Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors

  • Received: 01 January 2009 Accepted: 29 June 2018 Published: 01 April 2010
  • MSC : 34A34, 34C40, 34C45, 34D10, 35A20, 35C10, 37N25, 92B05.

  • A new approach to the problem of characterizing the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using the notion of slow invariant manifold is proposed. The problem of interest is addressed within the context of singular partial differential equations (PDE) theory, and in particular, through a system of singular quasi-linear invariance PDEs for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system's slow invariant manifold exponentially attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original biosystem dynamics on the slow manifold. The analyticity property of the solution to the invariance PDEs enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, examples of certain immobilized enzyme bioreactors are considered to elucidate aspects of the proposed context of analysis.

    Citation: Nikolaos Kazantzis, Vasiliki Kazantzi. Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors[J]. Mathematical Biosciences and Engineering, 2010, 7(2): 401-419. doi: 10.3934/mbe.2010.7.401

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  • A new approach to the problem of characterizing the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using the notion of slow invariant manifold is proposed. The problem of interest is addressed within the context of singular partial differential equations (PDE) theory, and in particular, through a system of singular quasi-linear invariance PDEs for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system's slow invariant manifold exponentially attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original biosystem dynamics on the slow manifold. The analyticity property of the solution to the invariance PDEs enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, examples of certain immobilized enzyme bioreactors are considered to elucidate aspects of the proposed context of analysis.


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