Propagation of KPP equations with advection in one-dimensional almost periodic media and its symmetry

Reaction-diffusion equations in unbounded domain are used to study the propagation phenomena of biological species. When propagation can happen in different directions, an interesting question arises: In which direction is propagation the fastest?For the one-dimensional KPP equation in almost periodic media with advection: {ut=(a(x)ux)(x)+b(x)ux+f(x,u)t > 0,x is an element of R, u(0,x)=u0(x)is an element of[0,1] is nonzero with compact support, (?) let omega(+) and omega(-) be the spreading speeds of (?) in the positive and negative directions respectively. The above question becomes this: Which is larger, omega(+) or omega(-)? In this paper, after establishing the existence of omega(+) and omega(-), we give a complete answer to this question: sgn(omega(-)-omega(+))=sgn(lim(x ->infinity)?1/x integral 0xb(s)/a(s)ds). (C) 2022 Elsevier Inc. All rights reserved.