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Patterns in soil organic carbon dynamics: integrating microbial activity, chemotaxis and data-driven approaches

Angela Monti, Fasma Diele, Deborah Lacitignola, Carmela Marangi

TL;DR

This work advances soil organic carbon dynamics by analyzing two reaction-diffusion chemotaxis models that generate stripe, spot, and hexagonal patterns when chemotaxis exceeds a critical threshold. It combines (i) linear stability analyses to derive instability thresholds $\beta^*$ and $\beta_c$, (ii) symplectic numerical integration to faithfully simulate pattern formation, and (iii) piecewise Dynamic Mode Decomposition ($p$DMD) to reconstruct complex spatiotemporal patterns from data with large computational gains. The findings demonstrate that $p$DMD accurately recreates chemotaxis-driven patterns and extends applicability beyond classical Turing patterns, paving the way for applying data-driven spatiotemporal models to experimental soil-microbial data. These insights can enhance predictions of SOC dynamics and inform sustainable agricultural practices, including carbon sequestration strategies and soil health management. The work also points to potential broader impacts in ecological modeling and circular economy applications, contingent on future validation with field data.

Abstract

Models of soil organic carbon (SOC) frequently overlook the effects of spatial dimensions and microbiological activities. In this paper, we focus on two reaction-diffusion chemotaxis models for SOC dynamics, both supporting chemotaxis-driven instability and exhibiting a variety of spatial patterns as stripes, spots and hexagons when the microbial chemotactic sensitivity is above a critical threshold. We use symplectic techniques to numerically approximate chemotaxis-driven spatial patterns and explore the effectiveness of the piecewice dynamic mode decomposition (pDMD) to reconstruct them. Our findings show that pDMD is effective at precisely recreating chemotaxis-driven spatial patterns, therefore broadening the range of application of the method to classes of solutions different than Turing patterns. By validating its efficacy across a wider range of models, this research lays the groundwork for applying pDMD to experimental spatiotemporal data, advancing predictions crucial for soil microbial ecology and agricultural sustainability.

Patterns in soil organic carbon dynamics: integrating microbial activity, chemotaxis and data-driven approaches

TL;DR

This work advances soil organic carbon dynamics by analyzing two reaction-diffusion chemotaxis models that generate stripe, spot, and hexagonal patterns when chemotaxis exceeds a critical threshold. It combines (i) linear stability analyses to derive instability thresholds and , (ii) symplectic numerical integration to faithfully simulate pattern formation, and (iii) piecewise Dynamic Mode Decomposition (DMD) to reconstruct complex spatiotemporal patterns from data with large computational gains. The findings demonstrate that DMD accurately recreates chemotaxis-driven patterns and extends applicability beyond classical Turing patterns, paving the way for applying data-driven spatiotemporal models to experimental soil-microbial data. These insights can enhance predictions of SOC dynamics and inform sustainable agricultural practices, including carbon sequestration strategies and soil health management. The work also points to potential broader impacts in ecological modeling and circular economy applications, contingent on future validation with field data.

Abstract

Models of soil organic carbon (SOC) frequently overlook the effects of spatial dimensions and microbiological activities. In this paper, we focus on two reaction-diffusion chemotaxis models for SOC dynamics, both supporting chemotaxis-driven instability and exhibiting a variety of spatial patterns as stripes, spots and hexagons when the microbial chemotactic sensitivity is above a critical threshold. We use symplectic techniques to numerically approximate chemotaxis-driven spatial patterns and explore the effectiveness of the piecewice dynamic mode decomposition (pDMD) to reconstruct them. Our findings show that pDMD is effective at precisely recreating chemotaxis-driven spatial patterns, therefore broadening the range of application of the method to classes of solutions different than Turing patterns. By validating its efficacy across a wider range of models, this research lays the groundwork for applying pDMD to experimental spatiotemporal data, advancing predictions crucial for soil microbial ecology and agricultural sustainability.
Paper Structure (10 sections, 1 theorem, 39 equations, 6 figures, 1 algorithm)

This paper contains 10 sections, 1 theorem, 39 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The spatial MOMOS model
  3. Chemotaxis-driven pattern formation in the MOMOS model
  4. The Mimura-Tsujikawa model
  5. Numerical reconstruction by data-driven techniques
  6. Patterns of stripes and spots for the spatial MOMOS model
  7. Hexagonal patterns for the Mimura-Tsujikawa model
  8. Applying the pDMD procedure
  9. Discussion and concluding remarks
  10. Spatial discretization

Key Result

Theorem 1

Let If $\beta>\beta^*$, then the spatially homogeneous equilibrium $P_e=(u_1,v_1)$ of model (eq:model) undergoes to chemotaxis-driven instability.

Figures (6)

  • Figure 1: Pattern of ' ' stripes' ' for the spatial MOMOS model - Top left plot: bifurcation diagram in the $(\beta,q)$ parameter space. The other parameters are fixed and have the following numerical values: $k_1 = 0.4, k_2 = 0.6, D_u = 10^{-3}, D_v = 10^{-3}, c = 10^{-3}$. The yellow area indicates the region in the parameter space $(\beta, q)$ where conditions \ref{['eq:chemo_condition_momos']} are verified whereas parameters in the green area do not satisfy these conditions. Top right plot: numerical solution at the final time $T = 80000$ for the state variable $u$. Bottom: time dynamics of the spatial mean of the solutions $u$.
  • Figure 2: Pattern of ' ' spots' ' for the spatial MOMOS model - Top left plot: bifurcation diagram in the $(\beta,q)$ parameter space. The other parameters are fixed and have the following numerical values: $k_1 = 0.4, k_2 = 0.6, D_u = 0.6, D_v = 0.6, c = 0.8$. The yellow area indicates the region in the parameter space $(\beta, q)$ where conditions \ref{['eq:chemo_condition_momos']} are verified whereas parameters in the green area do not satisfy these conditions. Top right plot: numerical solution at the final time $T = 5000$ for the state variable $u$. Bottom: time dynamics of the spatial mean of the solutions $u$.
  • Figure 3: Hexagonal spatial patterns for Mimura-Tsujikawa model - Top left plot: bifurcation diagram in the $(\beta,q)$ parameter space. The other parameters are fixed and have the following numerical values: $k_1 = 1, k_2 = 32, D_u = 0.0625, D_v = 1$. The yellow area indicates the region in the parameter space $(\beta, q)$ where conditions \ref{['eq:chemo_cond_ks']} are verified whereas parameters in the green area do not satisfy these conditions. Top right plot: numerical solution at the final time $T = 500$ for the state variable $u$. Bottom: time dynamics of the spatial mean of the solutions $u$.
  • Figure 4: Reconstructed stripes patterns (spatial MOMOS model). Left plot: pDMD reconstruction of the state variable $u$ at the final time $T = 80000$. The number of partition used in the pDMD algorithm is $N = 8$. Right plot: relative error behaviour \ref{['error_time']} over time.
  • Figure 5: Reconstructed spots patterns (spatial MOMOS model). Left plot: pDMD reconstruction of the state variable $u$ at the final time $T = 5000$, obtained with $N = 32$ partitions. Right plot: relative error behaviour \ref{['error_time']} over time.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof