Impulsive Differential Equations With Application In Agricultural Mathematical Model In Ecology

The impulsive differential equations have been deeply investigated during the past few years. Their theory is not only considerably richer than that of ordinary differential equations without impulses, but also more adequately represents many mathematical simulation of such processes and phenomena. For example, impulsive birth,impulsive vaccination, chemotherapeutic treatment of disease and population ecology. Now some qualitative properties such as oscillation, asymptotic behavior, and stability are investigated extensively by many authors. However these theories are hard to be applied and there are almost no developments in the study of global stability, and little has been done for the periodicity of nonautonomous impulsive differential equations with time delays, especially the study based on the topological degree approach. In this dissertation, multi-dimensional prey-predator systems, two single-species models with impulsive selective harvesting and logistic models with impulsive effects and time delays are established. Mathematically, a combination of approaches to discrete dynamics, continuous dynamics, impulsive dynamics, operator theory and numerical simulations are used to investigated dynamical behaviors including the existence and global stability of periodic solutions, permanence and extinction and all kinds of complexities. The main results obtained in this paper may be summarized as following.Based on experiments, Holling suggested three different kinds of functional responses for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. Some authors have investigated two-dimension predator-prey systems with functional response and impulsive. However little has been done for multi-dimensional predator-prey systems with functional response and systems with mixed functional response. In chapter 2, By using Floquet theorem , small amplitude perturbation method, and the method of comparison involving multiple Liapunov functions, these systems were investigated respectively. It was proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value and permanence condition is established. It is shown that multi-predator impulsive control strategy is more effective than the classical and singleone, and Holling II system is more sensitive to control parameter(impulsive period) than Holling IV system. Further,Computer simulations are carried to confirmed that high orders quasi-periodicity,intermittency and crises exist in mixed functional response system.In chapter 3, we consider a logistic fishery model and discuss the selective impulsive harvesting of fishes above a certain age or size by incorporating a time delay in the impulsive harvesting term. It is proved that there exists an asymptotically stable positive periodic solution x^t) when the catchability coefficient h is less than some critical value hj. It is concluded that h* and x*n are increasing with respect to /. Simulations shows that the selective harvesting is advantageous to the sustainability of the population. The conclusion that impulsive harvesting is superior to continuous one when intrinsic growth rate of population is less. Further, optimal impulsive harvesting policy in mature population for Gompertz model was investigated. For the two models, the maximum sustainable yields per unit time as the management objective are obtained.In chapter 4, we investigated periodic systems with periodic impulses and delays. Firstly, a delayed logistic system governed by impulsive effects is investigated. By using comparison theorem, it is proved that the system is permanent under some appropriate conditions. Further, a set of sufficient conditions which guarantee the existence, uniqueness and global attractivity of a positive periodic solution are obtained by using Bohl-Brower fixed point theorem and Lyapunov function. Secondly, we are interested in studying the positive periodic solution of a periodic predator-prey system with time delays and impulses. By using the method of coincidence degree, some sufficient conditions are obtained for the existence of at least one strictly positive periodic solution. Computer simulations are carried to confirmed the main theorems.