Towards a mathematical theory of behavioral swarms^*

¹ University of Granada, Departamento de Matemática Aplicada, Spain, Politecnico of Torino, and IMATI CNR, Pavia, Italy.
² Seoul National University, Department of Mathematical Sciences and Research Institute of Mathematics, Seoul 08826, Republic of Korea.
³ Korea Institute for Advanced Study, Hoegiro, 85, Seoul 02455, Republic of Korea.
⁴ Cadi Ayyad university, Faculty of Sciences Semlalia, LMDP, Morocco and uMMISCO (IRD-Sorbonne University), France.

^** Corresponding author: nicola.bellomo@polito.it

Received: 7 July 2020
Accepted: 19 October 2020

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. The second part of the paper shows how the modeling can be developed, beyond the Cucker-Smale approach. This will be illustrated with the aid of numerical simulations in swarms whose movement strategy is sensitive to individual social behaviors. Finally, the presentation looks ahead to research perspectives.