Multi-level Modeling

Chairs & Co-Chairs

Fatihcan Atay Max Planck Institute for Mathematics in the Sciences

Laura Hernandez Laboratoire de Physique Théorique et Modélisation (LPTM) UMR8089 CNRS-Université de Cergy-Pontoise

Overview

From genetic and social networks to the ecosphere, we face systems composed of many distinct units that display collective behavior on space and time scales clearly separated from those of individual units at the microscopic level. Among many others, we can mention cellular movements in tissue formation, flock dynamics, social and economic behavior in human societies, speciation in evolution. The complexity of such phenomena manifests itself in the non-trivial properties of the collective dynamics – emerging at the global, population level – with respect to the microscopic level dynamics.
Many answers and insights into such phenomena can and have been obtained by analyzing them through the lens of non-linear dynamics and out-of-equilibrium statistical physics. In this framework, the microscopic level is often assumed to consist of identical units. Heterogeneity is, however, present to varying extents in both real and synthetic populations. Therefore, the existing descriptions also need to encompass variability both at the level of the individual units and at the level of the environment they are embedded in, and to describe the structures that emerge at the population level.
Similarly, homogeneous environment (medium) is a useful approximation for studying collective dynamics. Yet hardly any real, either natural or artificial, environment is homogeneous, thus deeply influencing the structures, dynamics and fates of a population. The variability of the environment applies both on spatial and temporal scales. Examples include filaments and vortices in fluid media, patches and corridors in landscapes, fluctuating resources.
From a methodological point of view, such influences require, at least: the quantification of environmental heterogeneities at multiple levels of organization; the improvement of the formalization of heterogeneity; the identification of the heterogeneity features that are relevant to the population level and the study of population responses to changes in these heterogeneities.
Close interaction between nonlinear physicists and biologists, social scientists and computer scientists has proved to be a key ingredient for advances in handling these subjects.

Hierarchical levels of orgaization over a wide range of space-time scales are ubiquitous in the geosciences, the environment, physics, biology and socio-economic networks. They are a fundamental building block of our 4D world?s complexity. Scale invariance, or scaling for short, is a powerful tool to investigate these structures and to infer properties across scales, instead of dealing with scale-dependent properties. Whereas scaling in time or in space have been investigated in many domains, 4D scaling analysis and modeling are still relatively inchoate, yet indispensable to describe, estimate, understand, simulate and predict the underlying dynamics. Rather complementary to this approach, random dynamical system theory is also a powerful approach for grasping multilevel dynamics. This theory is likely to provide interesting generalizations of what we have learned from deterministic dynamical systems, particularly in the case of bifurcations. Other important domains of investigation are phase transitions, emerging patterns and behaviors which result when we move up in level in the complex 4D fields.