Interacting particle systems with mobility and demographic dynamics as biological models

eng 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 individual-based 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.