Applications Of Stage Structure And Impulsive Differential Equations In Population Models

Mathematical models of differential equations play an important role in describing population dynamic behavior.Mathematically,these models explain all kinds of population dynamic behaviors,which allows people to understand population dynamics scientifically so that some interactions of population can be intend to control.Especially, impulsive differential equations describe population dynamic models,which is more reasonable and precise on reflecting all kinds of change orderliness,since many life phenomena and human exploitation are almost impulsive in the natural world.In this dissertation, population dynamic models for pest management are established to consider several problems in population controls by means of the theory and method of impulsive differential equations.Dynamic behaviors,including the existence and stability of equilibriums,the existence of periodic solution and its global attractivity,the permanence and extinction of system,are investigated.The main results of this dissertation may be summarized as follows:In Chapter 3,two simple species models with stage-structure are formulated and investigated.In section 3.1,a stage-structured simple species model with time-delay and disease in the infant is studied.It is assumed that the simple species has two stages, immature and mature.And the time from immature to mature is a constant which is expressed by a time delay.There is a disease among the immature population,and the disease will only infect the immature population,while the mature population will never be infected.The existence and stability of some potential equilibriums and the effect of time-delay on the dynamics of the model are studied.In section 3.2,a stage-structured SI epidemic model is constructed and studied.In order to control the number of pest, some infected pests are impulsively released at fixed time,so as to the pest number can be suppressed below economical injury level.The sufficient condition for the existence and stability of the susceptible pest eradication periodic solution is obtained,and the sufficient condition for the permanence of the system is also obtained.The results provide some theoretical bases for pest management.In Chapter 4,two stage-structured predator-prey models are studied.In section 4.1, a Lotka-Volterra predator-prey model with stage structure in the prey is investigated.It is assumed that the prey,population has two stages,immature egg and mature pest.The predator population only capture mature pest,since immature prey is protected by the eggshell.In the model,the traditional Lotka-Volterra predation response is considered. The natural enemy(predator) is impulsively released to control the pest.The sufficient condition for the existence and stability of the pest eradication periodic solution is gotten; and the sufficient condition for the permanence of the system is also obtained.A threshold is obtained,which provide theoretical base for pest management.In section 4.2,a ratiodependent predator-prey model with stage-structure in the prey is studied.In this model, it is also assumed that the prey population has two stages,immature egg and mature pest. The predator only capture mature pest,since the immature is protected by the eggshell. And,the ratio-dependent predation response is considered.The results show that under some special situation,the pest population will never be controlled,no matter how many natural enemy(predator) is released.This result shows the difference between ratiodependent predation and other predation response,and it also explains why some pests are hard to be eradicated.In Chapter 5,a predator-prey model with disease in the prey and two impulses for integrated pest management is investigated.The prey(pest) population is divided into two classes:susceptible prey(susceptible pest)and infected prey(infected prey).The infected pest will not do harm to the crops.Thus,infected pest and predator are impulsively released at different fixed time to control the number of the pest.Based on the above assumptions,a predator-prey model with two impulses is constructed.By use of the Floquet theorem,small interruption and comparative skills,the sufficient condition for the global stability of the pest-eradication periodic solution is obtained,and the permanence of the system is also obtained.