Are there any sustained efforts to model existing ecosystems?
Such a model would require a fairly detailed and complete biological inventory as well as detailed knowledge of the species' interactions, and a detailed map of the physical and chemical makeup of the location or region...

I suggest reading about the The Simons Collaboration on Computational Biogeochemical Modeling of Marine Ecosystems (CBIOMES).1 This collaboration has produced ample research concerning the modeling of marine microbial communities in terms of spatial composition, metabolic interactions, nutrient flow, and seasonal variation.
Here are the research efforts highlighted on their website:

Developing New Computational Modeling Frameworks

Developing a New Trait-based Understanding of Microbial Communities Based on Metabolic and Elemental and Macromolecular Constraints (led by Zoe Finkel/Dal)
Constraint-based Modeling of Marine Microbial Community Metabolism and Physiology (led by John Casey/MIT)
Statistical Modeling of Microbial Communities: Niches, Traits, and Interactions (led by Andrew Irwin/Dal)
Thermodynamically Constrained Metabolic Networks for Ocean Modeling (led by Joe Vallino/MBL)
Models of Marine Microbial Biogeography and Biogeochemistry (led by Mick Follows and Stephanie Dutkiewicz/MIT)

Collecting and Compiling Relevant Datasets

Ocean Color and Biogeochemistry (led by Shubha Sathyendranath/PML)
Data and Tools to Define the Biogeography of Marine Microbes (led by Ginger Armbrust/UW)
Microbial Growth, Interactions and Biogeographies from ‘Omics’ Data (led by Jed Fuhrman/USC)

Bringing Together Models with Data

Advanced Statistical Analysis of Marine Microbial Systems (led by Christian Müller/Flatiron Institute)
Unifying Data and Models through Biogeography (led by Chris Follett/MIT)
Data Assimilative Modeling of Marine Ecosystems (led by Chris Edwards/UCSC)
Statistical Network Inference and Time Series Analysis (led by Jacob Bien/USC)


Concerning specific geographic sites, one of the best understood marine ecosystems is the San Pedro Channel off the coast of Los Angeles, which has been studied since 1998 as part of the SPOT project.2 Sampling efforts from this site include sensors that capture continuous in situ measurements of dissolved oxygen, fluorescence, temperature, salinity, and pressure, as well as monthly charters that collect seawater to measure nutrients (nitrate, nitrite, phosphate, silicate) and chlorophyll at 12 discrete water column depths. Peruse the publications section of the SPOT website to better understand the ecological and geochemical research that the SPOT project has enabled.
If you are further interested in the global effort to characterize and model the dynamics of marine microbial communities, I recommend a 2015 review by Fuhrman, Cram, and Needham3, which discusses methods to analyze microbial networks through space and time.

References

https://cbiomes.org/
https://dornsife.usc.edu/spot/
Fuhrman JA, Cram JA, Needham DM. Marine microbial community dynamics and their ecological interpretation. Nat Rev Microbiol. 2015 Mar;13(3):133-46.