Mathematical modeling of the learning process based on predator-prey interactions

Author(s):

DOI: https://doi.org/10.32782/2307-9770.2026.14.02.04

Paper Language: UKR

Download full-pdf

Abstract

The purpose of the article is to substantiate the possibility of applying modified predator–prey interaction models to the formal description of the learning process of a higher education student as a controlled dynamic system, considering the influence of external disturbances and internal psychophysiological states. The study employs methods of mathematical and simulation modeling. A hybrid model is proposed that integrates a second-order cybernetic learning model with a modified Lotka–Volterra framework. Two main state variables are introduced: the normalized level of knowledge acquisition and an integral indicator of disturbance intensity, which generalizes fatigue, cognitive overload, attentional instability, anxiety, and environmental influences. The model is extended by saturation mechanisms, background disturbances, controllable parameters, and inertia, which ensures its physical interpretability and stability. Numerical experiments are conducted using a state-space representation based on systems of differential equations. The simulation results reveal a three-stage structure of the learning process: an initial adaptation phase, a quasi-stationary regime, and a transition to a new steady state following improvements in learning conditions. It is shown that reducing disturbance intensity and increasing teaching effectiveness lead to a higher average level of knowledge acquisition and a decrease in the amplitude of fluctuations associated with fatigue and recovery. The phase trajectory in the “knowledge acquisition – disturbance intensity” space is proposed as an informative indicator of the quality of the learning environment and the effectiveness of pedagogical interventions. The scientific originality of the study lies in the integration of a second-order cybernetic learning model with a modified predator–prey interaction model to describe learning dynamics, where disturbances are represented as a separate dynamic loop with clear pedagogical interpretation, which has not been previously considered in such a form. The proposed model can be used to analyze and optimize the educational process, identify critical factors that reduce learning efficiency, and support the development of adaptive and personalized learning systems in higher education institutions.

Keywords

educational process of a higher education institution, second-order cybernetic model, Lotka–Volterra model, external and internal factors influencing the educational process, learning dynamics

References

  1. Semerikov S.O., Teplytskyi I.O. Osnovy kompiuternoho modeliuvannia u shkoli ta pedahohichnomu VNZ. URL: https://lib.iitta.gov.ua/id/eprint/704244/1/%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D1%8F.pdf (accessed 19.05.2026)
  2. Теплицький О. І., Теплицький І.О., Семеріков С.О., Соловйов В.М. Професійна підготовка учителів природничо-математичних дисциплін засобами комп’ютерного моделювання: соціально-конструктивістський підхід: монографія. Теорія та методика навчання фундаментальних дисциплін у вищій школі. Кривий Ріг:  Видавничий  відділ  ДВНЗ  «Криворізький  національний університет», 2015. Том  X. Випуск 1 (10): спецвипуск «Монографія  в журналі». 278 с.
  3. Plaskura P. Modelling of forgetting curves in educational e-environment. Information Technologies and Learning Tools. 2019. Vol. 71, No. 3. P. 1–11. DOI: 10.33407/itlt.v71i3.2368.
  4. Demianenko V. M. The model for adaptive learning systems of open education information environment. Information Technologies and Learning Tools. 2020. Vol. 77, No. 3. P. 27–38. DOI: 10.33407/itlt.v77i3.3603.
  5. Sikora Ya. B. Kibernetychnyi pidkhid do modeliuvannia navchalnykh system. Science, engineering and technology: global trends, problems and solutions: kolektyvna monohrafiia. Riga: Baltija Publishing, 2021. S. 54–57. DOI: 10.30525/978-9934-26-046-9-13.
  6. Ozhyndovych L. M. Application of problem / project-oriented technology training for students of technical and IT specialties on the example of the course “Website Content Management System”. Journal of Information Technologies in Education (ITE). 2019. No. 40. P. 80–94. DOI: 10.14308/ite000704.
  7. Stasiv M.-A., Luchkevych M., Khudyi A. Adaptive learning algorithms in a mobile application for foreign language learning. Information Systems and Networks. 2025. Vol. 18, No. 2. P. 129–136. DOI: 10.23939/sisn2025.18.2.129.
  8. Tabibian B., Upadhyay U., De A., Zarezade A., Schölkopf B., Gomez-Rodriguez M. Enhancing human learning via spaced repetition optimization. Proceedings of the National Academy of Sciences of the United States of America. 2019. Vol. 116, No. 10. P. 3988–3993. DOI: 10.1073/pnas.1815156116.
  9. Koedinger K. R., Carvalho P. F., Liu R., McLaughlin E. A. An astonishing regularity in student learning rate. Proceedings of the National Academy of Sciences of the United States of America. 2023. Vol. 120, No. 13. e2221311120. DOI: 10.1073/pnas.2221311120.
  10. Kumar A., Benjamin A. S., Heathcote A., Steyvers M. Comparing models of learning and relearning in large-scale cognitive training data sets. npj Science of Learning. 2022. Vol. 7, No. 1. Art. 24. DOI: 10.1038/s41539-022-00142-x.
  11. Xiao Q., Wang J. DRL-SRS: Reinforcement Learning for Personalized Scheduling in Spaced Repetition Systems. Applied Sciences. 2024. Vol. 14, No. 13. Art. 5591. DOI: 10.3390/app14135591.
  12. Liu S., Li Q., Shen X., Sun J., Yang Z. Automated discovery of symbolic laws governing skill acquisition from naturally occurring data. Nature Computational Science. 2024. Vol. 4, No. 5. P. 334–345. DOI: 10.1038/s43588-024-00629-0.
  13. Lu M., Marghetis T., Yang V. C. A. first-principles mathematical model integrates the disparate timescales of human learning. npj Complexity. 2025. Vol. 2, No. 1. DOI: 10.1038/s44260-025-00039-x.
  14. Koenig B. C., Spratt J. J., Sitzmann V., Bar-Sinai Y., Lippe P., Mohr A., Schaefer L., Pfaff T., Holl P. KAN-ODEs: Kolmogorov-Arnold Networks for learning ordinary differential equations. Computer Methods in Applied Mechanics and Engineering. 2024. Vol. 432. Art. 117397. DOI: 10.1016/j.cma.2024.117397.
  15. Wu R., Zong Z., Yu J., Zhang S., Yi Z. Attractors coexistence and representation research of Lotka–Volterra recurrent neural networks by matrix analysis theory. Neurocomputing. 2024. Art. 128040. DOI: 10.1016/j.neucom.2024.128040.
  16. Wesselkamp M., Moser N., Kalweit M., Boedecker J., Dormann C. F. Process-Informed Neural Networks: A Hybrid Modelling Approach to Improve Predictive Performance and Inference of Neural Networks in Ecology and Beyond. Ecology Letters. 2024. Vol. 27, No. 11. e70012. DOI: 10.1111/ele.70012.
  17. Wang H., Wu Q., Bao C., Ji W., Zhou G. Research on knowledge tracing based on learner fatigue state. Complex & Intelligent Systems. 2025. Vol. 11.  DOI: 10.1007/s40747-025-01831-x.