Delay induced Hopf bifurcation and its control in a fractional p53 protein oscillator model

The p53 protein is essential for cancer prevention. Research on p53 mathematical models is popular since its oscillatory dynamics were observed. However, there is little research on fractional-order p53 oscillators, as well as on Hopf bifurcation control. We therefore introduce fractional-order into a p53 oscillator model and propose a delayed state feedback bifurcation controller. The stability of positive equilibrium and the existence of Hopf bifurcation are studied analytically regarding delay as the bifurcation parameter in both controlled and uncontrolled scenarios. Numerical simulations are consistent with the analytical results. We reveal that a smaller fractional-order facilitates the stability of fixed point and a larger feedback gain advances the Hopf bifurcation point. The work may inspire cancer therapy by artificially controlling the occurrence of p53 oscillations.