| Abstract: | The period-$k$ solutions of population differential system with state-dependent impulsive effect are investigated by the theory of discontinuous dynamical system. Through $G$-function theory, the necessary and sufficient conditions are obtained for trajectory direction of a population differential system, and the results are better than the previous work. Also, the local stability of period-$k$ solutions is studied by the mapping structure and the theory of eigenvalue analysis. Furthermore, the existence of period-1 solution is investigated for a special impulsive population differential system, and the analytical condition is established. Finally, the trajectory of period-1 solution and the relationship between different parameters and the module of eigenvalues are illustrated. |
