Mathematical analysis of the $PO_4$-$DOP$-$Fe$ marine ecosystem model driven by 3-D ocean transport

TL;DR

This paper provides a mathematical analysis of the $PO_4$-$DOP$-$Fe$ marine ecosystem model, focusing on solution existence, parameter reconstruction, and the implications for model validation and uncertainty in parameter identification.

Contribution

It establishes the existence of solutions for the model and highlights parameter dependencies, offering insights into model validation and uncertainty reduction.

Findings

Existence of transient, stationary, and periodic solutions confirmed.

Four model parameters may be dependent, indicating uncertainty.

Mathematical conditions for solvability are broadly applicable.

Abstract

Marine ecosystem models are developed to understand and simulate the biogeochemical processes involved in marine ecosystems. Parekh, Follows and Boyle introduced the $PO_4$-$DOP$-$Fe$ model of the coupled phosphorus and iron cycles in 2005. Especially the part describing the phosphorus cycle ($PO_4$-$DOP$ model) is often applied in the context of parameter identification. The mathematical analysis presented in this study is concerned with the existence of solutions and the reconstruction of parameters from given data. Both are important questions in the numerical model's assessment and validation not answered so far. In this study, we obtain transient, stationary and periodic solutions (steady annual cycles) of the $PO_4$-$DOP$-$Fe$ model equations after a slight change in the equation modeling iron. This result confirms the validity of the solutions computed numerically. Furthermore, we present a calculation showing that four of the $PO_4$-$DOP$ model's parameters are possibly dependent, i.e. different parameter values might be associated with the same model output. Thereby, we identify a relevant source of uncertainty in parameter identification. On the basis of the results, possible ways to overcome this deficit can be proposed. In addition, the stated mathematical conditions for solvability are universal and thus applicable to the analysis of other ecosystem models as well.