Multiscale kinetic structures for the collective dynamics of living systems

This paper develops a conceptual extension of the Kinetic Theory of Active Particles, building upon the framework introduced in [N. Bellomo, D. Burini and J. Liao, New trends in kinetic theory toward the complexity of living systems, Math. Models Methods Appl. Sci.36 (2026) 341–397]. Living systems cannot be adequately described within classical single-scale paradigms, even when refined. To overcome this limitation, we introduce a Multiscale Kinetic Theory of Active Particles (MS-KTAP), in which a sub-microscopic scale of interacting entities is incorporated into the description of collective dynamics. In this framework, the activity variable is interpreted as an emergent quantity arising from lower-scale regulatory mechanisms and influenced by interactions across higher scales. The proposed framework captures key features of living systems — heterogeneity, adaptive decision-making, nonlinear and non-conservative interactions, spatial dynamics, and cross-scale feedback — within a unified mathematical structure. Competition and cooperation are thus described across multiple levels of organization. The first part of the paper derives the mathematical framework, while the second shows how specific models can be obtained. The paper concludes with perspectives on further developments, including possible integrations with scientific machine learning.