| In this paper, we discuss the coexistence for a two-species periodic competitive sys-tem with unidirectional impulsive dispersal and a single-species periodic system withbidirectional impulsive dispersal in two patches. We mainly discuss the permanence, theextinction, the existence and global stability of positive periodic solutions. By using com-parison theorem of impulsive diferential equation and other analysis methods, sufcientand realistic conditions on permanence and extinction are established. By utilizing thecoincidence degree theory, sufcient conditions are established for the existence of at leastone strictly positive periodic solution, and by using a suitable Lyapunov function, theuniqueness and global attractivity of positive periodic solution are obtained.The main contents and organization in this paper as follows:In section1, we present research background of impulsive diferential equation, andthen the research status and results are given, Finally, the organization of this paper isalso presented.In section2, for the convenience, we present some preliminaries.In section3, we discuss a periodic competitive system with unidirectional impulsivedispersal in two patches, and by using comparison theorem of impulsive diferential equa-tion and other analysis methods, sufcient and realistic conditions on permanence andextinction are established. Futher by utilizing the coincidence degree theory, sufcientconditions are established for the existence of at least one strictly positive periodic solu-tion, and by using a suitable Lyapunov function, the uniqueness and global attractivityof positive periodic solution are obtained. By the numerical simulation, our results areset up.In section4, we discuss a periodic single-species with bidirectional impulsive dispersalin two patches, by set up a Poincare map, sufcient conditions are established for the existence of at least one strictly positive periodic solution. By the numerical simulation,our results are set up. |
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