This research is inspired by Brown, Bruder and Kummel’s research project on the predator-prey interaction of aphids and ladybugs on yucca plants. An important feature of this study system is that it contains ants as a third species. Therefore, this ecological system is composed of a predator-prey relationship between the ladybugs and aphids, a competitive relationship between the ladybugs and ants, and a mutualistic relationship between the aphids and ants. Most existing mathematical models study one type of interaction or they focus on three species and study a tri-trophic food chain. We develop and analyze a new mathematical model that includes the predator-prey interaction as well as the competitive and mutualistic aspects of the system. The predator-prey interaction is described by a Rosenzweig-MacArthur model, which assumes logistic growth of the predator. To build a mathematical model for the competitive and mutualistic relationships, we use a modified Lotka-Volterra model and include terms representing competition and mutualism. Since the three-species model is substantially harder to analyze, we first study the three submodels, i.e. the predator-prey, competition, and mutualism model. Then we use the submodel results to explore the three-species model and the significance of its parameter values. With the help of Mathematica and MATLAB, we construct phase planes and time series plots, find the equilibria of the systems, and determine the stability of each equilibrium.
Bullying is defined as a specific type of aggression, in which the behavior is intended to harm or disturb, the behavior occurs repeatedly, and there is an imbalance of power. This results in significant psychological damage in the victim, but also in the bully. Studies report the number of bullied children in middle schools as between 4% and 82%. The goals of our study are to understand how bullying behavior spreads in a population of adolescents, and to examine the impacts of the most common bullying intervention strategies. We propose a compartmental model parametrized using data on the prevalence of bullying. We compute the basic reproductive number R0 and perform numerical simulations and a sensitivity analysis of the model. An extension of the simple model includes the most common intervention strategies. Numerical simulations suggest that the Traditional Disciplinary Approach, although commonly implemented, is the least effective of the intervention strategies we study.
If a process is truly random, how can we begin to understand its behavior? We turn to probability theory to model stochastic processes and to answer important questions about how these systems work. Specifically, we turn to Markov chains. Markov chains have the property that, given the current state, the past has no affect on the future. Both for finite and infinite chains, studying the expected value, G(x, y), is critical to get a sense of what’s going on. Expectation allows us to understand how the random walk behaves in various dimensions. Furthermore, the limiting process of the random walk gives us Brownian motion, which is a useful model in physics, biology, and finance.
Epidemics have long been studied in the realm of mathematical modeling, aiding in parameter estimation, determining effective intervention methods, and, in some cases, predicting epidemics. How do networks and network structure influence infectious diseases? Are certain networks better suited to naturally contain an epidemic where others are not? In this paper, we will explore one method of stochastic modeling and implement it into a network of sub-populations. We will conclude with interesting results on networks and their dynamics in disease propagation.
Modeling traffic flow has historically been tackled using a variety of different approaches. The microscopic method uses a system of ordinary differential equations to model the interaction of a single car with its surroundings. This model has fixed pa- rameter values for all driver behavior characteristics, such as reaction time and reaction strength. Given that these parameters represent driver behavior, it is more realistic to introduce some stochasticity into the model parameters and observe the effect on the resulting traffic conditions. In this paper, direct simulation results show that stochastically varied parameter values produce larger free-flow regions, faster stabilization to free-flow conditions, and reduced traffic intensity overall. Fundamental diagrams show that the model behaves as predicted by real traffic data.
The Cantor middle-thirds set is an interesting set that possesses various, sometimes surprising mathematical properties. It can be presented through ternary representation and obtained through an iterative process. This paper will discuss selected topological properties of the Cantor set, as well as its connection to fractal geometry. It will then discuss the existence of the Cantor set in a variety of artistic contexts.
Many American classrooms have adopted a strategy which mostly focuses on rote memorization to teach math to its students. While this may sometimes help students pass multiple choice exams, we argue that this focus will not help students develop problem solving and critical thinking abilities. We created different activities targeted towards a wide range of age groups to see how students would respond to difficult problems that require perseverance and critical thinking. Each activity fit in with the Common Core Standards laid out by the Colorado Department of Education. Different activities catered to different age groups, but all of them were meant to be challenging for the audience. The culminating activity was shaped around a classic problem of Hercules fighting the Hydra. Hercules cuts off a head of the multi-headed dragon, and a certain amount of heads grow back based on rules we will define. We asked the students how the Hydra would behave as Hercules continued to attack. To help them with this problem, we compared it to a special group of fast growing sequences called Goodstein Sequences.