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Dissolved oxygen (DO) is one of the main parameters to assess the quality of lake water. This study is intended to construct a parabolic distributed parameter system to describe the variation of DO under the ice, and identify the vertical exchange coefficient *K* of DO with the field data. Based on the existence and uniqueness of the weak solution of this system, the fixed solution problem of the parabolic equation is transformed into a parameter identification model, which takes *K* as the identification parameter, and the deviation of the simulated and measured DO as the performance index. We prove the existence of the optimal parameter of the identification model, and derive the first order optimality conditions. Finally, we construct the optimization algorithm, and have carried out numerical simulation. According to the measured DO data in Lake Valkea-Kotinen (Finland), it can be found that the orders of magnitude of the coefficient *K* varying from 10^{-6} to 10^{-1} m^{2} s^{-1}, the calculated and measured DO values are in good agreement. Within this range of *K* values, the overall trends are very similar. In order to get better fitting, the formula needs to be adjusted based on microbial and chemical consumption rates of DO.

In this paper, we present a framework to infer the possible transmembrane transport of intracellular substances. Considering four key enzymes, a modified fourteen-dimensional nonlinear hybrid dynamic system is established to describe the microbial continuous culture with enzyme-catalytic and genetic regulation. A novel quantitative definition of biological robustness is proposed to characterize the system's resilience when system parameters were perturbed. It not only considers the expectation of system output data after parameter disturbance but also considers the influence of the variance of these data. In this way, the definition can be used as an objective function of the system identification model due to the lack of data on the concentration of intracellular substances. Then, we design a parallel computing method to solve the system identification model. Numerical results indicate that the most likely transmembrane mode of transport is active transport coupling with passive diffusion for glycerol and 1, 3-propanediol.

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