Mathematical modeling and sparse identification of delay-driven dynamical systems with applications to gene regulatory networks

Delay-driven dynamics arise naturally in many biological and physical systems, where finite processing times, transport effects, and memory mechanisms significantly influence system behavior. Although such delays are often not directly observable, they play a critical role in determining stability, oscillations, and pattern formation. This dissertation investigates delay-driven dynamical systems through a combination of mechanistic modeling and data-driven system identification, with a focus on oscillatory gene regulatory networks. The first part of the dissertation develops and analyzes delayed reaction–diffusion models motivated by the Hes1 gene regulatory system. A spatiotemporal framework is proposed to describe the interaction between Hes1 mRNA and protein, explicitly incorporating nuclear–cytoplasmic structure, diffusion, and transcriptional delay. Rigorous stability analysis is performed, and explicit conditions are derived for delay-induced Hopf bifurcation, demonstrating that time delay is the key parameter governing the transition from stable steady states to sustained oscillations. The second part of the dissertation focuses on identifying delay-driven dynamics directly from data. A novel data-driven framework is developed by extending the Sparse Identification of Nonlinear Dynamics (SINDy) methodology to systems with distributed delay using the Linear Chain Trick. This approach enables the simultaneous identification of governing equations, effective delay, and delay distribution from time-series data while preserving interpretability. Numerical experiments demonstrate the robustness of the method under noise and sparse sampling. Finally, the proposed framework is extended to spatiotemporal systems, enabling the identification of delayed reaction–diffusion models from data. This work establishes a systematic approach for discovering distributed-delay dynamical systems and provides a foundation for data-driven modeling of spatiotemporal systems with memory.

Name thesis:degree_name

Ph.D. (Doctor of Philosophy)

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Physics (UMKC)

Grantor

University of Missouri--Kansas City

Year dc:date.issued

2026

Author dc:creator

• Alanazi, Mohammed Naif

Advisors dc:contributor.advisor

• Bani-Yaghoub, Majid
• Rulis, Paul Michael, 1976-

Language dc:language.iso

en_US

Handle dc:identifier.uri

https://hdl.handle.net/10355/112325

OAI identifier oai:identifier

oai:mospace.umsystem.edu:10355/112325