This research examines computer and mathematical modeling of anaerobic digestion in livestock waste recycling. Various modeling approaches, including ADM1, are analyzed, and the devel-opment of a proprietary computer program based on a self-developed mathematical model is described. The proposed mathematical model of anaerobic fermentation has demonstrated effi-ciency and reliability under industrial-scale conditions. It was developed using regression analysis and correlation methods, establishing empirical relationships between biogas yield and 20, 26 key process parameters continuously measured by biosensors in real production settings. Despite a limited static dataset (only five measurement points), the model exhibited a low average ap-proximation error of 5.5% and a high determination coefficient of 0.95, confirming its accuracy and high quality. The computer program is designed for online statistical data transmission via cellular communication and the Internet, enabling the processing of large volumes of data in ac-cordance with Big Data concepts. Its architecture ensures flexibility, scalability, and precision. Furthermore, the integration of machine learning technologies is proposed to enhance the pro-gram's predictive capabilities, leveraging Big Data analytics. The proposed computer–mathematical model is thus considered a practical, accurate, and scalable solution for monitor-ing and optimizing industrial-scale anaerobic digestion processes, combining empirical modeling with modern data processing technologies.

Introduction: Contemporary educational environments require educators to participate in professional networks that extend across institutional levels. This study presents a methodology for AI-supported cross-level professional collaboration involving pre-service teachers, secondary-school teachers, and college instructors, with the AI-Ustaz platform serving as the shared technological environment. Methods: The methodology comprised three sequential stages: theoretical–orientational, technological, and project–networked. A cluster quasi-experimental pretest–posttest design was implemented during the 2024–2025 academic year with 347 participants across nine educational institutions in Kazakhstan. Participants were allocated to an experimental group (n = 178) and a control group (n = 169). Cognitive, performative, and reflective components of networked-collaboration competence were assessed. Results: Descriptive analyses indicated greater pretest-to-posttest improvements in the experimental group across the cognitive, performative, and reflective components. At posttest, the experimental group required approximately 61% less time to prepare instructional materials than the control group and received higher expert ratings for content quality. Discussion: The findings suggest that AI-supported cross-level professional collaboration may provide a promising model for educational practice. However, the results should be interpreted cautiously because participants were nested within institutional clusters and cluster-adjusted analyses could not be performed..