Mathematical analysis of a marine ecosystem model with nonlinear coupling terms and non-local boundary conditions

Abstract

We investigate the weak solvability of initial boundary value problems associated with an ecosystem model of the marine phosphorus cycle. The analysis covers the model equations themselves as well as their linearization which is important in the model calibration via parameter identification. We treat both cases simultaneously by investigating a system of advection-diffusion-reaction equations coupled by general reaction terms and boundary conditions. We derive a weak formulation of the generalized equations and prove two theorems about its unique solvability provided that the reaction terms consist of Lipschitz continuous and monotone operators. In the proofs, we adapt different techniques (Galerkin approximation, Banach's Fixed Point Theorem) to the multi-dimensional model equation. By applying the general theorems to the problems associated with the phosphorus model we obtain results about existence and uniqueness of their solutions. Actually, by assuming a generalized setting the theorems establish the basis for the mathematical analysis of the whole model class to which the investigated phosphorus model belongs.