Periodic Trajectories of Nonlinear Circular Gene Network Models

Abstract

The article addresses the qualitative analysis of the two dynamical systems simulating circular gene network functioning. The equations of a three-dimensional dynamical system contain some monotonically decreasing smooth functions that describe negative feedback. A six-dimensional dynamical system consists of three equations with monotonically decreasing smooth functions and three equations with monotonically increasing smooth functions that characterize negative and positive feedbacks. In both models the process of degradation is described by smooth nonlinear functions. We construct invariants domains in order to localize cycles for both systems, show that each of the two systems has a unique stationary point in the invariant domain, and find the conditions for this point to be hyperbolic. The main result is the proof of existence of a cycle in the invariant subdomain from which the trajectories cannot pass to other subdomains obtained by discretization of the phase portrait. The cycles of three- and six-dimensional systems bound the two-dimensional invariant surfaces including the trajectories of the systems.