Phase portraits of two gene networks models

We construct mathematical models of functioning of two few-components gene networks which regulate circadian rhythmes in organisms by means of combinations of positive and negative feedbacks between components of these networks. Both models are represented in the form of non-linear dynamical systems of biochemical kinetics. It is shown that the phase portraits of both models contain exactly one equilibrium point each and in both cases for all values of parameters of these dynamical systems, eigenvalues of their linearization matrices at their equilibrium points are either negative or have negative real parts. Thus, these equilibrium points are stable. We construct their invariant neighborhoods and describe the behavior of trajectories of these systems. Biological interpretations of these results are given as well.