Open AccessFeature PaperArticle Dynamical Model for Stigeoclonium nanum in Thin-Layer Photobioreactors Considering Abiotic Losses and Logistics Constraints 1 División de Ingeniería Mecatrónica, Tecnológico Nacional de México/ITS Región de Los Llanos, Calle Tecnológico No. 200, Col. Oriente, Guadalupe Victoria, Durango 34700, Mexico 2 Postgraduate Program in Engineering Sciences, BioMath Research Group, Tecnólogico Nacional de Mexico/IT Tijuana, Blvd. Alberto Limón Padilla s/n, Tijuana 22454, Mexico 3 Departamento de Ingeniería Eléctrica y Electrónica, Tecnológico Nacional de México/IT Durango, Blvd. Felipe Pescador No. 1830, Oriente, Colonia Nueva Vizcaya, Durango 34080, Mexico 4 Secretaría de Ciencia, Humanidades, Tecnología e Innovación, Tecnológico Nacional de México/IT Valle del Guadiana, Km. 22.5 Carretera Durango-México, Villa Montemorelos, Durango 34371, Mexico * Author to whom correspondence should be addressed. Mathematics 2026, 14(12), 2050; https://doi.org/10.3390/math14122050 (registering DOI) Submission received: 1 May 2026 / Revised: 31 May 2026 / Accepted: 4 June 2026 / Published: 9 June 2026 Abstract This paper presents a mechanistic model for a thin-layer microalgal bioreactor cultivating Stigeoclonium nanum, with a comprehensive analysis of its dynamics and stability. Unlike most bioreactor studies that assume simple Monod or linear growth, our model rigorously explores the nonlinear interplay between logistic constraints and multiple nutrient limitations. We introduce a coupled Logistic–Monod system of nonlinear ordinary differential equations that captures sigmoidal transitions and steady states of Stigeoclonium nanum under simultaneous nitrogen and phosphorus depletion and incorporates abiotic nutrient removal to ensure mass conservation. Qualitative analysis proves positive invariance and boundedness of solutions using the Localization of Compact Invariant Sets method. Asymptotic stability of the biologically relevant equilibrium is established. Experimental validation in a thin-layer photobioreactor using three-fold cross-validation yielded high correlation coefficients (0.78–0.96) for biomass, nitrate, and phosphate concentrations, confirming predictive accuracy. The model thus provides a robust framework for process control and optimization in industrial-scale applications. Keywords: thin-layer bioreactor; microalgae; mathematical modeling; monod kinetics; logistic function; stability; positive invariance; boundedness; nitrate–nitrogen; orthophosphate–phosphorus MSC: 93C15; 34H05; 93C05; 93C10; 34A34 1. Introduction Microalgae have emerged as relevant biological systems for addressing contemporary environmental and energy challenges. Pioneering studies [ 1, 2] have demonstrated that these organisms possess a unique combination of traits, including high photosynthetic efficiency, metabolic versatility, and a remarkable ability to adapt to diverse environments. Recent research [ 3, 4] reports significant nutrient removal efficiencies, particularly nitrogen and phosphorus, from various types of effluents. Their potential for CO 2 fixation has been quantified across multiple cultivation systems, while their capacity to produce high-value compounds such as lipids, proteins, and pigments has been widely documented. Even in environmental applications, microalgae have shown promising results in wastewater treatment. Along this path, scientific research is necessary to explore better solutions to optimize the process. These applications have been optimized through the design of specialized photobioreactors [ 5] and their integration into circular economy processes [ 6, 7], enabling the valorization of the generated biomass into multiple products. The growth and productivity of microalgae critically depend on the availability of essential nutrients. Pioneering work [ 8] established the basic nitrogen and phosphorus requirements for various species, while more recent studies have delved into the assimilation mechanisms and their impact on biochemical composition [ 9]. Despite these advances, significant gaps persist in the modeling of less-studied species, such as Stigeoclonium nanum. Recent investigations have increasingly characterized the specific engineering potential of this microorganism; for