The Theory Of Impulsive Differential System And Its Applications In Biology Dynamic System

This dissertation is intended to make research on the asymptotic behavior of impulsivedifferential system, and on the applications of the impulsive differential system to thepopulation ecology and epidemic dynamics. The impulsive differential system is normallyused for the description of the rapid variation or rapid jumps of a certain types of motion atfixed or unfixed points of time;in the real sense, the system reflects the development ofprocesses in natural world, and many variations in the fields of scientific researches can bedescribed by using such a system. In recent years, impulsive differential system, being one ofthe focuses of interest, has been applied in such fields as population ecology and epidemicdynamics.After a brief introduction of the evolution process as well as its applications of the theoryof impulsive differential system, this dissertation elaborates major trends of the developmentof the population ecology and epidemic dynamics. With regard to the theory of impulsivedifferential system, this dissertation focuses on the asymptotic behavior of the impulsivedifferential system. After defining sub-linear delay differential system, researches on theperturbation of sub-linear delay differential system under linear impulse and non-linearimpulse, and on the nature of the solutions to the two systems thereof, enable the author toobtain the sufficient conditions for asymptotic attraction of all the solutions to sub-linearimpulsive delay differential system as well as non-linear impulsive delay differential system.Researches on the perturbation of forced sub-linear delay differential system under linearimpulse and non-linear impulse enable the author to obtain the sufficient conditions forasymptotic attraction of all the solutions to the forced sub-linear impulsive delay differentialsystem under linear impulsion as well as non-linear impulsion. These researches have, to agreat degree, enriched the theory of impulsive differential system.Based on Smith Model after improvement for single population under continuous inputof toxin, and on Volterra Model of three populations under continuous input of toxin, thisdissertation has improved and modified the models there above. Taking into account of thetoxins that the population that takes in from the food chain, the author has established themodels respectively for single population and three populations under the impact of impulsivedischarge of toxin. Researches on the dynamical behavior of the models enable the author toobtain the key factors as well as the critical vales for determining the existence andextinguishment of population;Researches also enable the author to obtain and prove in theory,the conditions for population to continue to live. It has been proved in this dissertation thatthere exist periodic solutions to the above-mentioned systems. All the conclusions have beenverified through computer simulation.After an analysis on the tremendous impact of epidemic diseases on human being aswell as on its trend of research, the author has established the SIR epidemic model underpulse vaccination with a periodic infection rate. Researches and analyses on the dynamicbehavior of the SIR epidemic model enable the author to have defined the basic reproductivenumber R0 , proved the existence of periodic infection-free solution, proved the globalasymptotic stability of periodic infection-free solution when R0 < 1, as well as the conditionsfor the persistence of diseases when R0 > 1, and verified the existence of positive periodicsolution of the SIR model. Standard bifurcation theory enables the author to obtain thepositive periodic solutions of the SIR system when R0 → 1?. All the conclusions in this sectionhave been simulated in computer.