Studies On Asymptotic Behaviors Of Two Classes Of Biological Dynamical Models

Chemostat models and epidemic models are two classes of very significant biolog-ical dynamical models. Chemostat is a laboratory device used for continuous culturemicroorganisms and chemostat models can be used to study the growth rule of microor-ganisms in a growth-limiting nutrient. And the changes of disease states of susceptiblepopulation, latent population, infectious population, recovered population are studied byepidemic models, which play an important role on controlling the spread of infectious dis-ease. On the basis of the two classes of biological dynamic models, this dissertation hassystematically studied the asymptotic behaviors of n-species chemostat models with dis-crete delays and finitely distributed delays, an epidemic model with different distributedlatencies, a general epidemic model with heterogeneous mixing and dispersal and an epi-demic patchy model with entry-exit screening, using ordinary differential equations anddelay differential equations, especially Lyapunov-La Salle principle, monotone dynamicalsystem theory and uniform persistence. The main contents and details in this dissertationare as follows:Firstly, n-species chemostat models with discrete delays and finitely distributed de-lays are studied. By employing the method of Lyapunov functionals and placing twoassumptions on the response functions, we prove that competitive exclusion holds undera generic condition for chemostat models with differential removal rates and discrete(alsofinitely distributed) delays. The results show that the competition outcome is completelydetermined by the species’ break-even concentrations: it is the species with the lowestbreak-even concentration that survives in the chemostat and drives other species extinct.Secondly,an SEIR epidemic model with different distributed latencies and generalnonlinear incidence is presented and studied. By constructing suitable Lyapunov func-tionals, the biologically realistic su?cient conditions for threshold dynamics are estab-lished. It is shown that the infection-free equilibrium is globally attractive when the basicreproduction number is equal to or less than one, and that the disease becomes globallyattractively endemic when the basic reproduction number is larger than one. The criteriain this chapter generalize and improve some previous results in the literatures.Thirdly, we formulate a general disease model that takes both heterogenous group-like mixing and patch-like dispersal into consideration. This model is particularly appli-cable to the spread of infectious diseases in China where urban villages located in citiesbring in complex contact patterns. Using monotone dynamical system theory and the u-niform persistence, we can prove that the spread of the infectious diseases depends onwhether the the basic reproduction number larger than one and the basic reproductionnumber relates to dispersal rates and heterogenous mixing rates. The range of the ba-sic reproduction number is obtained and we find that the basic reproduction number isnondecreasing with heterogenous mixing rates k by matrix theory.Finally, we discuss a multi-patch SEIQR epidemic model to investigate the long-term impact of entry-exit screening measures on the spread and control of infectious dis-eases. A threshold dynamics determined by the basic reproduction number ?0is estab-lished: the disease can be eradicated if ?0<1, while the disease persists if ?0> 1. Asan application, six different screening strategies are explored to examine the impacts ofscreening on the control of the 2009 influenza A(H1N1) Pandemic. We find that it iscrucial to screen travelers from and to high-risk patches and it is not necessary to imple-ment screening in all connected patches, and the minimum number of patches that shouldimplement screening depends critically on the dispersal rates and the successful detectionrate of screening measures.