Authors:James S. Cook et al. Abstract: The method of undetermined coefficients is used to solve constant coefficient nonhomogeneous differential equations whose forcing function is itself the solution of a homogeneous constant coefficient differential equation. In this paper, we show that the classical methods for tackling constant coefficient equations, including the method of undetermined coefficients, generalize to much wider class linear differential equations which, for example, include Cauchy-Euler type equations. This general method includes an explicit construction of the fundamental solution sets of such equations. We also briefly consider where this method can be applied by producing the most general second and third order differential equations that are polynomial in a first order differential operator. In addition, we provide a number of constant coefficient, Cauchy-Euler, and novel examples. PubDate: Fri, 30 Sep 2022 11:35:33 PDT

Authors:Maila Hallare et al. Abstract: We consider a three-species fish dynamical system in Chesapeake Bay consisting of the Atlantic menhaden as the prey and its two competing predators, the striped bass and the catfish. Building on our previous work in this system, we consider the issue of balancing economic harvesting goals (financial gain for fishermen) with ecological harvesting goals (non-extinction of species). In particular, we investigate the bionomic equilibria, maximum sustainable yield, and the maximum economic yield. Analytical computations and numerical simulations are employed to provide some mathematical guidance on fisheries management policies. PubDate: Tue, 23 Aug 2022 13:10:40 PDT

Authors:Frank H. Lynch Abstract: A mathematical model of competitive binding on a microarray in real-time yields a planar system of nonlinear ordinary differential equations. This model can be used to explore dimensionless formulation, linear approximation, and reduction. Real-time competitive binding is proposed as an uncommon approach to advance the study of planar systems of differential equations. PubDate: Fri, 11 Mar 2022 17:20:38 PST

Authors:Samantha Secor et al. Abstract: Starting with a toy climate model from the literature, we employ a system of two nonlinear differential equations to model the reciprocal effects of the average temperature and the percentage of glacial volume on Earth. In the literature, this model is used to demonstrate the potential for a stable periodic orbit over a long time span in the form of an attracting limit cycle. In the roughly twenty five years since this model appeared in the literature, the effects of global warming and human-impacted climate change have become much more well known and apparent. We demonstrate modification of initial conditions to understand how human activity could affect the model results. Although we too see the attracting limit cycle that yields a periodic orbit, we demonstrate that small perturbations in initial conditions can lead to extreme outcomes due to the presence of a nearby saddle point. We simulate the results over time to to highlight the critical nature of perturbations that in effect change the initial conditions and to determine how soon drastic climate events might take place. PubDate: Fri, 11 Mar 2022 17:05:43 PST

Authors:Michael C. Barg Abstract: This is an account of how a reading and writing project in an introductory differential equations course was transitioned to a professor-student research group collaborative project, in response to the global COVID-19 pandemic. Adapting on the fly to the ever-evolving pandemic, we collected data, estimated parameters in our models, and computed numerical solutions to SIR-based systems of differential equations. This is a description of what we did and how we found comfort in the project in this time of great uncertainty. The collaboration yielded successes and more questions than we had answers for, but the situation provided an opportunity of a lifetime for my students to engage in a real-world developing situation. PubDate: Sat, 13 Mar 2021 22:05:42 PST

Authors:Francesca Bernardi et al. Abstract: We suggest the use of historical documents and primary sources, as well as data and articles from recent events, to teach students about mathematical epidemiology. We propose a project suitable -- in different versions -- as part of a class syllabus, as an undergraduate research project, and as an extra credit assignment. Throughout this project, students explore mathematical, historical, and sociological aspects of the SIR model and approach data analysis and interpretation. Based on their work, students form opinions on public health decisions and related consequences. Feedback from students has been encouraging.We begin our project by having students read excerpts of documents from the early 1900s discussing the Indian plague epidemic. We then guide students through the derivation of the SIR model by analyzing the seminal 1927 Kermack and McKendrick paper, which is based on data from the Indian epidemiological event they have studied. After understanding the historical importance of the SIR model, we consider its modern applications focusing on the Ebola outbreak of 2014-2016 in West Africa. Students fit SIR models to available compiled data sets. The subtleties in the data provide opportunities for students to consider the data and SIR model assumptions critically. Additionally, social attitudes of the outbreak are explored; in particular, local attitudes towards government health recommendations. PubDate: Sat, 13 Mar 2021 22:00:35 PST

Authors:Glenn Ledder Abstract: Observed whale dynamics show drastic historical population declines, some of which have not been reversed in spite of restrictions on harvesting. This phenomenon is not explained by traditional predator prey models, but we can do better by using models that incorporate more sophisticated assumptions about consumer-resource interaction. To that end, we derive the Holling type 3 consumption rate model and use it in a one-variable differential equation obtained by treating the predator population in a predator-prey model as a parameter rather than a dynamic variable. The resulting model produces dynamics in which low and high consumption levels lead to single high and low-level stable resource equilibria, respectively, while intermediate consumption levels result in both high and low stable equilibria. The phase line analysis is made more transparent by applying a particular structure to the function that gives the derivative in terms of the state. By positing a consumption level that starts low, gradually increases through technological change and human population growth, and decreases as a result of public policy, we are able to tell a story that explains the unexpectedly rapid decline of some resources, such as whales, followed by limited recovery in response to conservation. The analysis also offers guidelines for how to establish sustainable harvesting for restored populations. We include a bifurcation analysis and suggestions for how to teach the material with three different levels of focus on the modeling aspect of the study. PubDate: Sat, 13 Mar 2021 21:50:25 PST

