  The aim of this paper is to prove results about the existence and stability
of multiple steady states in a system of ordinary differential equations
introduced by R. Lev Bar-Or to model the interactions between T cells and
macrophages. Previous results showed that for certain values of the parameters
these equations have three stationary solutions, two of which are stable. Here
it is shown that there are values of the parameters for which the number of
stationary solutions is at least seven and the number of stable stationary
solutions at least four. This requires approaches different to those used in
existing work on this subject. In addition, a rather explicit characterization
is obtained of regions of parameter space for which the system has a given
number of stationary solutions.
