This thesis focuses on ecological models of population dynamics and the traveling, migratory waves that can result when a stable state either displaces an unstable state, or displaces another stable state. We consider the effect of nonlocal interactions, where members of the species interact over a distance. This gives rise...
Recent years have witnessed success in using mathematical models to understand complex social phenomena. In this dissertation, we develop and apply two mathematical models in the area of urban productivity and political elections. ', 'First, we investigate the puzzling superlinear scaling behavior of many outputs in urban areas, such as...
In this work, we explore the utility of the three main types of neural networks: feed forward, convolutional, and recurrent. While using these networks, we develop a new way to model multiagent trajectory data, explore the use of multiple activation functions for neurons at each layer of a neural network,...
The goal of this thesis is to develop practical algorithms with sound theoretical properties for three different subfields of nonlinear optimization: (i) Optimization Algorithms for Supervised Machine Learning; (ii) Derivative-Free Optimization; and (iii) Distributed Optimization. As such, the thesis is divided into three main chapters.
The focus of Chapter 2...
This dissertation investigates a single-column model for Arctic energy balance in the limit as a smoothing parameter associated with ice-albedo feedback tends to zero. This limit is a common modeling approximation used in conceptual climate models, and we explore the implications of taking this limit on bifurcations associated with the...
Bacterial biofilms are aggregates of cells that adhere to nearly any solid-fluid interface. While many have harmful effects, such as industrial damage and nosocomial infections, certain biofilm species are now generating renewable energy as the fundamental components of Microbial Fuel Cells (MFCs). In an MFC, bacteria consume organic waste and,...
Vegetation in dryland environments is often patchy in response to water limitation. This patchiness can take the form of periodic patterning at length scales much larger than that of an individual plant. Instances of patterns resembling leopard spots and tiger stripes are widespread in dry regions of Africa, Australia, and...
Minimal mathematical models are used to understand complex phenomena in the physical, biological, and social sciences. This modeling philosophy never claims, nor even attempts, to fully capture the mechanisms underlying the phenomena, and instead offers insights and predictions not otherwise possible. Here, we build and explore minimal dynamical systems models...
Mixing due to cutting-and-shuffling is studied at a fundamental level using 2D mappings known as Piecewise Isometries (PWI) which can create beautiful mixing patterns. The PWI studied here splits a hemispherical shell (HS) into four curved triangular pieces that are rearranged to make a shuffled HS. Applying the PWI repeatedly...
A fractal is created by a Piecewise Isometry (PWI) which cuts a 2D object into pieces and rearranges those pieces to recreate the original object in a scrambled form. The PWI studied here splits a hemispherical shell into four pieces along cutting lines and rearranges them into a shuffled hemisphere....