Mathematical modelling of microbial-nutrient kinetics in biochar amended soils using backward differentiation formula

Abstract

Soil degradation and nutrient depletion pose significant challenges to sustainable agriculture, especially in intensively farmed areas with low organic matter inputs. The use of biochar in farmlands with minimal organic matter inputs aims to sequestrate carbon while also improving soil fertility, microbial activity, and crop yield. Microbial growth, governed by nutrient-mediated kinetics, crucial for soil health and productivity, is largely absent from current land surface models used in climate mitigation assessments. Here, to fill this gap, this study formulated a mathematical model that incorporates nonlinear microbial growth and nutrient uptake kinetics modulated by biochar, capturing feedback loops that influence soil health and productivity. An implicit technique that works especially well for stiff ODEs, the Backward Differentiation Formula (BDF), was used to solve the system in order to guarantee numerical stability and computational efficiency. The model's sensitivity analysis was evaluated using both the Morris and Sobol techniques to determine which factors had a significant impact on nutrient-driven microbial growth dynamics and consequent soil health outcomes. The Morris method revealed that microbial carrying capacity (𝑀_𝑀𝑎𝑥) and crop growth efficiency (𝛽) exert direct effects on yield, as indicated by high mean elementary effects (𝜇^∗). Meanwhile, the Sobol analysis confirmed 𝑀𝑀𝑎𝑥 as the most influential parameter via its high first-order index (𝑆₁ ≈ 0.38), while the total-order index (𝑆_𝑇) highlighted 𝛽 and crop decay rate (𝑑_𝑌) as critical drivers through their interaction effects (𝑆_𝑇 ≈ 0.42). The convergence of both methods on 𝑀_𝑀𝑎𝑥 as the primary determinant reinforces its central role in regulating microbial biomass and productivity. Discrepancies, such as 𝛽′𝑠 low 𝑆₁ but high 𝑆_𝑇, underscore the importance of considering parameter interactions. These findings demonstrate the value of integrating elementary effect and variance-based methods to enhance model simplification and inform targeted agricultural decision-making.