Over the years, the study of collective dynamics has attracted growing interest across different scientific disciplines. Aggregate motions can be easily observed in flock of birds, schools of fish, human crowds, but also cells exhibit collective behaviors in different biological processes characterizing the human body. One of the main features of collective cell migrations is the influence of a chemical signal, guiding the motion towards higher concentrations of chemicals.

In this talk I will present results and open problems of a general class of hybrid ODE-PDE models, gathering the advantages of multiscale descriptions (see [1] for the most general framework and references therein). In this context, cells are modeled as discrete entities and their dynamics is described by a system of second-order ODEs, while the chemical signal influencing the motion is modelled as a continuous concentration solving a diffusive equation. The particular coupling of the two scales raises some issues that have been analytically investigated over the last years. Concerning applications, I will present recent advancements on a hybrid mathematical model inspired by Cancer-on-chip experiments [2]. In particular,  a macroscopic model, based on a pressureless Euler-type system with nonlocal chemotaxis, has been rigorously derived from the underlying microscopic dynamics [3], and considered to simulate cell migration in the context of two competitive populations of cells [4]. Numerical simulations of the presented models will be shown, including one-dimensional and multidimensional scenarios.

[1] M. Menci, M. Papi, M. Porzio and F. Smarrazzo. On a coupled hybrid system of nonlinear differential equations with a nonlocal concentration. Journal of Differential Equations, Vol. 361, pp. 288-338, 2023.

[2] G. Bretti, E. Campanile, M. Menci, R. Natalini.  A scenario-based study on hybrid PDE-ODE model for Cancer-on-chip experiment , In: Problems in Mathematical Biophysics: A Volume in Memory of Alberto Gandolfi. Cham: Springer Nature Switzerland, 2024.

[3] R. Natalini and T. Paul. The mean-field limit for hybrid models of collective motions with chemotaxis. SIAM Journal on Mathematical Analysis, 55(2), 2023.

[4] M. Menci, R. Natalini, T. Tenna. Numerical study on a multi-dimensional pressureless Euler-type model with non-local interactions and chemotaxis for collective cell migration. Submitted, 2025.