioral variable which is modi ed by interactions. It is shown how the pioneering ideas proposed in Bellomo et al. [Towards a mathematical theory of behavioral swarms, ESAIM Control Optim. Calc. Var. 26 (2020) 125] can be developed towards modeling behavioral swarms within a quest towards a mathematical theory of living systems. The rst part of the paper presents a qualitative analysis of the emerging behaviors predicted by the model in aforementioned work. Some simulations follow to depict the said emerging behaviors. The last part of the paper is devoted to derive a new, more general theory in view of applications to model living systems.University of GranadaNational Research Foundation of Korea NRF-2020R1A2C3A0100388

Emergent behaviors of rotation matrix flocks

We derive an explicit form for the Cucker-Smale (CS) model on the special orthogonal group $\mathrm{SO}(3)$ by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and La Salle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the $\omega$-limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results

Towards a mathematical theory of behavioral human crowds

Nicola Bellomo acknowledges the support of the University of Granada, Project Modeling in Nature MNat from micro to macro, https://www.modelingnature.org.This paper has been partially supported by the MINECO-Feder (Spain) research Grant Number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Project PY18-RT-2422, A-FQM-311-UGR18, and B-FQM-580-UGR20. Livio Gibelli, gratefully acknowledges the financial support of the Engineering and Physical Sciences Research Council (EPSRC) Under Grants EP/N016602/1, EP/R007438/1. Annalisa Quaini acknowledges support from the Radcliffe Institute for Advanced Study at Harvard University where she has been a 2021-2022 William and Flora Hewlett Foundation Fellow. Alessandro Reali acknowledges the partial support of the MIUR-PRIN Project XFAST-SIMS (No. 20173C478N).The first part of our paper presents a general survey on the modeling, analytic problems, and applications of the dynamics of human crowds, where the specific features of living systems are taken into account in the modeling approach. This critical analysis leads to the second part which is devoted to research perspectives on modeling, analytic problems, multiscale topics which are followed by hints towards possible achievements. Perspectives include the modeling of social dynamics, multiscale problems and a detailed study of the link between crowds and swarms modeling.University of Granada, Project Modeling in Nature MNat from micro to macroSpanish Government RTI2018-098850-B-I00Junta de AndaluciaEuropean Commission PY18-RT-2422 A-FQM-311-UGR18 B-FQM-580-UGR20UK Research & Innovation (UKRI)Engineering & Physical Sciences Research Council (EPSRC) EP/N016602/1 EP/R007438/1Radcliffe Institute for Advanced Study at Harvard UniversityMinistry of Education, Universities and Research (MIUR) 20173C478

Emergent dynamics of various Cucker–Smale type models with a fractional derivative

In this paper, we demonstrate emergent dynamics of various Cucker–Smale type models, especially standard Cucker–Smale (CS), thermodynamic Cucker–Smale (TCS), and relativistic Cucker–Smale (RCS) with a fractional derivative in time variable. For this, we adopt the Caputo fractional derivative as a widely used standard fractional derivative. We first introduce basic concepts and previous properties based on fractional calculus to explain its unusual aspects compared to standard calculus. Thereafter, for each proposed fractional model, we provide several sufficient frameworks for the asymptotic flocking of the proposed systems. Unlike the flocking dynamics which occurs exponentially fast in the original models, we focus on the flocking dynamics that occur slowly at an algebraic rate in the fractional systems