Introduction to Reaction-Diffusion Equations

This book introduces some basic tools and discusses recent progress in reaction-diffusion models motivated by spatial ecology and evolution. The first part introduces the basics of the theory of monotone dynamical systems, with a focus on the single species model. The second part concerns the competition of multiple species. It includes detailed discussions on the competition model introduced by Dockery et al. (1998), their proof of the two species case, conjectures and recent progress on the N species case, evolutionarily stable strategies and other notions from adaptive dynamics, related results for stream populations, the basics of the theory of principal Floquet bundles, and elements of the theory of dynamical systems.The third part discusses a mutation-selection model introduced by Diekmann et al. (2005) concerning the competition of infinitely many species, Perthame and Souganidis' result (2016) on stationary solutions, and related results on stream populations (Hao et al. (2019)), as well as recent progress on the time-dependent problem, including uniqueness in constrained Hamilton-Jacobi equations (Calvez et al. (2020)).