Bifurcation analysis of a two-dimensional p53 gene regulatory network without and with time delay

In this paper, the stability and bifurcation of a two-dimensional p53 gene regulatory net-work without and with time delay are taken into account by rigorous theoretical analyses and numerical simulations. In the absence of time delay, the existence and local stability of the positive equilibrium are considered through the Descartes' rule of signs, the determinant and trace of the Jacobian matrix, respectively. Then, the conditions for the occurrence of codimension-1 saddle-node and Hopf bifurca-tion are obtained with the help of Sotomayor's theorem and the Hopf bifurcation theorem, respectively, and the stability of the limit cycle induced by hopf bifurcation is analyzed through the calculation of the first Lyapunov number. Furthermore, codimension-2 Bogdanov-Takens bifurcation is investigated by calculating a universal unfolding near the cusp. In the presence of time delay, we prove that time delay can destabilize a stable equilibrium. All theoretical analyses are supported by numerical sim-ulations. These results will expand our understanding of the complex dynamics of p53 and provide several potential biological applications.