instance, Lima et al. [ 13] evaluated dynamic nutrient removal trajectories in closed photobioreactors, while Mohd-Sadiq et al. [ 14] quantified macroscopic consumption rates in synthetic media for wastewater treatment and biopolymer production. However, comprehensive dynamic models that adequately capture its particular behavior are not yet available. Furthermore, taxonomic and physiological studies by Almomani [ 15] have revealed distinctive enzymatic and structural features in this species under nutrient stress, which complicates the direct extrapolation of models developed for other conventional microalgae [ 16]. This work proposes the development and validation of a comprehensive dynamic model for Stigeoclonium nanum that describes its growth and the simultaneous consumption of nitrate-nitrogen (N- NO 3 ) and orthophosphate-phosphorus (P- PO 4 ). The approach integrates recent advances in mathematical modeling with experimental data obtained in controlled environments. The methodology combines numerical simulation techniques with rigorous experimental protocols, providing a comprehensive framework for optimizing processes involving these species. The model validation follows standardized protocols, ensuring the reliability of the results and their applicability under real-world conditions. The developed model features a mechanistic structure that combines different kinetic approaches: a Monod-type function models nutrient assimilation, a Logistic equation captures the slowing of biomass growth as it approaches its maximum capacity, and exponential decay terms represent the abiotic degradation of N- NO 3 and P- PO 4 . This formulation allows for an adequate representation of the exponential and stationary phases observed in real cultures. For parameter fitting, biostatistical tools were employed using the lsqcurvefit algorithm in MATLAB R2025a ପ୍ପ, implemented under a nonlinear least-squares scheme. This was complemented by numerical filtering techniques and cross-validation using statistical goodness-of-fit metrics, such as the Root Mean Square Error (RMSE) and the coefficient of determination ( R 2 ). This integrated approach represents a significant advance over previous studies by providing a framework for modeling and optimizing systems based on Stigeoclonium nanum. 2. Experimental Thin-Layer Photobioreactor The experimental database was obtained from a thin-layer flat-panel photobioreactor constructed from 6 mm-thick transparent glass, measuring 60 cm in length, 60 cm in width, and 5 cm in thickness. It had an operating volume of 15 L and an effective depth of 4.17 cm. This configuration was selected for its ability to provide homogeneous culture conditions, especially in photosynthetic systems that require a uniform distribution of light and nutrients [ 19]. The dynamic growth behavior of Stigeoclonium nanum was analyzed under controlled conditions of illumination, agitation, and nutrient availability, including nitrates and phosphates, to develop a representative mathematical model. The microalgae used were Stigeoclonium nanum, isolated from the clarifier of the South Wastewater Treatment Plant (PTAR Sur) in Durango, Mexico. The inoculum was cultivated in BG11 medium (Blue-Green Medium), a standard formulation recommended by the Culture Collection of Algae and Protozoa [ 20]. This medium is widely used for microalgae cultivation due to its balanced nutritional composition and its ability to sustain stable growth under autotrophic conditions. It provided a controlled source of nutrients, allowing for the establishment of reproducible conditions for the kinetic study. Illumination was supplied by broad-spectrum LED lamps with an average intensity of 250 μ mol photons m − 2 s − 1 under a continuous photoperiod (24 h of light) [ 21]. The ambient temperature during the experiment was maintained between 22 and 25 °C. Agitation was achieved by bubbling air enriched with 1% of CO 2 at a rate of approximately 0.3 vvm (4.5 L/min), ensuring adequate gas transfer and medium homogeneity [ 22]. During the experimental period, daily samples were taken to measure the following variables: Biomass (mg/L): determined by filtration through grade 3 glass fiber filters (2 μ m porosity) and drying at 105 °C, following