Dynamical analysis and numerical simulation of a reaction-diffusion model for microbial decomposition of organic matter in 3D soil structure

¹ Inria, CNRS, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, Université Jean Monnet, ICJ UMR5208, Villeurbanne 69603, France; LMDP Laboratory, Cadi Ayyad University, B.P: 2390, Marrakesh, Morocco; UMMISCO, IRD, 93143 Bondy, Paris, France
² Centre for Remote Sensing Applications (CRSA), Mohammed VI Polytechnic University (UM6P), Lot 660, Hay Moulay Rachid, 43150, Ben Guerir, Morocco; LMDP Laboratory, Cadi Ayyad University, B.P: 2390, Marrakesh, Morocco; UMMISCO, IRD, 93143 Bondy, Paris, France
³ LMDP Laboratory, Cadi Ayyad University, B.P: 2390, Marrakesh, Morocco; UMMISCO, IRD, 93143, Bondy, Paris, France
⁴ UMMISCO, IRD, 93143, Bondy, Paris, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 20 July 2024
Accepted: 3 March 2026

Abstract

Microbial decomposition of organic matter in soil is a fundamental process in the global carbon cycle, directly influencing soil health, fertility, and greenhouse gas emissions. This paper presents a dynamic analysis and numerical simulation of a reaction-diffusion model that describes microbial decomposition of organic matter within a three dimensional soil structure. We investigate the interactions between Microbial Biomass (MB) and organic substrates, as well as the diffusion of various compounds through the soil matrix, using nonlinear parabolic partial differential equations. Our study provides proofs for the existence and uniqueness of solutions, as well as an analysis of asymptotic behavior. Notably, our investigation reveals the presence of a global attractor, where any solution, regardless of initial conditions, tends to converge. To illustrate the practical implications of our findings, we have developed a numerical tool to simulate the long-term behavior of the system with reasonable computational expense. This tool provides a visual proof of the global attractor for a validated set of biological parameters in a real sandy loam soil sample captured using 3D tomography imagery.