Mathematics

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Subject: search.filters.subject.collective dynamics

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    The emergence of lines of hierarchy in collective motion of biological systems
    (Institute of Physics, 2023-06-29) Greene, James M.; Tadmor, Eitan; Zhong, Ming
    The emergence of large-scale structures in biological systems, and in particular the formation of lines of hierarchy, is observed at many scales, from collections of cells to groups of insects to herds of animals. Motivated by phenomena in chemotaxis and phototaxis, we present a new class of alignment models that exhibit alignment into lines. The spontaneous formation of such ‘fingers’ can be interpreted as the emergence of leaders and followers in a system of identically interacting agents. Various numerical examples are provided, which demonstrate emergent behaviors similar to the ‘fingering’ phenomenon observed in some phototaxis and chemotaxis experiments; this phenomenon is generally known to be a challenging pattern for existing models to capture. A novel protocol for pairwise interactions provides a fundamental alignment mechanism by which agents may form lines of hierarchy across a wide range of biological systems.
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    Multi-scale problems on collective dynamics and image processing
    (2014) Tan, Changhui; Tadmor, Eitan; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Multi-scale problems appear in many contexts. In this thesis, we study two dif- ferent subjects involving multi-scale problems: (i) collective dynamics, and (ii) image processing. For collective dynamics, we concentrate on flocking models, in particular, Cucker-Smale and Motsch-Tadmor systems. These models characterize the emergent behaviors of self-organized dynamics. We study flocking systems in three different scales, from microscopic agent-based models, through mesoscopic kineitc discriptions, to macroscopic fluid systems. Global existence theories are developed for all three scales, with the proof of asymptotic flocking behaviors. In the macroscopic level, a critical threhold phenomenon is addressed to obtain global regularity. Similar idea is implemented to other fluid systems as well, like Euler-Poisson equations. In the kinetic level, a discontinuous Galerkin method is introduced to overcome the numerical difficulty due to the precence of δ -singularity. For image processing, we apply the idea of multi-scale image representation to construct uniformly bounded solutions for div U = F. Despite the fact that the equation is simple and linear, it is suprisingly true that its bounded solution can not be constructed through a linear procedure. In particular, the Holmholtz solution is not always bounded. A hierarchical construction of the bounded solution of the equation is proposed, borrowing the idea from image processing. We also present a numerical implementation to deal with the highly nonlinear construction procedure. Solid numerical result verifies that the constructed solution is indeed uniformly bounded.