Traveling waves solutions of isothermal chemical systems with decay

This article studies propagating traveling waves in a class of reaction–diffusion systems which include a model of microbial growth and competition in a flow reactor proposed by Smith and Zhao [17], and isothermal autocatalytic systems in chemical reaction of order m with a decay order n, where m and n   are positive integers and m≠nm≠n. A typical system in autocatalysis is A+2B→3BA+2B→3B (with rate k1ab2k1ab2) and B→CB→C (with rate k2bk2b), where m=2m=2 and n=1n=1, involving two chemical species, a reactant A and an auto-catalyst B   whose diffusion coefficients, DADA and DBDB, are unequal due to different molecular weights and/or sizes. Here a is the concentration density of A, b that of B and C   an inert chemical species. The two constants k1k1 and k2k2 are material constants measuring the relative strength of respective reactions.It is shown that there exist traveling waves when m>1m>1 and n=1n=1 with suitable relation between the ratio DB/DADB/DA, traveling speed c   and rate constants k1k1, k2k2. On the other hand, it is proved that there exists no traveling wave when one of the chemical species is immobile, DB=0DB=0 or n>mn>m for all choices of other parameters.