Authors:Iordanka N. Panayotova et al. Abstract: We present an inquiry-based project that is designed for a mathematical modeling class of undergraduate junior or senior students. It discusses a three-species mathematical model that simulates the biological interactions among three important fish species in the Chesapeake Bay: the prey Atlantic menhaden and its two competing predators, the striped bass and the non-native blue catfish. The model also considers the following ecological issues related to these three species: the overfishing of menhaden, the invasiveness of the blue catfish, and the harvesting of blue catfish as a method to control the population. A series of modeling scenarios are considered based on some simplifying assumptions to demonstrate the application of theoretical concepts to actual fisheries in the Chesapeake Bay. Analysis involves elementary skills such as finding the roots of polynomial equations, computing eigenvalues and eigenvectors, and some advanced topics such as Routh-Hurwitz criteria and the Hartman-Grobman Theorem. Numerical simulations via MATLAB are utilized to produce graphical simulations and analyze long-time behaviors. Our model predicts that if no serious measures are taken to prevent the spread of the invasive blue catfish, the native predator species will be seriously affected and may even become extinct. The model also shows that linear harvesting is sufficient to limit the growth of the invasive catfish population; however, it is not sufficient to save the striped bass from becoming extinct. The results of this study illustrate the fundamental ecological principle of competitive exclusion, according to which two competing species that attempt to occupy the same niche in an ecosystem cannot co-exist indefinitely and one of the two populations will either go extinct or will adapt to fill a different niche. PubDate: Sat, 13 Mar 2021 21:50:15 PST

Authors:Chinenye Ofodile Abstract: Today's world is global. However, despite increasing numbers and diversity of participants in Study Abroad programs, only 10% of U. S. college students get that experience. There is an ever-growing need for students to become aware of and experience other cultures, to understand why others think and act differently. Internationalization is the conscious effort, begun nearly 40 years ago, to integrate an international, intercultural, and global dimension into the purpose, functions, and delivery of post-secondary education.Albany State University began a Global Program Initiative in the 1990s. In 2016, we extended into mathematics the curriculum innovations of this program. The result has engaged students in a serious way, both in mathematical modeling and in cultural research. We have introduced students to new skills of research and presentation. For the past few years we have offered one section of Calculus II in traditional mode and one in internationalized mode, and we have compared results. In this article, we give details of the process and highlight the success of the program. We end with more recent examples from Spring 2020. PubDate: Sat, 13 Mar 2021 21:50:05 PST

Authors:James S. Sochacki Abstract: A neural cell or neuron is the basic building block of the brain and transmits information to other neurons. This paper demonstrates the complicated dynamics of the neuron through a numerical study of the Hodgkin-Huxley differential equations that model the ionic mechanisms of the neuron: slight changes in parameter values and inputted electrical impulses can lead to very different (unexpected) results. The methods and ideas developed for the ordinary differential equations are extended to partial differential equations for Hodgkin-Huxley networks of neurons in one, two and three dimensions. PubDate: Mon, 30 Nov 2020 16:00:53 PST

Authors:Mel Henriksen et al. Abstract: Specifications-based grading (SBG) is an assessment scheme in which student grades are based on demonstrated understanding of known specifications which are tied to course learning outcomes. Typically with SBG, students are given multiple opportunities to demonstrate such understanding. In undergraduate-level introductory ordinary differential equations courses at two institutions, SBG has been found to markedly decrease students’ self-reported anxiety related to the course as compared to traditionally graded courses. PubDate: Sun, 13 Sep 2020 14:05:31 PDT

Authors:Younes KarimiFardinpour Abstract: The growing importance of education equity is partly based on the premise that an individual's level of education directly correlates to future quality of life. Educational equity for differential equations (DEs) is related to achievement, fairness, and opportunity. Therefore, a pedagogy that practices DE educational equity gives a strong foundation of social justice. However, linguistic barriers pose a challenge to equity education in DEs. For example, I found myself teaching DEs either in classrooms with a low proficiency in the language of instruction or in multilingual classrooms. I grappled with a way to create an equity educational environment that supported students both with social justice mathematics and satisfaction for the students. According to the 2000 NCTM (National Council of Teachers of Mathematics) Principles and Standards for School Mathematics, mathematics instructors should strive to meet the demands of Principles and Standards. However, meeting the standards such as the Equity Principle can be a daunting task. DE student diversity in language proficiency might enrich the educational environment, but it also raises challenges. This brief report summarizes a way to build an equity learning environment by providing visualization support. PubDate: Mon, 25 Mar 2019 21:09:31 PDT

Authors:Haynes Miller Abstract: Since 2013 I have been traveling to Haiti as part of the "MIT- Haiti Initiative." This initiative, led by Professor Michel DeGraff of the MIT Department of Linguistics and Philosophy, aims to encourage active learning strategies, enabled by technology when possible and appropriate, and strongly stresses the importance of the use of the one language spoken by all Haitians, namely Haitian Creole (or Kreyòl). To use a Haitian metaphor, these three components form the three stones on which the cook-pot of our educational approach rests. We have focused our attention on the higher education sector.In this note I will begin with a review of the educational landscape that forms the background for our efforts. Then I will describe the work of the MIT-Haiti Initiative and some of the efforts undertaken in Haiti by participants in our workshops. I will discuss the mathematical material developed for these workshops and what we learned in leading them, and then describe the findings of a site visit to a campus of the State University of the Haiti. I will end by discussing how the typography of equity spelled out by Rochelle Gutiérrez (Teaching for Excellence and Equity in Mathematics, 2009) applies in the Haitian educational setting. PubDate: Mon, 25 Mar 2019 21:09:21 PDT

Authors:Dan Flath Abstract: For thousands of years the population of Earth increased slowly, while per capita income remained essentially constant, at subsistence level. At the beginning of the industrial revolution around 1800, population began to increase very rapidly and income started to climb. Then in the second half of the twentieth century as a demographic transition began, the birth and death rates, as well as the world population growth rate, began to decline. The reasons for these transitions are hotly debated with no expert consensus yet emerging. It's the problem of economic growth. In this document we investigate a mathematical model of economic growth proposed by Michael Kremer in 1993. PubDate: Mon, 25 Mar 2019 21:09:12 PDT

Authors:Selenne Bañuelos et al. Abstract: The United States has proven to be and remains a dual political party system. Each party is associated to its own ideologies, yet work by Baldassarri and Goldberg in Neither Ideologues Nor Agnostics show that many Americans have positions on economic and social issues that don't fall into one of the two mainstream party platforms. Our interest lies in studying how recruitment from one party into another impacts an election. In particular, there was a growing third party presence in the 2000 and 2016 elections. Motivated by previous work, an epidemiological approach is taken to treat the spread of ideologies and political affiliations among three parties, analogous to the spread of an infectious disease. A nonlinear compartmental model is derived to study the movement between classes of voters with the assumption of a constant population that is homogeneously mixed. Numerical simulations are conducted with initial conditions from reported national data with varying parameters associated to the strengths of political ideologies. We determine the equilibria analytically and discuss the stability of the system both algebraically and through simulation, parameters are expressed to stabilize a co-existence between three parties, and numerical simulations are performed to verify and support analysis. PubDate: Mon, 25 Mar 2019 21:09:00 PDT

Authors:Jessica Deters et al. Abstract: How does a lie spread through a community' The purpose of this paper is two-fold: to provide an educational tool for teaching Ordinary Differential Equations (ODEs) and sensitivity analysis through a culturally relevant topic (fake news), and to examine the social justice implications of misinformation. Under the assumption that people are susceptible to, can be infected with, and recover from a lie, we model the spread of false information with the classic Susceptible-Infected-Recovered (SIR) model. We develop a system of ODEs with lie-dependent parameter values to examine the pervasiveness of a lie through a community.The model presents the opportunity for the education of ODEs in a classroom setting through a creative application. The model brings a socially and culturally relevant topic into the classroom, allowing students who may not relate with purely technical examples to connect with the material. Including diverse perspectives in the discussion and development of mathematics and engineering will enable creative and differing approaches to the worlds' problems. PubDate: Mon, 25 Mar 2019 21:08:46 PDT

Authors:Lorelei Koss Abstract: This article surveys how SIR models have been extended beyond investigations of biologically infectious diseases to other topics that contribute to social inequality and environmental concerns. We present models that have been used to study sustainable agriculture, drug and alcohol use, the spread of violent ideologies on the internet, criminal activity, and health issues such as bulimia and obesity. PubDate: Mon, 25 Mar 2019 21:08:36 PDT

Authors:Therese Shelton et al. Abstract: Cholera is an infectious disease that is a major concern in countries with inadequate access to clean water and proper sanitation. According to the World Health Organization (WHO), "cholera is a disease of inequity--an ancient illness that today sickens and kills only the poorest and most vulnerable people\dots The map of cholera is essentially the same as a map of poverty." We implement a published model (Fung, "Cholera Transmission Dynamic Models for Public Health Practitioners," Emerging Themes in Epidemiology, 2014) of a SIR model that includes a bacterial reservoir. Bacterial concentration in the water is modeled by the Monod Equation in microbiology. We investigate the sensitivity of the models to some parameters. We use parameter values for cholera in Haiti that are consistent with the ranges in meta-analysis by Fung and other sources. We show the results of our numerical approximation of solutions. Our goal is to use this system of nonlinear ordinary differential equations to raise awareness among the mathematics community of the dynamics of cholera. We discuss the enhancement of undergraduate experiences by motivating learning with a real-world context in social justice implications of global health. PubDate: Mon, 25 Mar 2019 21:08:20 PDT