Browsing by Subject "pattern formation"

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    A Generalized Method for A Posteriori and A Priori Error Estimates for Homoclinic Orbits in Reversible Systems
    (2023-05) Jankovic, Sally
    Coherent structures in pattern-forming systems, such as pulses and spikes, are often mathematically represented as homoclinic orbits. We here present a generalized method for finding such homoclinic solutions to 2nd-order ODEs using a posteriori data derived from approximate solutions to boundary value problems on truncated intervals. We then show in the opposite direction that the a priori existence of a non-degenerate homoclinic implies the existence of a family of solutions to a Dirichlet boundary-value problem, with an explicit lower bound on domain size necessary to obtain a solution. In each direction, we also provide explicit error estimates as a function of truncation error. We lastly apply our method to find a family of homoclinics in the Lengyel-Epstein system and compute a minimum domain size for the existence of finite-domain solutions using the a priori proof.
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    Mathematical Modeling of Cell Migration: Mechanisms in Dictyostelium discoideum.
    (2023-03) Felix, Bryan
    This study aims to understand the biochemical pathways involved in the cytoskeleton of Dictyostelium discoideum, particularly the self-organization process of actin structures. While previous models have explored protein dynamics in various contexts, they tend to oversimplify the underlying biochemical mechanisms. This study presents two extended models that offer updated insights into the chemical interactions and feedback mechanisms of Dictyostelium discoideum. The first model explores how bistability emerges from the topology of the underlying network, while the second model focuses on the mechanisms of filopodia initiation and the role of membrane curvature in their emergence.