Food Chain Reaction-Diffusion Models Arising From Unstirred Chemostat

It is observed that mathematics is more and more involves researches for every field of current sciences, including Biology. A lot of mathematical models are successfully established for various life phenomena already. Under help of theories and methods from the modern mathematics, significant achievements are ceaselessly obtained for researches of Biology. Now, the so called Mathematical Biology becomes a active branch of modern sciences.Chemostat is a laboratory apparatus where microorganisms can be grown and all relevant parameters (temperature, PH, etc) can be controlled. Chemostat plays an important role in microbiology, being used as a model of a simple lake, in the commercial production of microorganisms and as a model for waste water treatment. In the beginning, a Chemostat is characterized by constant input nutrient to a well-stirred vessel and therefore its contents are spatially homogeneous. Most works on Chemostat in 1970's and 1980's were of these ODEs models. However, a real case should be space-dependent. If we remove the well-mixed hypothesis and allow a nutrient gradient, this led to a type model often referred to as the un-stirred chemostat, where the continuous culture is not assumed to be well stirred, and the nutrient and populations are assumed to diffuse within the medium in the culture vessel, and thus, the model equations take the form of a system of parabolic partial differential equations for the nutrient and microorganisms population concentrations.The so-called plug flow reactor is another kind of bio-reactor different from Chemostat. Basically, it is a tube through which nutrient containing medium moves at constant velocity nutrient and nutrient-consuming microbial populations diffuse and are carried by the flow. In such environments, nutrient and microorganism population are spatially inhomogeneous and are subjected to fluid motions, thus gradients of nutrient concentration are present. Consequently, the ability of cells to move in a random fashion will play an important role to determine the long time behavior of a population as well as the ultimate population size.Since 1990's, a lot of papers have been contributed to the subjects of un-stirred Chemostat and plug flow reactors, especially those of competing microorganism models. This thesis will deal with another kind of models-food chain models in Chemostat. The food chain models in Chemostat exhibit a rich set of dynamical behaviors, such asiiilocal or global bifurcations of equilibria and chaos with respect to Chemostat control parameters. However, because of the complexity of the models, most studies of them were obtained via ODE methods and based on numerical simulations. In this thesis, we consider reaction-diffusion models for food chains in un-stirred Chemostat and flow reactors. Base on the theories on nonlinear analysis and nonlinear PDEs, especially those of reaction-diffusion equations and corresponding elliptic equations, we will study the dynamical behaviors of populations, such as coexistence and stabilities of positive steady states, persistence and asymptotic behavior of populations, and existence of positive periodic solutions under time-dependent environs. The tools used here include bifurcation theory, fixed point theory of Topology, and general theories of nonlinear elliptic and parabolic equations. The main results obtained in this thesis axe summarized as follows.Chapter 1 deals with a reaction-diffusion system of a food-chain model in an un-stirred Chemostat. The main result is that under suitable conditions stable to small perturbationthe, the model does indeed allow for the possibility of coexistence in the form of a stable equilibrium, where both populations are represented in positive concentrations throughout the Cemostat. By using the bifurcation theorem from simple eigenvalues we show the local coexistence of steady-states with some second bifurcation. Moreover, we prove the stability for the local coexisting solutions by means of the perturbation theorem for linearized operators a...