Discrete Lotka-volterra System Persistence And The Existence Of Periodic Solution And Global Attractivity,

In the first chapter, we firstly state the necessity for the study of biological models. Then, we introduce some known competition and predator-prey models of Lotka-Volterra type. Finally, some basic definitions and the related initial conditions of those studied system are given.In the second chapter, the permanence for an autonomous delaycompetition model of Lotka-Volterra type is considered by means of Liapunovfunctionals. In 1974, May first proposed the discrete two-species competitionmodel of Lotka-Volterra type. Since then, the discrete two-species competitionmodel of Lotka-Volterra type has attracted great attention and interest of manyauthors, and it has been found that the system can demonstrate quite rich andcomplicated dynamics. In real situations, however, time delays always existand they should be taken into account in modeling. Therefore, several recentpapers considered the discrete delay two-species competition model ofLotka-Volterra type. Due to the possible change of the coefficients in realworld, Qinqin Zhang and Zhan Zhou considered the permanence of thenonautonomous two-species competition model of Lotka-Volterra type withdelays, and the necessary and sufficient conditions for the permanence wereobtained. Our model includes all those models mentioned earlier. Wendi Wangstudied the global stability of discrete population models with time delays andfluctuating environment. Though those models studied by Wcndi Wang includeour model in this chapter, during our proof, we dismiss the conditionsinf a(k)>0 for i=1,2. Therefore, our conclusion improves thecorresponding ones.In the third chapter, we discuss a discrete Lotka-Volterra competition system with m-species. We first obtain the persistence of the system. Assuming that the coefficients in the system are periodic, we obtain the existence of a periodic solution. Moreover, under some additional conditions, the periodic solution is globally attractive. Our results can be reduced to those for two-species system. At the same time, our results not only can bereduced to those for the scalar equation when there is no coupling, but also improve and complement some in the literature.Finally, in the fourth chapter, by constructing Liapunov functional, we study a discrete two-dimensional predator prey-system with a finite number of delay, the similar conclusions to the continuous case are obtained. That is, the time delays are harmless for permanence of the solution of the system.