This article studies propagating traveling waves in a class of reaction-diffusion
systems which model isothermal autocatalytic chemical reactions as well as microbial
growth and competition in a flow reactor. In the context of isothermal autocatalytic
systems, two different cases will be studied. The first is autocatalytic chemical reaction
of order m without decay. The second is chemical reaction of order m with a decay of
order n, where m and n are positive integers and m >n≥1. A typical system in autocatalysis
is A+2B→3B and B→C involving two chemical species, a reactant A and an
auto-catalyst B and C an inert chemical species.

The numerical computation gives more accurate estimates on minimum speed of
traveling waves for autocatalytic reaction without decay, providing useful insight in
the study of stability of traveling waves.

For autocatalytic reaction of order m=2 with linear decay n=1, which has a particular
important role in chemical waves, it is shown numerically that there exist multiple
traveling waves with 1, 2 and 3 peaks with certain choices of parameters.