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We prove homogenization for reaction–advection–diffusion equations with KPP reactions, in the time-periodic spatially stationary ergodic setting, and find an explicit formula for the homogenized dynamic. We also extend this result to models with non-local diffusion and KPP reactions. 

We study two types of divergence-free fluid flows on unbounded domains in two and three dimensions—hyperbolic and shear flows—and their influence on chemotaxis and combustion. We show that fast spreading by these flows, when they are strong enough, can suppress growth of solutions to PDE modeling these phenomena. This includes prevention of singularity formation and global regularity of solutions to advective Patlak-Keller-Segel equations on R 2 \mathbb {R}^2 and R 3 \mathbb {R}^3 , confirming numerical observations by Khan, Johnson, Cartee, and Yao [Involve 9 (2016), pp. 119–131], as well as quenching in advection-reaction-diffusion equations.