Published electronically September 27, 2018DOI: 10.1137/18S01720XM3 Challenge Introduction
Authors: Michael Vronsky, Ryan Huang, Daniel Wang, Justin Ju, and Joanne Yuan (Los Altos High School, Los Altos, CA)Sponsor: Carol Evans (Los Altos High School, Los Altos, CA)
Published electronically October 10, 2018DOI: 10.1137/18S016643
Authors: Mackenzie Jones (The University of Akron), Alison Frideres (Simpson College), and Bailey Kramer (University of Wisconsin-Stout)Sponsor: Keith Wojciechowski (University of Wisconsin-Stout)
Abstract: Near Menomonie, Wisconsin the lakes suffer from algae blooms during the warm summer months. Mathematical models describing the cyanobacteria population dynamics are studied with the intent of analyzing the conditions under which the populations grow and stabilize. Two models are considered, one for forecasting the population as the lake turns toxic from excess biomass after a flushing event occurs, and the other for establishing an algal bloom stability condition. These models are proposed for consideration to test the effectiveness of solutions put forth to ameliorate the algal bloom problem.
Published electronically November 8, 2018DOI: 10.1137/17S015860
Author: Wei (Maggie) Wu (Scripps College)Sponsor: Deanna Needell (University of California, Los Angeles)
Published electronically November 26, 2018DOI: 10.1137/18S016655
Authors: Caitlin Shener, Brian Oceguera, and Sally Lee (UCLA)Sponsor: Mason Porter (UCLA)
Abstract: We develop an approach for exploring questions regarding language use and connections between people in social networks. In particular, we investigate community structure and language usage in the network composed of the most followed ninety-nine users in Twitter. Our goal is to measure the relation between a community of users and the words employed by those users. We accomplish the investigation using k-means clustering to group users based on word choice, and we use modularity maximization and InfoMap clustering to find communities based on network connections. Our study illustrates how to mathematically analyze and interpret language use within social network structure.
Published electronically November 30, 2018DOI: 10.1137/18S017181
Authors: Kristen Lawler (Marist College), and Christian Schmidt (The College of Saint Scholastica)Sponsor: John Alford (Sam Houston State University)
Abstract: Whooping cranes (Grus americana) became endangered in the late 1800s when population numbers dwindled to as low as only 15 wild birds. Today, the wild population has significantly rebounded, but remains threatened. The last surviving migratory flock of whooping cranes travels to Texas every year for the winter. When the cranes arrive they primarily forage for wolfberries, but as the berry population decreases eventually the cranes' diet switches to one of mainly blue crabs. Understanding how abiotic conditions such as drought and water level fluctuations affect these food resources is important to wildlife managers who are tasked with the protection of the cranes. In this paper, we formulate a dynamic model that predicts the net energy intake of the whooping crane with parameters that may be used to control both the abiotic and biotic characteristics of the ecosystem inhabited during winter. We optimize the model to establish the maximum net energy intake of the crane over one season in ANWR and determine the point at which a crane will switch from foraging for berries to foraging for blue crab. This switching point gives insight to the cranes' foraging behavior, allowing us to have a better idea of what steps will be most effective moving forward if human action is needed for conservation of the species. Furthermore, the complete development and analysis within this paper will aid in determining whether or not sufficient resources exist in a crane's territory for it to sustain the winter and prepare for a successful flight back to breeding grounds.
Published electronically November 30, 2018DOI: 10.1137/18S01717X
Author: Mark Dibbs (University of Louisiana-Lafayette)Sponsor: Amy Veprauskas (University of Louisiana-Lafayette)
Abstract: In this paper, we develop a density dependent model to describe sperm whale population dynamics in the Gulf of Mexico. For this model, we consider the stability of the extinction equilibrium and prove the existence and uniqueness of a positive equilibrium. We then examine the stability of the positive equilibrium and substantiate the results with numerical simulations using Matlab. Next, we consider the effect of a disturbance, such as the Deepwater Horizon oil spill, on the sperm whale population in the Gulf of Mexico. Describing a disturbance as an event that results in reductions in survival rates for a certain period of time, we examine the recovery time of a sperm whale population following a disturbance, which we define to be the length of time it takes the population to return to a certain percentage of its asymptotic equilibrium value. Through testing various recovery threshold values, reductions in survival rates, and lengths of time over which the survival rates are reduced, we find that the recovery time is sensitive to each of these variables; depending on the values of these three quantities, the recovery time can last anywhere between a few years and many centuries.
Published electronically December 6, 2018DOI: 10.1137/18S017053
Author: Aaron Yeiser (MIT)Sponsor: Alex Townsend (Cornell University)
Abstract: When numerically solving partial differential equations (PDEs), the first step is often to discretize the geometry using a mesh and to solve a corresponding discretization of the PDE. Standard finite and spectral element methods require that the underlying mesh has no skinny elements for numerical stability. Here, we present a novel spectral element method, that is numerically stable on meshes that contain skinny quadrilateral elements, while also allowing for high degree polynomials on each element. Our method is particularly useful for PDEs for which anisotropic mesh elements are beneficial and we demonstrate it with a Navier-Stokes simulation. This method is presented without analysis and demonstrated using with Laplace-type operators. Code for our method can be found at https://github.com/ay2718/spectral-pde-solver.