Properties Of A Class Of Quasi-homogeneous Vector Field In R~3

It is well-known that the planar dynamical systems and simple homogeneous vector fields (such as twice homogeneous vector fields) of three dimensional Eu-clidean space have very abundant results, while based on spatial complexity, there are still many problems to be solved. In this paper, we utilize equivalence re-lation between the quasi-homogeneous vector fields and the homogeneous vector fields in R3, and draw lessons from the research methods of the homogeneous vec-tor fields. By the investigation of tangent vector field Qt(u) which induced by the quasi-homogeneous vector fields in R3and planar vector fields which induced by the tangent vector fields Qt(u) in Πi, we simplify the investigation of geometric prop-erties of the quasi-homogeneous vector fields in R3. At last, we analyze geometrical and biological significance of three species competition model which has different growth rates, getting the largest population growth is going to survive while the smallest population growth will extinct in this competition system.