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

TL;DR

This paper proves the existence and uniqueness of solutions for a complex marine ecosystem model involving nonlinear coupling and non-local boundary conditions, providing a mathematical foundation for analyzing such models.

Contribution

It establishes general theorems on weak solvability for coupled advection-diffusion-reaction systems with non-local boundary conditions, applicable to marine phosphorus cycle models.

Findings

Proved unique solvability of the model equations under Lipschitz and monotonicity conditions.

Developed a weak formulation and applied Galerkin and fixed point methods.

Provided a mathematical basis for further analysis of marine ecosystem models.

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.