the protocol described in [ 23]. Nitrate-nitrogen concentration (N- NO 3 ): measured using adapted colorimetric techniques [ 24]. Orthophosphate-phosphorus concentration (P- PO 4 ): determined using the modified Taussky and Shorr colorimetric method [ 25]. The experimental design spanned 16 days, with daily monitoring of the aforementioned variables. Three independent growth kinetics experiments were performed, and a three-fold cross-validation was applied: in each fold, two kinetic runs were used for fitting, and the remaining one was used for validation. This strategy allowed for the evaluation of the system’s dynamic behavior reproducibility and ensured the robustness of the resulting mathematical model. Figure 1 presents the general flowchart of the experimental process developed for the study of Stigeoclonium nanum growth. 3. Mechanistic Model Formulation The proposed model is formulated using an iterative mechanistic approach, drawing on structures of biological and ecological systems reported in the literature. Throughout the process, diverse model versions were evaluated, with varying degrees of complexity. The final selection combines elements that best represent the temporal evolution of biomass and nutrients, with parameter bounds restricted to biologically plausible ranges verified by biochemical engineering experts. Based on the above-mentioned literature and exploratory considerations, specifically: (i) models lacking the Logistic regulation term for the dynamics of biomass, (ii) formulations based on univariate limiting nutrients, and (iii) alternative multiplicative Monod-type couplings. Each variant was calibrated and compared based on parsimony and predictive capability, using RMSE and R 2 metrics via cross-validation. Furthermore, biological coherence—defined by the properties of positivity, invariance, and boundedness—was an essential selection criterion, as these properties are fundamental for the subsequent formal mathematical qualitative properties analysis. The novel proposed model combines additive Monod-type assimilation for N- NO 3 and P- PO 4 with Logistic self-regulation for biomass. This structure proved to be the most parsimonious configuration that successfully reproduced both the exponential growth phase and the culture’s stabilization while maintaining the required structural and dynamical properties. The model describes the growth of microalgal biomass, given by the state variable x 1 ( t ) ; x 2 ( t ) represents the dynamics of N- NO 3 ; and x 3 ( t ) represents the dynamics of P- PO 4 consumption. The interactions are governed by the following system of nonlinear ordinary differential equations: x ˙ 1 = ρ 1 x 1 x 2 φ 2 + x 2 + ρ 2 x 1 x 3 φ 3 + x 3 + α 1 x 1 1 − x 1 σ , (1) x ˙ 2 = − ρ 3 x 1 x 2 φ 1 + x 1 − α 2 x 2 , (2) x ˙ 3 = − ρ 4 x 1 x 3 φ 1 + x 1 − α 3 x 3 , (3) the first expression of Equation ( 1) describes biomass growth ( x 1 ) combining Monod-type assimilation terms for nutrients ( x 2 , x 3 ) with a Logistic regulation term (last expression). To our knowledge, no model with these characteristics has been reported for photobioreactors. At this point, it is important to clarify the kinetic mechanisms of the governing equations with respect to the parameter φ 1 , which represents an initial biomass-dependent coefficient, conceptually aligned with Contois-type formulations rather than a classical substrate-limiting Monod constant. This mathematical structure explicitly accounts for crowding effects, specific metabolic down-regulation, and light self-shading inside the thin-layer photobioreactor. Biologically, as microalgal biomass x 1 ( t ) increases, the parameter φ 1 constrains the specific nutrient uptake rate per unit biomass, preserving coherence during the culture’s stabilization phase. 3.1. Abiotic Removal Terms and Mass Conservation Distinct from standard biological uptake, the linear decay terms − α 2 x 2 and − α 3 x 3 of Equations (2) and (3) represent abiotic loss mechanisms essential for the mass balance in photobioreactors. As described by [ 27], purely biological models often fail to close the mass balance because they ignore physico-chemical phenomena induced by high pH, such as ammonia stripping (volatilization) and phosphate precipitation. Therefore, these exponential decay terms account for the mass transfer to gaseous or solid phases, ensuring that the parameters ρ 3 and ρ 4 represent true biological consumption rather than aggregated losses. 