Abstract
The dramatic loss of biodiversity constitutes a critical threat to achieving a sustainable future for humanity. It is essential to understand the underpinnings of the spatial dimensions of biodiversity, and how reductions of area and homogenization of habitats can lead to reduced diversity of plant species first, and species higher in the food chain that rely on them. Mathematical models have become a central tool to explore ecological models and serve as a crucial tool to explore hypotheses. Reciprocally, ecology has brought to light a number of deep open mathematical questions that stimulate new developments. We review here a few mathematical challenges and open problems related to biodiversity in spatial ecology. In particular, we show how including spatial heterogeneities in complex nonlinear ecological systems can lead to a rich phenomenology that is still poorly understood, both mathematically and ecologically. Habitat fragmentation and complex, non-convex geometries can also lead to complex patterns, and the determinant of coexistence in complex geometries offers deep questions to the applied mathematician and ecology. We also argue for the inclusion of complex spatial phenomena and processes in models, and review some findings and challenges associated with the fine modeling of fire spread.