Modelling Soil as a Living System: Feedback between Microbial Activity and Spatial Structure

TL;DR

The paper addresses how soil's physical space and microbial metabolism coevolve, proposing a cellular automaton where soil sites, nutrients, microbes, and empty sites interact with rates $\sigma$ and $\theta$, and where reproduction necessitates a two-step nutrient pathway producing a carrying capacity $\mathcal{C}$. The approach combines mean-field analysis with 2D and 3D lattice simulations to show that spatial structure can sustain a stable living state and allow parasite coexistence, while revealing power-law soil clustering near nutrient-maximizing regimes and coherent oscillations in 3D. Key contributions include linking soil aggregate statistics to nutrient production and demonstrating how spatial heterogeneity supports resilience and coexistence beyond what is predicted by non-spatial models. The work highlights the importance of spatio-temporal dynamics in soil ecology and provides a framework for studying how microbial processes sculpt and are shaped by the soil structure, with implications for understanding soil health and ecosystem stability.

Abstract

Soil is a complex, dynamic material, with physical properties that depend on its biological content. We propose a cellular automaton model for self-organizing soil structure, where soil aggregates and serves as food for microbial species. These, in turn, produce nutrients that facilitate self-amplification, establishing a cyclical dynamic of consumption and regeneration. Our model explores the spatial interactions between these components and their role in sustaining a balanced ecosystem. The main results demonstrate that (1) spatial structure supports a stable living state, preventing population collapse or uncontrolled growth; (2) the spatial model allows for the coexistence of parasitic species, which exploit parts of the system without driving it to extinction; and (3) optimal growth conditions for microbes are associated to diverse length scales in the soil structure, suggesting that heterogeneity is key to ecosystem resilience. These findings highlight the importance of spatio-temporal dynamics of life in soil ecology.

Figures (8)

• Figure 1: A): Schematic diagram of the model. S,E,N,M represent Soil, Empty, Nutrient, and Microbe states respectfully. Arrows indicate transitions, and the text above them the rates. B): Snapshots of the 2D lattice at different time steps, showing dynamic spatial behaviour.
• Figure 2: A): Phase diagram of the attractors of the mean-field model over soil filling rate $\sigma$ and microbe death rate $\theta$. B): Attractors of the corresponding red points in A), in state space $S$, $E$, $M$ represent Soil, Empty and Microbe populations. Fixed points are represented by circles, while limit cycles (oscillations) can be seen as closed loops.
• Figure 3: A): Phase diagram over soil filling rate $\sigma$ and microbe death rate $\theta$ for a 2D lattice. B) Snapshots of the long-term behaviour on the lattice for different parameter values.
• Figure 4: Cluster size distribution calculated on a 2D lattice with soil filling rate $\sigma$=0.53, $L$=4096, for increasing microbe death rate $\theta$. A): Empty site cluster size distributions: supercritical (A1), critical (A2), and subcritical (A3). B): Filled cluster size distributions, subcritical (B1), critical (B2) and supercritical (B3).
• Figure 5: Plots of soil boundary fraction,i.e, the number of soil boundaries, normalized by the maximum possible number (A) and nutrient production rate (B) vs microbe death rate $\theta$ for different values of soil-filling rate $\sigma$ show a correlation between the soil power-law point (marked with an arrow), the nutrient production maximum, and the soil boundary maximum.