Modeling Nutrient-Release Dynamics of Organic
and Inorganic Fertilizers Using Differential Equations: A Comparative Analysis of Fast and Slow Release Kinetics
Jean de Dieu Niyomugabo1* 
, Victoire Ayingeneye2 ,
Angelique Nyiramahoro2 , and Marie Ange Cyuzuzo3
Abstract. Optimizing nutrient-use efficiency (NUE) in agricultural systems requires a precise mechanistic understanding of how different fertilizer types release their nutrients over time. Differential equation-based kinetic models provide a rigorous mathematical framework for describing, comparing, and predicting nutrient-release dynamics across diverse fertilizer formulations from rapidly solubilizing inorganic salts to microbially-mediated organic amendments and polymer-incapsulated controlled-release fertilizers (CRFs). This review and analytical study systematically compile, parameterizes, and comparatively evaluates eight kinetic model classes — zero-order, single first-order, double first-order (two-pool), logistic/sigmoidal, Weibull, Korsmeyer–Peppas, Fickian diffusion, and the Stanford–Smith model against published laboratory incubation and release-experiment datasets for nine representative fertilizer types. Model performance was assessed using the root mean squared error (RMSE), Nash–Sutcliffe efficiency (NSE), Akaike Information Criterion (AIC), and Percent Bias (PBias). Results demonstrate that inorganic fertilizers (urea, ammonium nitrate) are best described by zero-order or fast first-order models (k₁ ≈ 0.82–0.94 d⁻¹), whereas organic amendments (poultry manure, compost, green manure) require double first-order two-pool models to capture the heterogeneous labile and recalcitrant nitrogen fractions (k₁ = 0.09–0.41 d⁻¹; k₂ = 0.003–0.012 d⁻¹). Polymer-coated CRFs exhibit three-stage kinetics with an initial lag phase best modeled by logistic or diffusion equations (k ≈ 0.03–0.05 d⁻¹). Temperature sensitivity, expressed through modified Arrhenius relationships and Q₁₀ factors, significantly modulates rate constants and must be incorporated into predictive frameworks for field-scale application. The study provides a unified model comparison matrix, parameterized rate constants, and guidelines for selecting appropriate kinetic frameworks to support fertilization scheduling, environmental nutrient-loss modeling, and the design of next-generation CRF formulations.
Keywords: controlled-release fertilizer, differential equation, first-order kinetics, mineralization, nitrogen use efficiency, nutrient synchrony, Weibull model, two-pool model, Arrhenius, soil incubation