Memory-driven bifurcation analysis of a multi-order fractional biological system with coupled genetic-epigenetic regulation

This study develops a fractional-order biological model incorporating genetic and epigenetic regulation to investigate the dynamics of cellular stress responses. The model employs Caputo fractional derivatives to capture multi-scale memory effects, representing fast molecular interactions and slower epigenetic processes. A nonlinear incidence function governed by gene products and epigenetic feedback introduces complex coupling, enabling rich bifurcation behavior. The mathematical well-posedness of the system is established, ensuring existence, uniqueness, positivity, and boundedness of solutions. Equilibrium and stability analyses are performed using the basic reproduction number R0, while sensitivity analysis identifies key parameters influencing system dynamics. Bifurcation analysis reveals transcritical and Hopf transitions driven by memory effects and regulatory feedback. To obtain accurate solutions, the Laplace Residual Power Series (LRPS) method is proposed and compared with the classical RPS method. Numerical results demonstrate improved convergence and stability of LRPS for fractional systems. Overall, the proposed framework provides a unified approach to studying memory-driven biological dynamics under genetic and epigenetic control.