The Recurrent Motions Of Competing Systems

The solutions of three competition systems, the automorphic solutions of nonautonomous Lotka-Volterra competition equations and a global conver-gengce result for strongly monotone map with positive translation invariance are studied in this thesis which consists of four chapters, and the first chapter is introduction.In chapter 2, the Field-Noyes model are considered. There are two possibilities if we consider only the case that the positive equilibrium of the system p is hyperbolic. Either p is asymptotically stable or it is unstable with a one dimensional stable manifold. Both cases can occur, depending on the values of the parameters. Condition for the existence of a nontrivial periodic orbit are given in the second case by the knowledge of type-K order and competitive systems.In chapter 3, the two-dimensional nonautonomous Volterra-Lotka competition equations are considered which are continuous almost automorphic in time. Conditions for the existence of an asymptotically stable almost automorphic solution with positive components are given. Accordingly, the result of Ahmad are generalized. In chapter 4, We consider a community of three mutually competing species modeled by the Lotka-Volterra system, the geometric analysis is generalised to higher dimensions to determine the dynamical behaviour on a neighbourhood of the coordinate two-skeleton, and the geometric analysis leads to 33 partial classication of the dynamics.In chapter 5, we show that strictly monotone map which have a certain translation-invariance property are so that all orbits converge to a unique equilibrium. The result may be seen as a dual of a well-known theorem of Mierczynski for systems that satify a conservation law. In the last section, we consider periodic cooperative systems, give the condition that the Poincare have translation-invariance property, so all the solutions are asymptotic periodic solutions.