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Modeling And Stability Analysis For The Chemostat Systems

Posted on:2004-04-02Degree:DoctorType:Dissertation
Country:ChinaCandidate:Z P QiuFull Text:PDF
GTID:1100360095452345Subject:Control theory and control engineering
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Chemostat model is one of the most significant models in Mathematical biology. The Chemostat is an important device used for growing micro-organisms in a continuous cultured environment, and a medium of great importance between principles and applications. It has been widely applied to the study of the increase in different populations of micro-organisms and their interactive law. In addition, it has also been applied to the management and prediction of the ecology system, especially the marine ecology, and the control of the environment pollution.In the light of the recent work in biological models, especially in the chemostat models, the dissertation provides a systematic study on the asymptotical behaviour of some chemostat models built by delay or diffusion differential equations. The main contents and results in this dissertation are as follows:i) The global asymptotic behavior of the Chemostat model with the Beddington-DeAngelies functional responses and time delays is studied. The conditions for the uniform persistence of the competing populations are obtained via uniform persistence of infinite dimensional systems. Then the global asymptotical stability of the positive equilibrium of the model with time delays is proved via monotone dynamical systems. Our results imply that mutual interference in a species may result in coexistence of the two competing species and demonstrate that those time delays do not influence the competitive outcome of the organisms.ii) The asymptotic behaviour of the Chemostat model with mutual interference or without mutual interference is studied. For the two models with delay, the uniform persistence of the models are both proved under the conditions of the existence of the positive equilibrium. Moreover, under those conditions, the global stability of the positive equilibrium is proved for the two models without delays.iii) The asymptotic behaviour of the Chemostat model with predator-prey populations and delays is studied. Sufficient conditions for the global attractivity of boundary equilibrium are obtained via fluctuation lemma, and sufficient conditions for uniform persistence of this model are obtained via uniform persistence of infinite dimensional systems.iv) The Chemostat model with diffusion and two-nutrients is considered. Sufficient conditions for uniform persistence of this model are obtained via uniform persistence of infinite dimensional discrete dynamical systems. Then one can easily obtain the existence of periodic solution of the Chemostat model.v) The asymptotical behavior of population models with diffusion is studied. Firstly sufficient conditions for the existence of periodic solution are obtained by comparison theory of reaction-diffusion differential equations; secondly sufficient conditions are established, under which the models admits a positive periodic solution which attracts all positive solutions. Then we apply the general theory to some types of population models with diffusion and periodic coefficients. Thus some earlier results of population models with delays are extended to diffusion population models.Finalh. the asymptotic behaviour of the Chemostat model with two-nutrient and diffusion is further studied. The global attractivity of the periodic solution is proved under the unique existence of the periodic solution.vi) The asymptotic behavior of flow reactor models with two-nutrient are considered. Different diffusion coefficients of the population and nutrients, the death rates of the population and the velocity exist in the flow reactor are introduced in these models. In complementary case, sufficient conditions for uniform persistence and extinction of the population are obtained by the theory of uniform persistence of infinite dimensional dynamical systems. Especially for the model with equal diffusion coefficients and zero death rates, the global attractivity of the unique positive steady-state solution is proved. In substitutable case, sufficient conditions for uniform persistence and extinction of population...
Keywords/Search Tags:Chemostat, Uniform Persistence, Stability, Periodic Systems, Reaction-Diffusion Systems, Delay Systems, Monotone Theory, Dynamical Systems
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