Impulsive And Delayed Effects On Population Control

Mathematical models of differential equations play an important role in describing population dynamic behavior. Mathematically, these models explain all kinds of population dynamic behavior, which allows people to understand population dynamics scientifically so that some interactions of population can be intend to control. Delayed impulsive equations, which consider the effects of both the present state and the passed state on the behavior of dynamical system, is more suitable to describe the population dynamical system. In this dissertation, population dynamic models are established to consider several problems in pathogen controls and population controls by means of the theory and method of delayed functional differential equations and impulsive differential equations. We investigate dynamic behavior including the stability of equilibrium, the existence of periodic solution, the permanence and extinction of system. The main results of this dissertation may be summarized as follows:In Chapter 2, some basic theories are provided for delayed equations and impulsive equations.In Chapter 3, three predator-prey models with time delay and functional response are investigated. In section 3.1, a Ivlev predator-prey model with time delay and impulsive interruption in the predator is studied. The prey population is divided into two classes, the immature prey and mature prey. The time from immature to mature is a constant, and is expressed with a time delay. The functional response is Ivlev type, and the predator only capture the mature prey. We get the sufficient condition for the global attractivity of the pest-eradication periodic solution, and the condition for the permanence of the system. In section 3.2, a predator-prey model with time delay and impulsive harvesting strategy is studied. We impulsively release the prey at fixed time, and harvest the predator continuously. We get the condition for the global attractivity of the predator-eradication periodic solution, and we also obtain the condition for permanence of the system. Our result provide some theoretical base for exploitation of biological resources. In section 3.3, a stage-structured Gomportz predator-prey model with time delay and impulsive interruption in the prey is studied. The predator population is divided into two classes, the immature predator and the mature predator. Only the mature predator has the ability of capture the prey. We impulsively capture the prey, and the growth rate for the prey is Gomportz type, while the predation response is type Holling II. Our main purpose is to studied the effect of impulsive capturing of prey on the predator population. The sufficient condition for the global attractivity of the predator-eradication periodic solution is obtained, and we also get the condition for the permanence of the system. Some numerical results are done to show the dynamical behaviors of the system.In Chapter 4, pest management models are studied. In section 4.1, two SI epidemic models are investigated. One continuous control system and one impulsive control system are used to control the number of pest, by using endemic. To the continuous system, we get the sufficient condition for the existence and stability condition for the equilibriums. To the impulsive system, a sufficient condition for the existence and stability of periodic solution are obtained. The results are also verified by simulations. In section 4.2, a pest-pathogen model is formulated. The pest is divided into two classes, the susceptible and infective. Infected pest can release pathogen cells to infect the susceptible. By some suitable assumptions, the partial differential equation system is transformed to ordinary differential equation system. The sufficient condition for the susceptible pest eradication periodic solution is obtained as our main result.In Chapter 5, a stage-structured and periodic time dependent predator-prey model is studied. Since there are many factors in the natural world are periodic, it is necessary to study the effect of periodic variable environment on the behaviors of population model. In our model, the prey population is divided into two classes, the immature prey and mature prey. The predator is omnivorous, and it can capture both of the immature prey and the mature prey at different predation rate. The predator population is also being impulsively harvested. The condition for the permanence of the system is obtained. We get the threshold, above which the predator population will die out due to excessive harvesting.