Ecology Of Infectious Diseases With Time Delay System Stability And Hopf Bifurcation

Recently, extensive research on the single population dynamics model with delays,andwith the introduction of the systems of two or more interacting species and the delayedsystems. It further enriched the research content of the population dynamics,and promotedpopulation dynamic development. This paper considers di?erent time delayed phenomenonof ecological significance, and study they how to in?uence the stability of the positive equi-librium.Chapter II and chapter III study two eco-epidemiological models with di?erent de-lays.Against two time delayed phenomenon of ecological significance,by regarding the delayτas a parameter. Its dynamics are studied in terms of local analysis and Hopf bifurcationanalysis. By analyzing the associated characteristic equation, it is found that Hopf bifurca-tion occurs whenτcrosses some critical values. Using the normal form theory and centermanifold argument, the explicit formulae which determine the stability, direction and otherproperties of bifurcating periodic solutions are derived.ChapterIV ,based on chapter II,introduced new delay and considered the reduce tothe probability of the susceptibility predator ,which due to they are infected by Infectedpredators in the past time,namely the spread of disease have a time delay.The conditionsof existence and locally asymptotically stable of equilibrium is studied ,by constructingapprppriate Lyapunov fonctionelle the global stability of the equilibrium is investigated.