  We are concerned with the problem of decomposing the parameter space of a
parametric system of polynomial equations, and possibly some polynomial
inequality constraints, with respect to the number of real solutions that the
system attains. Previous studies apply a two step approach to this problem,
where first the discriminant variety of the system is computed via a Groebner
Basis (GB), and then a Cylindrical Algebraic Decomposition (CAD) of this is
produced to give the desired computation. However, even on some reasonably
small applied examples this process is too expensive, with computation of the
discriminant variety alone infeasible. In this paper we develop new approaches
to build the discriminant variety using resultant methods (the Dixon resultant
and a new method using iterated univariate resultants). This reduces the
complexity compared to GB and allows for a previous infeasible example to be
tackled. We demonstrate the benefit by giving a symbolic solution to a problem
from population dynamics -- the analysis of the steady states of three
connected populations which exhibit Allee effects - which previously could only
be tackled numerically.