3.2. Maximum Carrying Capacity Constraint ( σ ) The logistic term accounts for self-limitation due to physical constraints (e.g., light shading). However, to maintain thermodynamic consistency, the carrying capacity σ is not defined as an arbitrary constant but is conceptually linked to the initial nutrient availability ( S 0 ). Following the theoretical framework proposed by [ 28], σ effectively acts as a stoichiometric boundary ( σ ≈ Y X / S · S 0 ). This formulation prevents spurious biomass accumulation profiles; mathematically, if nutrient availability approaches zero ( S 0 → 0 ), the carrying capacity vanishes ( σ → 0 ), thereby inhibiting the logistic growth component in the absence of substrates. This approach is conceptually aligned with variable-yield chemostat models [ 18], in which the dynamic coupling between growth bounds and resource limitations determines the system’s stability. We show the dynamic relationships between biomass and the nutrients involved in the model in Figure 2. The central node represents microalgal biomass dynamics, which are directly influenced by its own logistic growth, represented by the green arrow. Biomass and nutrients (N- NO 3 and P- PO 4 ) interactions are indicated with the brown arrows, i.e., contributions to biomass growth derived from nutrient utilization and nutrient consumption by the biomass, represented by Monod or Decay functions. Red arrows indicate additional losses not associated with growth, such as degradation processes or loss through other non-cellular mechanisms. The model assumes that biomass growth is limited by factors such as resource availability and the system’s maximum capacity, while nutrients are reduced by both biomass assimilation and independent processes. This graphical representation provides a clear visualization of the interaction between the variables and facilitates the understanding of the dynamic model before addressing its mathematical formulation. 3.3. Model Fit Evaluation Criteria The quality of the mathematical model’s fit to the experimental data was evaluated using the following statistical criteria, which are standard in the validation of dynamic models: Root Mean Square Error (RMSE): quantifies the average difference between the model’s predictions and the experimental values. A low RMSE indicates that the model accurately reproduces the magnitude of the observed data; it is especially useful for evaluating pointwise fit in time series [ 30]. Coefficient of Determination ( R 2 ): measures the degree of linear correlation between the model outputs and the experimental data. A value close to 1 suggests that the model explains a significant proportion of the observed variability, making it a standard metric for comparing the performance of mechanistic models [ 31]. Comparison of simulated trajectories: allows for the identification of systematic deviations that numerical metrics may not capture, such as time lags or errors in the shape of the curves. This visual inspection complements the quantitative analysis and is particularly valuable when modeling nonlinear biological processes. 3.4. Parameter Calibration and Validation Scheme Numerical integration was performed through a first-order explicit Euler scheme. This approach is numerically stable for the present model due to the smooth, non-stiff nature of the Monod-type kinetics and the relatively slow time scales of microalgal growth. A constant step size of Δ t = 0.01 days (approximately 14.4 min) was selected based on a temporal convergence test. To ensure numerical robustness, this step size was explicitly verified by cross-validating the integration scheme against a higher-order, variable-step Runge–Kutta solver ( ode45). This cross-validation is a standard practice in microalgal bioprocess modeling to ensure high precision in simulation outcomes and to successfully eliminate numerical artifacts [ 32]. The evaluation demonstrated negligible deviation between solvers, ensuring that the discretization error remained negligible relative to the experimental uncertainty. The