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

Abstract

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∈R,u(0,x)=u0(x)∈[0,1] is nonzero with compact support, let ω+ and ω− be the spreading speeds of (⁎) in the positive and negative directions respectively. The above question becomes this: Which is larger, ω+ or ω−? In this paper, after establishing the existence of ω+ and ω−, we give a complete answer to this question: sgn(ω−−ω+)=sgn(limx→∞⁡1x∫0xb(s)a(s)ds).