Investigation of properties of processes characterized by logistic curves, using probabilistic cellular automata

D.P. Dimitrichenko

---

Abstract: One of the most successful processes studied is the development of systems that are either limited in area (populations), have limited development potential (technologies), or are associated with a small market size (product producers). These processes have a similar mathematical description, which involves constructing the corresponding logistic curves. Despite the obvious external differences between the listed processes, their underlying similarities are revealed through the cybernetic approach. The processes of development of biological, technical, and economic systems, which are described using C-shaped curves with the resources limited, testify to the cybernetic nature of the interaction mechanism within these systems.
Aim. The paper aims to identify system-wide patterns and mechanisms of development under limited resources.
Research methods. A one-dimensional probabilistic cellular automaton has been used to study developmental processes under resource constraints. The automata, which possess a scalar degree of development, compete for shared resources.
Result. The use of one-dimensional cellular automata allowes us to easily obtain interpretable results and analyze the impact of different dominance conditions on the internal diversity of the system.
Conclusions. The analysis of the structures allows to identify the dependence of the internal diversity of the system on the choice of dominant representatives during the competition for resources. In addition, the study predicts regarding the nature of changes in the system’s structure under conditions not only of internal constraints, but also of the presence of another system that implements the “predator-prey” relationship.

Keywords: saturated growth, intrasystem interaction mechanism, limited resources, exponential growth, logistic curve, one-dimensional cellular automaton, state probability, direction of dominance, internal diversity

For citation. Dimitrichenko D.P. Investigation of properties of processes characterized by logistic curves, using probabilistic cellular automata. News of the Kabardino-Balkarian Scientific Center of RAS. 2025. Vol. 27. No. 6. Pp. 142–156. DOI: 10.35330/1991-6639-2025-27-6-142-156

References

1. Novikov D.A. Kibernetika: navigator. Istoriya kibernetiki, sovremennoe sostoyanie, perspektivy razvitiya [Cybernetics: Navigator. History of Cybernetics, Current State, Development Prospects]. Moscow: LENAND, 2024. 160 p. (In Russian)
2. Riznichenko G.Yu. Matematicheskie modeli v biofizike i ekologii [Mathematical Models in Biophysics and Ecology]. Izhevsk: Institut komp’yuternykh issledovaniy, 2003. 184 p. (In Russian)
3. Khaykin S. Neyronnye seti: polnyi kurs [Neural Networks: Comprehensive Course]. Moscow: Vil’yams, 2006. 1104 p. (In Russian)
4. Beer S. Mozg firmy [The brain of the firm]. Moscow: Librokom, 2009. 416 p. (In Russian)
5. Foster R. Obnovlenie proizvodstva: atakuyushchie vyigryvayut [Innovation: Attacker’s Advantage]. Moscow: Progress, 1987. 272 p. (In Russian)
6. Al’tshuller G.S. Tvorchestvo kak tochnaya nauka [Creativity as an Exact Science]. Moscow: Sov. radio, 1979. 184 p. (In Russian)
7. Viner N. Kibernetika, ili Upravlenie i svyaz’ v zhivotnom i mashine [Cybernetics, or control and communication in the animal and the machine]. Moscow: Nauka, 1983. 343 p. (In Russian)
8. Viner N. Kibernetika i obshchestvo [Cybernetics and Society]. Moscow: AST, 2019. 288 p. (In Russian)
9. Ashby W.R. Vvedenie v kibernetiku [An Introduction to Cybernetics]. Moscow: Izd-vo inostrannoy literatury, 1959. 430 p. (In Russian)
10. Toffoli T., Margolus N. Mashiny kletochnykh avtomatov [Cellular Automata Machines]. Moscow: Mir, 1991. 284 p. (In Russian)
11. Gladkov L.A., Kureichik V.V., Kureichik V.M. Geneticheskie algoritmy [Genetic Algorithms]. Moscow: FIZMATLIT, 2010. 368 p. (In Russian)
12. Netrusov A.I., Kotova I.B. Mikrobiologiya [Microbiology]. Moscow: Akademiya, 2009. 352 p. (In Russian)
13. Enatskaya N.Yu. Teoriya veroyatnostei: uchebnik dlya vuzov [Probability Theory: textbook for universities]. Moscow: Yurayt, 2025. 204 p. (In Russian)

Information about the author

Dmitriy P. Dimitrichenko, Candidate of Technical Sciences, Researcher, Department of Neural Networks and Machine Learning, Institute of Applied Mathematics and Automation – branch of the KabardinoBalkarian Scientific Center of the Russian Academy of Sciences;
89 A, Shortanov street, Nalchik, 360000, Russia;
dimdp@rambler.ru, ORCID: https://orcid.org/0000-0003-2399-3538, SPIN-code: 3272-3520