Jonathan Potts - Aggregation-diffusion in heterogeneous environments

Описание: This lecture was part of the Workshop on "Modelling Diffusive Systems: Theory & Biological Applications (MoDiS)" held at the ESI January 26 to 30, 2026.

Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical way. However, most existing studies do not account for the effect of the underlying environment on organism movement. In reality, the environment is often a key determinant of emergent space use patterns, albeit in combination with collective aspects of motion. Here, I will present work towards this end, studying aggregation-diffusion equations in a heterogeneous environment in one spatial dimension. Under certain assumptions, it is possible to find exact analytic expressions for the steady-state solutions when diffusion is quadratic. Minimising the associated energy functional across these solutions provides a rapid way of determining the likely emergent space use pattern, which can be verified via numerical simulations. This energy-minimisation procedure is applied to a simple test case, where the environment consists of a single clump of attractive resources. Here, self-attraction and resource-attraction combine to shape the emergent aggregation. Two counter-intuitive findings emerge from these analytic results: (a) a non-monotonic dependence of clump width on the aggregation width, (b) a positive correlation between self-attraction strength and aggregation width when the resource attraction is strong.