It is common to model such a network with a set of coupled ordinary differential equations (ODEs) or SDEs, describing the reaction kinetics of the constituent parts. Suppose that our regulatory network has  nodes, and let  represent the concentrations of the  corresponding substances at time . Then the temporal evolution of the system can be described approximately by
where the functions  express the dependence of  on the concentrations of other substances present in the cell. The functions  are ultimately derived from basic principles of chemical kinetics or simple expressions derived from these e.g. Michaelis–Menten enzymatic kinetics. Hence, the functional forms of the  are usually chosen as low-order polynomials or Hill functions that serve as an ansatz for the real molecular dynamics. Such models are then studied using the mathematics of nonlinear dynamics. System-specific information, like reaction rate constants and sensitivities, are encoded as constant parameters.
By solving for the fixed point of the system:
for all , one obtains (possibly several) concentration profiles of proteins and mRNAs that are theoretically sustainable (though not necessarily stable). Steady states of kinetic equations thus correspond to potential cell types, and oscillatory solutions to the above equation to naturally cyclic cell types. Mathematical stability of these attractors can usually be characterized by the sign of higher derivatives at critical points, and then correspond to biochemical stability of the concentration profile. Critical points and bifurcations in the equations correspond to critical cell states in which small state or parameter perturbations could switch the system between one of several stable differentiation fates. Trajectories correspond to the unfolding of biological pathways and transients of the equations to short-term biological events. For a more mathematical discussion, see the articles on nonlinearity, dynamical systems, bifurcation theory, and chaos theory.