Codimension-3 Bifurcation in the p53 Regulatory Network Model

In this paper, a p53-Mdm2 mathematical model is analyzed to understand the biological implications of feedback loops in a p53 system. Results show that the model can undergo four types of codimension-3 Bogdanov–Takens bifurcations, including cusp, saddle, focus and elliptic. Specifically, we find new phenomena including the coexistence of four positive equilibria, two limit cycles, the coexistence of three stable states (two stable equilibria and one stable limit cycle, or three stable equilibria), a heteroclinic loop enclosing a smaller stable limit cycle and a larger stable limit cycle. These findings extend the understanding of the complex dynamics of the p53 system, and can provide some potential biological applications.