Study of growth models formulated by random differential equations. Computational methods and simulation

• Autores: Cristina Pérez Diukina
• Directores de la Tesis: Rafael Jacinto Villanueva Micó (dir. tes.) , Juan Carlos Cortés López (dir. tes.)
• Lectura: En la Universitat Politècnica de València ( España ) en 2025
• Idioma: inglés
• Tribunal Calificador de la Tesis: José Carlos Valverde Fajardo (presid.) , Almudena Sánchez Sanchez (secret.) , Hugo Alexander de la Cruz Cancino (voc.)
• Enlaces
  • Tesis en acceso abierto en: RiuNet
• Resumen

  • Many mathematical models attempt to describe the dynamics of physical phenomena using differential equations. The scientific community acknowledges that these models are powerful tools for studying complex problems. However, when applying them to real situations, it is essential to consider uncertainty. This uncertainty may arise from the lack of knowledge of all the factors determining the process's dynamics due to its complexity or from measurement errors introduced when collecting the samples needed to use these models in practical scenarios.

    From this perspective, in the context of differential equations, two main approaches can be distinguished: Stochastic Differential Equations (SDEs) and Random Differential Equations (RDEs).

    In SDEs, uncertainty is introduced directly into the equation through a stochastic process. This process, referred to as noise, is often Gaussian in nature, such as white noise.

    On the other hand, RDEs consider the direct randomization of the corresponding deterministic differential equation. In this case, the model data i.e., initial conditions, source terms, and/or model parameters are treated as random variables or stochastic processes. Unlike SDEs, RDEs do not assume a probabilistic pattern in the formulation of the differential equation.

    For this reason, the RDE-based approach offers greater flexibility in handling uncertainty. The distributions of the model data are determined from the observed data whose dynamics (Uncertainty Quantification) one seeks to capture. Nevertheless, to rigorously apply RDE theory, it is necessary to assume a certain regularity in its formulation, such as continuity (or piecewise continuity) of its inputs.

    In growth models, this assumption is reasonable, since changes typically occur gradually; however, growth patterns with sudden changes -characterized by discrete jumps- can also be addressed through RDEs, as illustrated throughout this thesis.

    This work also aims to broaden the scope of RDEs by extending their applicability beyond continuous trajectories to incorporate random jumps, which serve as a critical tool for modeling real-world phenomena characterized by abrupt and unpredictable changes. By allowing for discontinuous dynamics, this approach provides a more comprehensive framework for capturing scenarios where sudden shifts occur, such as the administration of multiple doses of a drug in pharmacokinetics, where the concentration of a substance in the body experiences sharp increases at specific intervals, or environmental shocks that significantly influence biological or ecological systems, such as the influence of allelochemicals.

    Furthermore, this research delves into the modeling of systems with random domains, where the periodicity of an input function is itself uncertain, introducing an additional layer of complexity and realism. This is particularly relevant in contexts where external conditions or constraints evolve unpredictably over time, such as the movement of cells in a photobioreactor.