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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 adapting and 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 {\em 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.