The Dynamics Of Multi-Community Biological Competition Models

In this thesis, by using classical results of stability, qualitative theory in diferentialequation, combining the linearization technique with the method of computer numericalsimulation, we study two classes of biological competition models, and obtain a series ofconclusions related to these two diferent models, such as the existence, positivity anduniform boundedness, the existence and stability of the equilibrium are also studied.Specifcally, we divide this thesis into three parts, and the main results are as follows:In the frst chapter, we recall the background of biological competition models,studies that we have made, and the major works in this paper. Moreover, we listmathematical basic notions, defnitions and preliminaries used in this paper.In the second chapter, we study two-patch model with diferent difusivities ofdiferent species, and gain corresponding conditions of existence, boundedness of so-lutions. Moreover, we obtain the results of local stability of zero equilibrium andboundary equilibriums. Finally, by using implicit function theorem, we obtain the cor-responding existence conclusion of coexistence equilibria. Also, we proceed numericalsimulations using the Matlab to verify the result of coexistence equilibria is correct.Biologically, in a fast difusion environment, assume one species has smaller difusivitythan another species, then the species will win the competition if it has greater spatialvariation in its birth rate than another competing species.In the third chapter, we study the existence, positivity and boundedness of solu-tions of three-patch model. In addition, we obtain the results of the local stability ofzero equilibrium and boundary equilibriums using Hurwitz criterion, also we list somenumerical examples. Finally, by using implicit function theorem, we obtain the cor-responding existence conclusion of coexistence equilibria. Also, we proceed numericalsimulations using the Matlab to verify the result of coexistence equilibria is correct. Bi-ologically, if the birth rates of two species are greatly diferent, one species will becomeextinct, and the other species will reach a steady status as time delays. If the birthrates of two species are same or similar, two species will reach a steady coexistencestatus as time delays.