The influence of decay in dynamic models of gene regulatory networks

Abstract

We investigate the influence of decay rates on the dynamics of three-dimensional gene regulatory networks modeled by nonlinear differential equations with logistic sigmoidal regulatory functions. Particular attention is paid to the existence, location, and stability of equilibrium points in systems involving numerous interacting parameters. An explicit analytical representation of a central equilibrium is obtained under suitable relations between the regulatory weights, thresholds, and decay coefficients, allowing its location to be determined without solving the corresponding nonlinear algebraic system numerically. General formulas for the characteristic polynomial of the linearized system are derived, and spectral conditions for instability are established for several classes of regulatory interactions. We demonstrate how variations in decay rates can change the number and stability of equilibria, transform oscillatory dynamics, and influence the emergence or suppression of complicated attractors. The possibility of chaotic behavior is investigated through eigenvalue configurations, Lyapunov exponents, and the Kaplan-Yorke dimension. Numerical examples illustrate the sensitivity of the dynamics to decay parameters, including the suppression of previously observed chaotic behavior. The results provide an analytical framework for investigating equilibrium configurations, stability transitions, and complex dynamics in parametrized gene regulatory networks.