Autocatalytic reactions are chemical reactions in which at least one of
the products is also a reactant. This "feed-back loop" yields a system of nonlinear
polynomial equations. A famous example is called the Brusselator [1], [2]. Although the
equations describing the classic Brusselator are trivial, the basic idea generalizes to more
interesting situations [1], such as
a + x1
^{2}*y1 - b*x1 - x1 + dx (x2 - x1),
b*x1 - x1
^{2}*y1 + dy (y2 - y1),
a + x2
^{2}*y2 - b*x2 - x2 + dx (x1 - x2),
b*x2 - x2
^{2}*y2 + dy (y1 - y2)
We will examine the systems of equations that result from two- and three-dimensional configurations
of interacting Brusselators. We have up to eight equations in eight variables and up to twelve
parameters. We find that all are solvable with Dixon resultant methods. We will describe how
various implementations of Groebner Bases fail on all but the simplest cases. We will show other
examples of autocatalytic reactions.
[1] http://en.wikipedia.org/wiki/Autocatalytic_reaction
[2] Shaun Ault, http://fordham.academia.edu/ShaunAult/Papers/83373/Dynamics-of-the-Brusselator