Non-local hybrid models for collective dynamics

From a mathematical point of view, living systems such as cell aggregates, crowd and swarms, can be regarded as collections of particles characterized by a proper behavior, and by the ability to sense and actively interact with the other individuals and the surrounding environment. In particular, living particles do not respond passively to the rules of inertia but are able to change their individual behavior and affect the collective evolution of the system. In general, there exists several approaches able to describe the collective dynamics of living individuals. First, in microscopic/individual-based models, the single components are represented as localized/discrete units and suitable rules describe their individual behavior and mutual interactions. These approaches thus account for the intrinsic granularity of living systems. However, the usually large amount of individuals involved in interesting self-emergent patterns (such as in morphogenesis and cancer evolution, as well as in pedestrian evacuations) makes it difficult to recover usable synthetic quantitative information about the whole aggregate from the reproduction of the individual behavior. On the other hand, a macroscopic/continuous modelling approach is conversely based on the assimilation of the system to a whole entity distributed in space. The evolution of the collectivity is given by means of non linear conservation laws which directly provide the evolution of average quantities such as density and flux. In this respect, these techniques are able to overcome the above highlighted critical issues posed by an individual-based approach. However, by describing a living system as a whole via phenomenological constitutive relationships rather then by an actual one-to-one interaction basis, the behavior of single entities is not accessible. It thereby results hard to reproduce complex evolutions observable at the aggregate level (for instance, pattern formations) which are generated by microscopic/individual phenomena (such as, cell phenotypic transitions in biological systems, or individual choice of own motion mode in crowd dynamics). In these situations, in order to overcome the difficulties posed by purely micro/macro approaches, it can be convenient to opt for an hybrid modelling approach, i.e., to differentiate the individuals in more groups with specific properties and behavior, and to use different descriptive instances for each subsystem. In particular, the coupling of models based on both localized/discrete and distributed/continuous formulations, might allow to take the advantages of both classical techniques. Taking into account these considerations, the thesis is organized in two parts, respectively dedicated to the illustration of hybrid modelling techniques developed over the course of my Ph.D. to capture the evolution of specific biological systems and pedestrian dynamics. More specifically, we first focus on biological systems whose evolution is regulated by cell phenotypic differentiation processes (e.g., tumor growth and invasion, and zebrafish posterior lateral line development), and then on pedestrian dynamics affected by different types of human perceptions of surrounding individuals. In particular, in both applications, the dynamics of both cells and pedestrians is defined through a phenomenological description of their velocity, which is given by the superposition of a directional contribution and non-local interaction terms accounting for the ability of living particles to perceive and consequently react to the presence of individuals located at a certain distance from them.