Front propagation for integro-differential KPP reaction–diffusion equations in periodic media

We study front propagation phenomena for a large class of KPP-type integro-differential reaction–diffusion equations of order α∈(0,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,2)$$\end{document} in oscillatory environments, which model various forms of population growth with periodic dependence. We show that, under an exponential rescaling, the solution develops an isotropic advancing front and converges locally uniformly to zero beyond the front and to the periodic stationary state behind the front. The results are the most general available in this general setting.