Interacting particle systems with mobility and demographic dynamics as biological models

Interacting particle systems serve as a fundamental tool for modeling and understanding
the behavior of complex biological systems. By focusing on how individual agents interact,
these models provide valuable insights into emergent phenomena like spatial organization,
collective behavior, and evolutionary dynamics. In this thesis, we use individualbased
models to investigate systems of active and passive particles, exploring their collective
dynamics and structural organization.

More in detail, we first study (Chapter 2 of the thesis) the spatial distribution and dynamics
of Brownian particles, incorporating stochastic birth-death processes, active movement,
and spatial constraints. Using numerical simulations, we study how critical parameters,
including diffusion rates, activity levels, and reproduction rates, affect phenomena like
phase transitions, clustering, and motility-induced phase separation. These findings help
to understand some underlying mechanisms that drive the organization of biological systems
on microscopic scales.

The second study (Chapter 3) broadens this analysis to multi-type systems, emphasizing
the balance between competition and coexistence in binary particle mixtures. By
combining computational simulations with theoretical models inspired by Lotka–Volterra
dynamics, we explore how factors such as mobility, random demographic variations, and
interaction rules influence population stability and species dominance.

The third study (Chapter 4) explores how game-theoretical interactions influence spatial
population dynamics and stability. We analyze the conditions that promote coexistence,
competitive exclusion, or dominance, focusing on the role of environmental factors and
interaction rules in determining these outcomes.

Overall, this thesis demonstrates the versatility and effectiveness of interacting particle
systems as a framework for studying complex biological phenomena. By connecting theoretical
approaches with practical applications, this thesis contributes via simple models,
to analyzing some real-world biological systems.