摘要
Biological experiments clarify that p53-Mdm2 module is the core of tumor network and p53 oscillation plays an important role in determining the tumor cell fate.In this paper,we investigate the effect of time delay on the oscillatory behavior induced by Hopf bifurcation in p53-Mdm2 system.First,the stability of the unique positive equilibriurm point and the existence of Hopf bifurcation are investigated by using the time delay as the bifurcation parameter and by applying the bifurcation theory.Second,the explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem.In addition,the combination of numerical simulation results and theoretical calculation results indicates that time delays in p53-Mdm2 system are critical for p53 oscillations.The results may help us to better understand the biological functions of p53 pathway and provide clues for treatment of cancer.
基金
The authors express gratitude to the anony imnous referee for his/her helpful Sug-gestions and the partial supports of the National Natural Science Foundations of China(Grants Nos.11762022,11562021 and 11572278)
the youth academic and technical leaders of Yunnan Province(No.2019HB015).
