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Explore aggregation-diffusion equations as mathematical tools for modeling biological aggregations in heterogeneous environments through this 24-minute conference lecture. Discover how collective movement mechanisms of organisms link to their emergent space use patterns while accounting for environmental effects that significantly influence organism movement and spatial distribution. Learn about analytical approaches to studying aggregation-diffusion equations in one-dimensional heterogeneous environments, including exact analytic expressions for steady-state solutions under quadratic diffusion assumptions. Examine energy functional minimization techniques for determining emergent space use patterns and their verification through numerical simulations. Investigate a test case involving a single clump of attractive resources where self-attraction and resource-attraction combine to shape aggregation patterns. Uncover two counter-intuitive findings: the non-monotonic relationship between clump width and aggregation width, and the positive correlation between self-attraction strength and aggregation width under strong resource attraction conditions. Gain insights into advanced mathematical modeling techniques that bridge theoretical mathematics with practical biological applications in understanding how organisms distribute themselves in complex environments.
