The Application Of Impulsive Differential Equations In Mathematical Modelling Of Population Ecology Management

Fairly rich results have been made for the theories of impulsive differential equations for almost thirty years. However, these theories are hard to be applied and there are almost no developments in the study of global stability. Therefore, it is of value in theory and of significant in practice to investigate the application of impulsive differential equations in each field. Scholars have been using continuous or discrete equations to study the population dynamic models, while ignoring the external disturbance. However, in the natural world, many biological phenomena and the optimization and control of some biological phenomena are impulsive. In this dissertation, population dynamic models with impulsive effect are established concerning some actual problems of population ecology management and the meaning of impulse in these problems. Mathematically, a combination of approaches to discrete dynamics, continuous dynamics, impulsive dynamics, operator theory and numerical simulations are used to investigate dynamical behaviors including the existence and global stability of periodic solutions, permanence and extinction and all kinds of complexities. The mathematical models are presented based on biological meanings and the results we obtained can be used to provide reliable decisive basis for the production. The main results obtained in this dissertation may be summarized as the following:In Chapter 2, population dynamic models with impulsive effect at fixed moment concerning pest management are established and analyzed. Two-prey one-predator system with impulsive effect is investigated concerning augmentation of natural enemies. By using Floquet theory, two-pest eradication periodic solution is shown to be globally asymptotically stable. Further the conditions for the permanence of the system are obtained by comparison theorem and analytic method, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining two species are also given. In addition, continuous release natural enemies is compared with pulse release natural enemies. A predator-prey model with mutual interference and impulsive effect concerning integrated pest management is studied. The conditions for the permanence or extinction are given. Moreover, the existence of positive periodic solution which arises from the pest-eradication periodic solution is proved by bifurcation theory, and the validity of integrated pest management is discussed. When the unique positive periodic solution loses its stability, numerical simulation shows there is a sequence of bifurcation, leading to a chaotic dynam-ics. A Holling I predator-prey model with mutual interference and impulsive effect concerning integrated pest management is also studied. The conditions for the permanence or extinction are given. Finally, the forced system is compared with the corresponding continuous system and to be found the forced system has different dynamical behaviors with different range of initial values which are inside or outside the unstable limit cycle of the unforced continuous system.In Chapter 3, a periodic predator-prey system with mutual interference and impulsive effect is considered. The linear stability of semi-trivial periodic solution is shown by applying Floquet theory. Some conditions for the permanence of the system are obtained. Further the bifurcation theory, operator theory and Lyapunov-Schmidt series expansion are used to show the existence of the positive periodic solution.In Chapter 4, a seasonal seed transfer mathematical model is presented and the effect of seasonal seed transfer on the growth of the species is discussed. Using the corresponding difference equations of impulsive differential equations and the Lyapunov method, a globally asymptotically stable positive periodic solution is shown.In Chapter 5, impulsive harvest models for bioeconomic management of renewable biological resources are investigated. A single species which obeys Gompertz growth law and a stage-structured population are discussed respectively. For t...