Analysis Of Hopf Bifurcation For Delay Model About Pierce’s Disease

With the development of science and technology, ecological destruction, and people’s living environment is polluted, there has been an increasing number of new bacteria. The emergence of these bacteria contributes a great threat to people’s production and life. Research on the growth of these bacteria is an important basis for our work to prevent disease.Pierce’s disease is a fatal disease of grapevines which results from an infection by the plant pathogen, and does a great harm to grape varieties. Pierce’s disease model reflects the growth of these bacterial, and it will block the movement of water in the xylem vessels of the plant. They can make wine deteriorated and pose a serious threat to the development of wine industry. Study on the growth of these bacterial will help us prevent in advance and reduce losses.In fact, the regeneration of the population needs a time course. In other words the rate of the population at one time depends not only on population this moment, but also the past. Based on these considerations, we construct a population growth model about these bacteria with time delay, and we introduce harvesting and immigration. First,according to eigenvalue distribution analysis method, we obtain the stability of the model in equilibrium solution, and the sufficient and necessary conditions for the occurrence of the Hopf bifurcation. Then we use center manifold theorem and normal form to obtain the direction of Hopf bifurcation and the formula that can judge the stability of the periodic solution. Finally, combining the global Hopf bifurcation theorem and the composite matrix theory to prove the periodic solution can be extended globally.