Stability And Bifurcation Analysis Of Two Delayed Gene Regulation Models

Since the twentieth century,systematic biology has undergone tremendous changes,and gradually developed into an important discipline of today's life sciences.Understanding the mechanism of gene regulation network has also become one of the important subjects of systematic biology research.The gene regulatory network is one of the most basic and important biological networks,which composeds of many genes,proteins,small molecules,and their mutual regulation.Time delay is inevitable in gene regulation networks as the process of transcription,translation,protein formation in gene regulation.On the other hand,due to the uneven distribution of the concentration of the gene product,the diffusion phenomenon in the biological cell is also inevitable.In particular,the time delay and diffusion can affect the dynamic behavior of the system.Therefore,studies on the time delay system and the time delay system with diffusion have very important significance.In this paper,we study the dynamic behavior of two delayed gene regulation models: a synchronized oscillations in cellular system with multiple time delays model and a gene regulatory network mediated by small noncoding RNA with time delays and diffusion model.The work of this paper is as follows:In chapter 2,we discuss the dynamics of synchronized oscillations in cellular systems with multiple time delays.By suitable transformation and certain assumptions on multiple time delays,the double-positive feedback loop(PP)model and the double-negative feedback loop(NN)model are reduced to four dimensional nonlinear delay differential equations with two delays,respectively.By furthur introducing two regulation functions,the PP model and NN model can be integrated into a new model.Through linearizing the new model at the positive equilibrium and analyzing the associated characteristic equation,the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated.Furthermore,combining the normal form method and the center manifold theorem,the explicit formulas which determine the direction of the bifurcation and the stability of the bifurcated periodic solutions are derived.Finally,numerical simulations are carried out to support our theoretical analysis.Results show that Hopf bifurcation occurs when time delay through some critical values.In addition,we found that ? play a definitive role in inducing hopf bifurcation,whereas s hardly induce hopf bifurcation.In chapter 3,a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied.Choosing the sum of delays as the bifurcation parameter,the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation.Finally,the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorems for partial functional differential equations.Finally,numerical simulations are carried out to support our theoretical analysis.It is shown that the sum of delays can induce hopf bifurcation and the diffusion we incorporate into the system can effect the amplitude of the system.Furthermore,if the diffusion coefficients of protein and mRNA are suitably large,there only exist spatially homogeneous periodic solutions.But both the spatially homogeneous and inhomogeneous bifurcating periodic solutions can occur if the diffusion coefficients of protein and mRNA are suitably small.Particularly,the effect of sRNA diffusion coefficient on model is much less than the other two,which suggests that sRNA diffusion coefficient is more robust.In chapter 4,we give the summary and prospect of the full text.