Dynamics Analysis Of A Hematopoietic Stem Cell Model

This thesis is devoted to the process of produce white blood cells by hematopoietic stemcells. Under certain assumptions, the process is converted into a differential system with state-dependent delays.The main purpose of this thesis is to analyze the properties of dynamics ofthe system. Thus, we analyze the dynamical behaviors in several steps.Firstly, due to the complexity of the system, we investigate the stability of the trivialequilibrium point. For this purpose, we analyze the forward invariance and boundness ofsolutions. Then we turn to analyze the properties of the trivial equilibrium point by discussingthe characteristic equation of formal linearization with the delays frozen at the trivialequilibrium point. Finally, we establish some sufficient conditions ensuring asymptoticalstability as well as instability of the trivial equilibrium point by analyzing the distribution ofzero solutions of the characteristic equation.Secondly, we distinguish several cases to investigate the dynamical properties of positiveequilibrium points in a large part due to the fact that it involves two state-dependent delays.The stability analysis of positive equilibrium points is based on freezing delays at theequilibrium points. Moreover, we also examine the existence of local Hopf bifurcation atpositive equilibrium points. Based on center manifold reduction and normal form theory, weobtain some important criteria concerning properties of Hopf bifurcated periodic solutions..Finally, Some numerical simulation via mathematic software are provided to support ourtheoretical results, and to extend the understanding and research on the nature of the model.Also the nature of the model can be given an explanation in biology, we got conclusions ofthe model in biological meaning.