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Monday July 14, 2025

Austin Orgeron Advanced Topic Exam
Nonlinear diffusion in fragmented habitats
- Austin Orgeron,
Time: 9:00 Room: Dirac 216
Abstract/Desc: Habitat fragmentation — often driven by human development — disrupts natural environments and alters how species move and interact. In such fragmented landscapes, habitats can be represented as networks of interconnected patches, where individuals disperse and compete for limited space. We model these dynamics using a nonlinear random walk that accounts for the habitat’s heterogeneous structure. Stronger competitors tend to accumulate in central, well-connected patches, while weaker ones are displaced toward the periphery. Analytical results and simulations reveal how the interplay between competition and network structure drives this spatial sorting. In large networks, segregation becomes more pronounced: strong individuals disappear from peripheral patches, and weak ones from central regions — a pattern that could foster evolutionary divergence. We further extend the model to multiple species by incorporating generalized logistic growth with Allee effects, capturing how low-density populations benefit from cooperation. This richer framework demonstrates how both competition and cooperation shape species distribution and persistence in fragmented habitats.

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This Week in Mathematics