Applications Of Impulsive Differential Equations In Biological Economics

The theories and methods of impulsive differential equations have achieved great de-velopment and become a whole system in recent thirty years.Mathematical models of differential equations play an important role in describing population dynamic behaviors. Mathematically,these models explain all kinds of population dynamic behaviors,which allow people to understand population dynamics scientifically so that some interactions of populations can be intend to control.Especially,impulsive differential equations de-scribe population dynamic models,which are 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,we investigate the dynamical behaviors of the stage-structured delay predator-prey systems with fixed impulsive mo-ments,the stage-structured single population systems with state-dependent impulses,and the epidemic systems with fixed moments impulses.By using the theories and methods of impulsive,discrete and continuous dynamical system,arithmetic operators,optimiza-tion and numerical simulation,we study locally and globally asymptotieal stability of the boundary periodic solutions,and the permanence of the these systems.The thesis is arranged by four chapters.In Chapter 1,we introduce concisely the present development of the relevant subjects about population and epidemic dynamics as well as the main work done in this thesis. Moreover,we give some definitions and fundamental theories of differential equations and impulsive differential equations.In Chapter 2,based on the background of biological resource management in agri-culture,we investigate two stage-structured delay predator-prey systems.In Section 2.1, we construct a predator-prey system with stage structure on predator and impulsive per-turbations on prey,the predating products is used to increase the predator's constitution. We obtain that the predator-extinction boundary periodic solution is globally attractive, the permanent condition of the investigated system is also obtained.In Section 2.2,we present a delay predator-prey system with stage structure on predator and impulsive perturbations on prey,the predating products is used to increase the predator's ability of birth.We obtain that the predator-extinction boundary periodic solution is globally attractive.We also derive the permanent condition of the investigated system.In Chapter 3,based on the background of pest management in agriculture,we inves-tigate a predator-prey system and two epidemic models.In section 3.1,We construct a delay predator-prey system with stage structure on prey and impulsive perturbations on predator.By using comparison theorem of impulsive differential equations and mathe-matical analysis methods,we obtain that the prey-extinction boundary periodic solutions are globally attractive,the permanent condition of the investigated system is also ob-tained.In section 3.2,we present an epidemic model with impulses,that is,spraying biological pesticides and releasing infected pests at the same fixed moments,which causes an epidemics in the susceptible pests,and the pests lose the ability of destroying emble-ments.By using Floquet theorem and small amplitude perturbation technique,we obtain the critical value for the globally asymptotical stability of susceptible pest-extinction boundary periodic solution.In section 3.3,we consider an epidemic model with spraying biological pesticides and releasing infected pests at different fixed moments,which causes an epidemics in the susceptible pests,and the pests lose the ability of destroying emble-ments,so we come to the objectives of pest management.By using Floquet theorem and small amplitude perturbation technique,we also have the critical value for the globally asymptotical stability of susceptible pest-extinction boundary periodic solution.in Chapter 4,based on the background of the fishing management,we investigate a stage-structured single population system with state-dependent harvesting.By using the fixed point theory,we prove the existence,uniqueness of periodic solutions of the investigated system,and they are orbitally stable.