How can we validate and improve this model for self-organised limestone-marl alternations?

Our goal is to discern chemical self-organisation in sediments from astronomical forcing of the
palaeoclimate. In modelling these two effects, we started with an attempt to reproduce the main
results from a paper by Ivan L'Heureux (2018)¹ - who formulated a system of five stiff
differential equations, for calcite and aragonite compositions, for the concentrations of calcium
and carbonate ions in the pore water and for the porosity, as functions of depth and time.
Reproducing the self-organized steady-state distributions along depth required a major effort,
mostly with regard to understanding what triggers numerical instabilities, but was finally
successful. Currently, we have not yet succeeded in reproducing oscillations, that L'Heureux
predicted for high initial and boundary sediment porosity, without requiring an external force.
It is essential that we are able to determine for which inital and boundary conditions oscillations
should occur. We have formulated two questions that we want to share with the audience in
order to seek help. These are our questions:
1) Do the five differential equations describe the underlying physics adequately?
2) Our current software implementation of the five differential equations does not yield any
oscillations, is that a flaw on our side, or does this agree with mathematical insights?

1. "Diagenetic Self-Organization and Stochastic Resonance in a Model of Limestone-Marl
Sequences" by Ivan L'Heureux (2018). https://doi.org/10.1155/2018/4968315