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Personal Information

My name is Ben Barros and I am from Wahiawa, Hawaii. I am currently attending Lindenwood University in St. Charles, Missouri and I will graduate in May of 2016 with a B.A. in Mathematics with certification in 9-12 Education. After I graduate, I plan on finding a job teaching high school mathematics and sharing my enthusiasm for mathematics with my students!

Research Topic

Food webs are an effective way to model ecosystems, but in application food webs are often extremely complex. In an effort to understand the food web of a native Hawaiian ecosystem, we define several new types of graphs to help with the study of ecosystems. A graph is a set of vertices (points) and edges (lines) that represent a relationship between two vertices. Competition graphs and (1,2)-step competition graphs show predator and prey relationships. Based on these structures, weights are added to the arcs (directed edges) and edges of different types of directed graphs and graphs. The introduction of weighted arcs and edges allows for empirical data to be used while working with these types of graphs (see Figures 1 and 2). We also define a directed graph called the benefit digraph that allows us to study the relationship between species that strictly (1,2)-compete. The ultimate goal is to obtain tools that better measure population changes in an ecosystem.

Weighted Food Web
Figure 1: Weighted Food Web
Weighted Predator Overlap Graph
Figure 2: Weighted Predator Overlap Graph

Milestones & Goals

Week 1

  • Complete education module, "The Biology and Mathematics of Food Webs", by Thursday, 6/3/2015
  • Attend the talk at 1:00 PM in CU 401
  • Search for information on any ecosystems in Hawaii. Look for information about who eats what. Find out how much of a species' energy comes from what type of species. Prepare initial findings for Monday, 6/8 meeting.

Week 2

  • Meeting Monday 6/8, with mentor.
  • Begin reading Kaitlyn Ryan's master's thesis, which gives information regarding the biological background, math background, and some statistical information on this topic. Make notes and questions for Thursday's meeting.
  • Begin reading "A Characterization of Competition Graphs"
  • Participate in the talk and luncheon at 11:30 on 6/11, CU 401
  • Mentor meeting on Thursday, 6/11 (go over questions and comments on Katie's paper; possibly questions on competition graph paper)
  • Start learning LaTeX
  • Update Wiki a end of the week or during the week to reflect on what has been accomplished

Week 3

  • Mentor meeting on Tuesday, 6/16
  • Attend the talk at 1:00 PM on Tuesday, 6/16, CU 401
  • Begin to read the paper, "The (1,2)-step Competition Graph of a Tournament";
  • Start to compare the competition graphs and (1,2)-step competition graphs by using some small food webs from the initial food webs packet, and by creating your own small food webs examples.
  • Attend the Responsible Conduct of Research Workshop from 9:00 AM to 1:00 PM on Thursday, 6/18, lunch is provided
  • Meet with mentor for a short meeting after Thursday's workshop
  • Try to use LaTeX to write up some of your findings
  • Find a way to draw a small food web on the computer; when successful, find a way to import it into your LaTeX manuscript.
  • Update Wiki at end of the week or during the week to reflect on what has been accomplished

Week 4

  • Begin to look at competition graphs and (1,2)-step competition graphs for your focus area.
  • Develop questions that can be asked in your focus area regarding the graphs
  • Continue to write up your findings in LaTeX and create figures
  • Attend the luncheon at 11:30 AM on Thursday, 6/25
  • Meet with mentor on Thursday, 6/25
  • Update Wiki at end of the week or during the week to reflect on what has been accomplished

Week 5

  • Have a draft of a 8-10 minute talk on what has been learned so far by 11:00 AM Monday, 6/29, and be prepared to show it to your mentor. You should have some things that you can use already in LaTeX.
  • Continue, as time allows, to work on the first and second bullet points of Week 4, and expand by beginning to answer these questions
  • Present 8-10 minute mini-presentation to peers and mentors on Thursday, 7/2
  • Update Wiki at end of the week or during the week to reflect on what has been accomplished

Week 6

  • Continue working on the first bullet point and LaTeX updates
  • Meet with mentor on Monday, 7/6; evaluate where you are on research and make plans for how to focus the rest of the research for the next 2-3 weeks.
  • Meet with mentor on Thursday, 7/9 to see how plans are going since Monday
  • Go to working lunch at 11:30 AM on Thursday, 7/9
  • Update Wiki at end of the week or during the week to reflect on what has been accomplished

Week 7

  • There should be some conjectures that have been made in your area by now. