报告题目：Non-homogeneous reaction-diffusion equations
EHESS, PSL Research University – Paris
报告摘要： In this series of lectures, I will present three topics on reaction-diffusion equations emphasizing the effects of spatially inhomogeneous features.
1. The effect of domain shape on reaction-diffusion equations. I will discuss some reaction-diffusion equations of bistable type motivated by biology and medicine. The aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves. I will present various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve new qualitative results for some non-linear elliptic and parabolic partial differential equations. Much of the qualitative information is gleaned thanks to the moving plane and sliding methods in the spirit of earlier work with Louis Nirenberg. I report here on joint work with Juliette Bouhours and Guillemette Chapuisat.
2. The effect of diffusion on a line on Fisher KPP propagation. I will present a system of equations describing the effect of including a line (the “road”) with a specific diffusion on biological invasions in the plane (the “field”). Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. I will discuss several results such as the asymptotic shape of propagation, or further effects such as transport, reaction on the road or the influence of various parameters. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.
3. Predators-prey model with competition: emergence of ter r itor iality and packs in animal behavior . I report here on a series of joint works with Alessandro Zilio about systems of predators interacting with a single prey. This system is related to models arising in multi-phase Bose-Einstein condensates or spatially distributed chemical reactions. We analyze the situation of predators like wolves that can divide up into several hostile packs. The questions are to understand the conditions under which predators segregate into packs, whether there is an advantage to have such hostile packs, and to compare the various territory configurations that arise in this context. Mathematically, we focus on the analysis of stationary states, stability issues, and asymptotics of the system when the competition parameter becomes unbounded.