Models for coupled active--passive population dynamics: mathematical analysis and simulation

In this dissertation, we study models for coupled active--passive pedestrian dynamics from mathematical analysis and simulation perspectives. The general aim is to contribute to a better understanding of complex pedestrian flows. This work comes in three main parts, in which we adopt distinct perspectives and conceptually different tools from lattice gas models, partial differential equations, and stochastic differential equations, respectively. In part one, we introduce two lattice models for active--passive pedestrian dynamics. In a first model, using descriptions based on the simple exclusion process, we study the dynamics of pedestrian escape from an obscure room in a lattice domain with two species of particles (pedestrians). The main observable is the evacuation time as a function of the parameters caracterizing the motion of the active pedestrians. Our Monte Carlo simulation results show that the presence of the active pedestrians can favor the evacuation of the passive ones. We interpret this phenomenon as a discrete space counterpart of the so-called drafting effect. In a second model, we consider again a microscopic approach based on a modification of the simple exclusion process formulated for active--passive populations of interacting pedestrians. The model describes a scenario where pedestrians are walking in a built environment and enter a room from two opposite sides. For such counterflow situation, we have found out that the motion of active particles improves the outgoing current of the passive particles. In part two, we study a fluid-like driven system modeling active--passive pedestrian dynamics in a heterogenous domain. We prove the well-posedness of a nonlinear coupled parabolic system that models the evolution of the complex pedestrian flow by using special energy estimates, a Schauder's fixed point argument and the properties of the nonlinearity's structure. In the third part, we describe via a coupled nonlinear system of Skorohod-like stochastic differential equations the dynamics of active--passive pedestrians dynamics through a heterogenous domain in the presence of fire and smoke. We prove the existence and uniqueness of strong solutions to our model when reflecting boundary conditions are imposed on the boundaries. To achieve this we used compactness methods and the Skorohod's representation of solutions to SDEs posed in bounded domains. Furthermore, we study an homogenization setting for a toy model (a semi-linear elliptic equation) where later on our pedestrian models can be studied.