Stabilityand HOPF Bifurcation Analysis In A Blood Clees Model With Time-delay

Blood cells have played a vital role in the bodies of mammals. Therefore, the body must carefully regulate their production. In normal animal and human bodies, blood cells are constantly regenerated, while the number of various blood cells basically unchanged, which is kept by a series of dynamic equilibrium, such as generation of the blood cells, release, survival, removal or death. Once this balance has been destroyed, a variety of blood diseases may occur. Change of the number of mature blood cells usually indicates the occurrence of some diseases. To establish and study mathematical models of blood cells, can help people understand the inherent mechanisms of blood cells, such as mature, and predict their behaviors.In this paper, a blood cells model with time-delay is considered. The model describes the known physiological processes leading to the production of mature blood cells. Studying the dynamic properties of the model will help us understand the processes leading to the production of mature blood cells and interpretation the causes of some blood diseases. In this paper, the main task is to research the blood cells model with time-delays from the perspective of stability and bifurcation, and we can obtain the sufficient conditions for stability of the number of mature blood cells, and the number of blood cells appears periodic change when some parameters change.First of all, we proved that the existence of equilibrium of the system, then the sufficient conditions of the stability and the existence of Hopf bifurcations at the equilibrium are obtained by analyzing the distribution of the characteristic values. Furthermore, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. At last, several numerical simulations are carried out using Matlab soft.