Mathematical Analysis of Marine Ecosystem Models
This work is concerned with the mathematical formulation of marine ecosystem models. Marine ecosystem models are systems of advection-diffusion-reaction equations. The influence of biogeochemical reactions (e.g. consumption of nutrients, growth, decomposition) is modeled by reaction terms which are, in general, nonlinear functions of all regarded concentrations. The reaction terms additionally include parameters (e.g. growth rates). Parameter identification, i.e., the determination of parameter values, enables the model's adjustment to the ecosystem in question. Parameter identification is a mathematical optimization problem. In applications, the model equation and the parameter identification problem are solved numerically. A satisfying assessment of numerical solutions requires information about the continuous model. In this work, we investigate the continuous equation of a general ecosystem model and the corresponding parameter identification problem. As a result, we obtain the existence of transient and periodic solutions. To prove the existence of transient solutions, we rely on standard methods (Galerkin's method, Banach's Fixed Point Theorem) which we adapt to the nonlinearly coupled systems of equations and the two types of reaction terms. In the periodic case, we investigate models which are solved by the constant zero function. The existence result ensures that, in addition, nontrivial periodic solutions exist. Concerning parameter identification, we prove the existence of optimal parameters for quadratically integrable data. In addition, we indicate first and second order optimality conditions and formulate the first order condition as an optimality system. We apply all theoretical results to the PO4-DOP model which is important for testing purposes. By means of a numerical test based on a two-dimensional version of the PO4-DOP model, we investigate uniqueness of numerically computed periodic solutions.
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