  In this paper we describe a combined combinatorial/numerical approach to
studying equilibria and bifurcations in network models arising in Systems
Biology. ODE models of the dynamics suffer from high dimensional parameters
which presents a significant obstruction to studying the global dynamics via
numerical methods. The main point of this paper is to demonstrate that
combining classical techniques with recently developed combinatorial methods
provides a richer picture of the global dynamics despite the high parameter
dimension.
  Given a network topology describing state variables which regulate one
another via monotone and bounded functions, we first use the Dynamic Signatures
Generated by Regulatory Networks (DSGRN) software to obtain a combinatorial
summary of the dynamics. This summary is coarse but global and we use this
information as a first pass to identify "interesting'' subsets of parameters in
which to focus. We construct an associated ODE model with high parameter
dimension using our {\em Network Dynamics Modeling and Analysis} (NDMA) Python
library. We introduce algorithms for efficiently investigating the dynamics in
these ODE models restricted to these parameter subsets. Finally, we perform a
statistical validation of the method and several interesting dynamical
applications including finding saddle-node bifurcations in a 54 parameter
model.
