Dynamic Analysis Of A Kind Of The Biological Reserve Models With Delay

In recent years, because of the environment pollution and the excessive catch and hunt, many species are on the edge of dying out, which, if not addressed effectively, would lead to the extinction. To avoid such situations, establishment of protection zone is an approach widely applied.The nature reserve in our country has been developing fast to a considerable scale. However, there is still a certain gap with the developed countries. Therefore, more efforts are still needed. The studies at home and broad into the population quantity in protection zone initially all took Logistic growth model to describe the features of the dynamic changes, which has posed many disadvantages. Though some researchers has taken the influence of the scale of the protection zone, their studies cannot reflect the influence of the mature period on the population growth, which still leaves a gap with the actual situation. Thus the time lag is introduced into the study to make the system more close to the actual situation. With the development of the biomathematics, delay differential equation has been widely applied in various biological models. We can use that to depict the system depending on both of the current and the history states.In this thesis, we firstly introduces a time delay parameter that representing the mature period, and analyses the stability of zero solution and the existence of positive equilibrium into the system of the population of the first level protecting zone. Based on this foundation, with mature period as the bifurcation parameter, using the distribution of zero theorem and analyzing the distribution of the roots of the characteristics of the transcendental equation correspondent to the system, the author discusses the stability of positive equilibrium solution and the existence of Hopf bifurcation. Secondly, to study the direction of the Hopf bifurcation and the stability of periodic solution, the author uses the normative approach and the center manifold theorem to deduce several corresponding approximate expressions. Finally, this paper chooses a group of specific parameter values, using the Matlab to do the numerical modeling to support the analyzed results.