Dynamic Modeling And Analysis Of Several Typical Biological Systems

Mathematical biology is a new interdiscipline of biology by studying biological problems with mathematical methods.It is mainly a theoretical study of mathematical methods which is related to biology.Mathematical biology offers a new powerful research tool to study the problem of biology.Dynamic modelling and analysis of biology system is one of the important directions of Mathematical biology.Namely,the relationship between variables in a biological system is expressed by differential equations or other forms,and it can carry out the dynamic analysis and numerical simulation for this dynamic model.The advantages of dynamic modelling are to replace the complicated and even unfulfillable experiments,greatly reduce the cost and improve the research efficiency.With the background of mathematical biology,this dissertation mainly researchs the dynamic modelling and analysis of several typical biological systems.The concrete works include:Firstly,for the questions of modelling of Hepatitis B Virus(HBV),when the viruses infect of target cells,a body's immune response is not instantaneous,and needs some time,therefore,we propose a delayed differential equations dynamical model of HBV.The model not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes.In the following,we first discuss the existence of two boundary equilibria and one positive(infection)equilibrium.We then analyze the global stability of the two boundary equilibria,the local asymptotic stability and Hopf bifurcation of the positive equilibrium and also the stability of the bifurcating periodic solutions.Moreover,we perform numerical simulations to illustrate some of the theoretical results we obtain and also illustrate how the factors such as the time delay,immune response to infected cells and viruses and the proliferation of hepatocytes affect the dynamics of the model under time delay.Secondly,for the questions of stochasticity in the process of HBV infection,we investigate the stochastic dynamic model with delayed immune response.This is because the patients are easily affected by external noise or internal stochastic disturbance.Baesd on this stochastic dynamic model,we study how the noise affected the dynamic characteristic of virus model: verify that there is a unique global positive solution for this model with any positive initial value,and establish the sufficient conditions for extinction and persistence of the model.Further,we perform a couple of numerical simulations to illustrate our theoretical analysis re-sults,and find many interesting results.For instance,when the basic reproduction number is bigger than 1,and the specific condition is satisfied,small intensities of white noises may result in approximate periodic solutions while the solution of the corresponding deterministic system is asymptotically stable.However,when the intensities of white noises are large,the approximate periodic solutions of the stochastic system will be destroyed,and the solutions become unstable.When the basic reproduction number in deterministic and stochastic system are all bigger than 1,and the specific condition is satisfied,we find that the infected state will exist in the deterministic system,but in the stochastic system,the disease will become extinct due to the influence of white noises.Thirdly,with the question of the Delta-Notch dependent boundary formation in the drosophila large intestine,we propose a kind of high-dimensional dynamic model based on boundary formation in the drosophila large intestine.In this high-dimensional dynamic model,we arrange all cells in a regular M ?N lattice,each site(cell)being a hexagon and having at most six neighboring cells.That is,we have NC(28)M ?N/2 cells for an M ?N lattice.The dynamic model is high-dimensional,so it can't be analyzed in theory,we firstly simplify the model into two cells,and study the equilibria and stability.In the following,we investigate the parameter sensitivity and perturbation analysis by numerical simulations,and obtain sensitive parameters in this model.We get the consistent results when compare to experimental result by selecting appropriate parameters.Finally,for the question of one-dimension time series of movement counts,we explore a new approach based on phase space reconstruction to analyse the behavioural responses to calorie restriction in the mice.We reconstruct the phase space of each mouse in each group and divide the whole time course into ten periods and thus obtain the change trends of attractors in the ten periods.Further,we compute the invariant of attractors(correlation dimension and Kolmogorov entropy)and size from period 1 to period 10,plot the corresponding three-dimensional diagram to explore the change trends from period 1 to period 10 under different levels of calorie restriction,and predict the development tendency.Researching the dynamic modeling and analysis of the biological system of viral infection can help doctors understand the factors that govern the infectious disease progression and offering insights into developing treatment strategies and guiding antiviral drug therapies.Modeling and analysis of the Delta-Notch dependent boundary formation in the Drosophila large intestine can further understand the mechanism of development and by researching the movement counts of mice can help us realize the different behavioural responses to different level calorie restriction in the mice.The research work in this paper provides new methods and application cases for the research field of mathematical biology,and has a positive significance to promote the development of this field.