Based on mathematical ecological models, this report reviews the impact of spatially heterogeneous environments on the persistence and distribution of biological populations. The report aims to elucidate the interplay between population density and key factors, including diffusion coefficients, resource availability, and habitat structure. The study first investigates the ecological consequences of dispersal strategies within environments characterized by uneven resource distribution, demonstrating the monotonic dependence of peak population densities on diffusion rates. Furthermore, analysis of a consumer-resource system indicates that under resource-limited conditions, the ecosystem converges to a globally asymptotically stable state of coexistence. Building on these findings, the report additionally explores the constraints imposed by domain geometry on the spatial patterning of populations.

In this talk we present different modeling approaches to describe and analyse the dynamics of large pedestrian crowds. We start with the individual microscopic description and derive the respective partial differential equation (PDE) models for the crowd density. Hereby we are particularly interested in identifying the main driving forces, which relate to complex dynamics such as lane formation in bidirectional flows. We then analyse the time-dependent and stationary solutions to these models, and provide interesting insights into their behavior at bottlenecks. We conclude by discussing how the Bayesian framework can be used to estimate unknown parameters in PDE models using individual trajectory data.