simulation horizon was set to T = 100 days to evaluate long-term asymptotic behavior. To attenuate experimental noise while preserving the essential underlying dynamics, a Savitzky–Golay filter [ 33] with a 13-point window and a second-degree polynomial was applied to the raw biomass, N- NO 3 , and P- PO 4 time series. Parameter estimation was carried out using the nonlinear least-squares solver lsqcurvefit in MATLAB R2025aପ୍ପ. Search bounds were established based on biologically plausible ranges reported in the literature to ensure the physical consistency of the kinetic constants. A three-fold cross-validation scheme was implemented to assess the model’s predictive robustness, using RMSE and R 2 as performance metrics. The system’s positivity is analytically guaranteed (see Section 3.5). At each iteration, a numerical projection onto R ≥ 0 was applied to ensure that the numerical trajectories remain strictly within the biologically feasible domain Ω . Given that the vector field f ( x ) is well-conditioned and the integration step Δ t is several orders of magnitude smaller than the system’s time constants, the discrete-time approximation reliably preserves the qualitative properties of the continuous model. 3.5. Experimental Data Before parameter fitting, the internal consistency of the three experimental kinetic runs ( κ 1 – κ 3 ) was evaluated to verify common trends and rule out anomalies. In all three trials, the biomass exhibited a sigmoidal-type increase, transitioning to a stationary phase toward the end of the cultivation period. Meanwhile, N- NO 3 and P- PO 4 showed a sustained decrease, with the sharpest drop occurring during the first few days. The trajectories shown in Figure 3, Figure 4 and Figure 5 correspond to raw (unfiltered) measurements from the three runs ( κ 1 – κ 3 ); no smoothing or denoising was applied for this visualization. Any preprocessing (e.g., Savitzky–Golay smoothing) is used exclusively prior to parameter estimation and is described in Section 3.4. 3.6. Error Bars for Experimental Data To illustrate the experimental variability inherent in the raw data, Figure 6, Figure 7 and Figure 8 display the observed error bars for biomass, N- NO 3 , and P- PO 4 concentrations, respectively. These error bars represent the standard deviation across replicate measurements at each kinetic run ( κ 3 — κ 3 ) and provide a quantitative measure of experimental uncertainty. Notably, the magnitude of these error bars reflects both biological variability and potential measurement errors. Careful experimental design is crucial to minimize these sources of variability; however, the observed error bars also enhance our ability to discern subtle differences between kinetic runs, thereby supporting the statistical robustness of subsequent model validation and parameter estimation analyses. 3.7. Parameter Fitting Results Unless stated otherwise, all metrics in this section are computed against the original (unfiltered) data. Filtering is used only as a preprocessing step for parameter estimation. The three-fold cross-validation results for kinetics runs ( κ 1 , κ 2 , and κ 3 ) are organized as follows: Case 1: Training with κ 1 and κ 2 ; validation with κ 3 . Case 2: Training with κ 1 and κ 3 ; validation with κ 2 . Case 3: Training with κ 2 and κ 3 ; validation with κ 1 . The statistical metrics confirm the model’s high predictive capacity. Notably, the R 2 values for x 3 (P- PO 4 ) consistently exceeded 0.95, indicating that the additive Monod-type coupling effectively captures the limiting role of phosphorus in Stigeoclonium nanum growth. While the biomass ( x 1 ) shows slightly higher RMSE values due to inherent biological variability in batch cultures, the overall correlation remains robust ( R 2 ≈ 0.80 ), validating the model for process optimization. 4. Qualitative Mathematical Analysis To complement the numerical validation, we analyze the qualitative properties of the system ( 1)–( 3). The objective is to verify that the dynamics are biologically admissible and mathematically well-posed. Specifically, we: (i) prove the positive invariance of R + 3 , (ii) establish boundedness via the Localization of Compact Invariant Sets (LCIS) method, (iii) ensure existence and uniqueness of solutions, and (iv) characterize the stability of equilibrium points. Establishing such qualitative properties is essential for guaranteeing global stability in nonlinear models of biological resource utilization, as demonstrated in classical chemostat studies [ 34]. A sufficient condition for R + 3 to be positively invariant is that the vector field f ( x ) does not point outward on the boundary planes of the first octant. This is satisfied if: ∀ j ∈ { 1 , 2 , 3 } , f j ( x ) x j = 0 , x ℓ ≥ 0 ∀ ℓ ≠ j ≥ 0 . (4) Evaluating the system ( 1)–( 3) on each boundary plane, we observe: On the plane x 1 = 0 : x ˙ 1 = f 1 ( 0 , x 2 , x 3 ) = 0 ≥ 0 . On the plane x 2 = 0 : x ˙ 2 = f 2 ( x 1 , 0 , x 3 ) = 0 ≥ 0 . On the plane x 3 = 0 : x ˙ 3 = f 3 ( x 1 , x 2 , 0 ) = 0 ≥ 0 . Since f is continuous and locally Lipschitz on R + 3 (as all denominators φ j + x j are strictly positive for x j ≥ 0 ), any trajectory starting in R + 3 remains in R + 3 for all t ≥ 0 . This confirms the mathematical coherence and biological consistency of the model, ensuring that biomass and nutrient concentrations remain non-negative at all times. 4.1. Bounds of the Dynamical Model To characterize the system’s global behavior, we apply the LCIS method. We define a bounded region in the state space that all trajectories eventually enter and are confined to. Let x 1 max , x 2 max , x 3 max ∈ R > 0 be upper bounds for an axis-aligned box of the system ( 1)–( 3): K ( h ) : = [ 0 , x 1 max ] ∩ [ 0 , x 2 max ] ∩ [ 0 , x 3 max ] ⊂ R + 3 . (5) Theorem 1(Localization and Boundedness) . We name the set K ( h ) a compact and positively invariant set for the system (1)–(3). For any solution starting in K ( h ) , the variables satisfy:0 ≤ x 1 ( t ) ≤ x 1 max , 0 ≤ x 2 ( t ) ≤ x 2 max , 0 ≤ x 3 ( t ) ≤ x 3 max , (6) withx 1 max = σ ρ 1 + ρ 2 α 1 + 1 , x 2 max = x 2 ( 0 ) , x 3 max = x 3 ( 0 ) , (7) where γ 2 : = ρ 3 + α 2 and γ 3 : = ρ 4 + α 3 . 4.2. Existence and Uniqueness of Solutions To ensure the mathematical well-posedness of the system, we analyze the existence and uniqueness of its trajectories. This property guarantees that for each initial condition, there is a unique physical evolution of the biological process. Consider the system x ˙ = f ( x ) on the domain R + 3 . Each component of the vector field f ( x ) consists of polynomial and rational terms. Since all parameters φ j are strictly positive, the denominators φ j + x j never vanish in R + 3 . Thus, f is continuous, and by the Cauchy–Peano theorem [ 16], a local solution exists for any initial condition x ( t 0 ) ∈ R + 3 . To establish uniqueness, we verify the Lipschitz condition on the compact invariant set K defined in ( 5). A sufficient condition for f to be Lipschitz on K is that it is C 1 and its Jacobian J f is bounded on K. For f 1 , we have: ∂ f 1 ∂ x 1 ≤ ρ 1 + ρ 2 + α 1 + 2 α 1 x 1 max σ , ∂ f 1 ∂ x 2 ≤ ρ 1 x 1 max φ 2 , ∂ f 1 ∂ x 3 ≤ ρ 2 x 1 max φ 3 . Similar bounds apply to f 2 and f 3 . Since all partial derivatives are bounded on K, there exists a constant M > 0 such that ∥ J f ( ξ ) ∥ ≤ M for all ξ ∈ K . By the mean value theorem, f is Lipschitz on K with constant L : = M . According to the above-mentioned Picard–Lindelöf theorem, there exists a unique local solution. Furthermore, since K is positively invariant (as proved in Section 1), trajectories cannot exit the compact set, ensuring that solutions exist globally for all t ≥ t 0 . 4.3. Equilibrium Points and Local Stability Analysis Firstly, to assess local stability, we compute the general Jacobian matrix of the system: J ( x 1 , x 2 , x 3 ) = ∂ f 1 ∂ x 1 ρ 1 x 1 φ 2 + x 2 ρ 2 x 1 φ 3 + x 3 − ρ 3 φ 1 x 2 ( φ 1 + x 1 ) 2 − ρ 3 x 1 φ 1 + x 1 + α 2 0 − ρ 4 φ 1 x 3 ( φ 1 + x 1 ) 2 0 − ρ 4 x 1 φ 1 + x 1 + α 3 . (11) Secondly, for any biologically feasible equilibrium point, it must hold that x 2 = 0 and x 3 = 0 , which yields α 1 x 1 ( 1 − x 1 / σ ) = 0 , leading to two equilibrium points: Extinction Equilibrium: E 1 = ( 0 , 0 , 0 ) . Evaluating the Jacobian at the origin, we obtain that the eigenvalues are λ 1 = α 1 , λ 2 = − α 2 , and λ 3 = − α 3 . Given that α 1 > 0 , the extinction equilibrium E 1 is a saddle point and is therefore unstable. Saturation Equilibrium: E 2 = ( σ , 0 , 0 ) . 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