| Bifurcation analysis is one of the important research contents of ordinary differential equations and dynamical systems.Bifurcation is a phenomenon that qualitative properties and topological structure of the system change suddenly with continuous change of parameters.It generally changes stability of the equilibrium point of system and generates new dynamic behaviors,which often occurs in nonlinear systems.Gene regulation network is a typical nonlinear system that describes the interaction between DNA,RNA and protein in cells,in which the variation of composition concentration can be described by ordinary differential equations established according to the biochemical reaction principle.In this thesis,the gene regulatory network associated with tumor suppressor protein p53 is studied,and bifurcation analysis is performed theoretically and numerically.Firstly,for a high-dimensional p53 gene regulation network,the AUTO software package in XPP is used to analyze its bifurcation,and it is found that there are abundant bifurcation phenomena,such as saddle node bifurcation and Hopf bifurcation,which lead to abundant dynamics of p53 expression,such as bistable and tristable.On the other hand,since oscillatory dynamics of p53 play an important role in system,Hopf bifurcation can make the system transform from stable state to oscillation,and time delay is an important condition for Hopf bifurcation.In gene regulatory networks,time delay in transcription and translation is unavoidable.Based on the stability theory,we analyze influence of time delay on Hopf bifurcation in the six-dimensional p53 gene regulatory network with time delay,and give conditions of Hopf bifurcation,and further analyze direction of Hopf bifurcation by using central manifold theorem,and the further numerical simulation results are consistent with the theoretical results.Through the above two aspects of research,we have a further profound understanding of the dynamics of p53 gene regulatory network,and the bifurcation phenomenon in the network has enriched the nonlinear dynamics theory. |
