Cross diffusion in biology: derivation, analysis and entropy structure

In the first part of this talk, we focus on the proof of the existence of global-in-time weak solutions to the Shigesada-Kawasaki-Teramoto cross-diffusion system in population dynamics for an arbitrary number of population species. This model was first studied by Shigesada, Kawasaki and Teramoto in 1979 in the context of population dynamics describing competing species in a heterogeneous environment, and has since then attracted some attention in the context of the mathematical analysis of cross-diffusion phenomena, segregation effects and pattern formation. We proved that global existence for this model follows under a detailed balance or weak cross-diffusion condition, where the detailed balance condition is related to the symmetry of the mobility matrix in the formal gradient-flow structure, which mirrors Onsager's principle in thermodynamics. The second part of this talk deals with different derivation techniques of the Shigesada-Kawasaki-Teramoto cross-diffusion system and other related cross-diffusion models from the microscopic level. We present how to link at the formal level the entropy structure of this cross-diffusion system satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. Moreover, we present a rigorous proof of a many-particle limit from a moderately interacting stochastic many-particle system to the cross-diffusion model using techniques of K. Oelschläger. Finally, we describe how to generalize and extend these approaches and discuss some open questions.