Author(s)

Anael Verdugo

Department of Mathematics, California State University, Fullerton, USA.

This paper presents a bifurcation study of a mRNA-protein network with negative feedback and time delay. The network is modeled as a coupled system of N ordinary differential equations (ODEs) and N delay differential equations (DDEs). Linear analysis of the stable equilibria shows the existence of a critical time delay beyond which limit cycle oscillations are born in a Hopf bifurcation. The Poincaré-Lindstedt perturbation method is applied to the nonlinear system, resulting in closed form approximate expressions for the amplitude and frequency of oscillation. We confirm our perturbation analysis results by numerically simulating the full 2N-dimensional nonlinear system for N = 2, 7, 15, and 30. Our results show that for small perturbations the equilibrium undergoes a supercritical Hopf and the system exhibits stable periodic solutions. Furthermore, our closed form numerical expressions exhibit the importance of the network’s negative feedback and nonlinear effects.

Bifurcation Analysis, Hopf, Delay Differential Equations

Verdugo, A. (2023) Bifurcation Analysis of a Nonlinear Genetic Network Model with Time Delay. Journal of Applied Mathematics and Physics, 11, 2252-2266. doi: 10.4236/jamp.2023.118146.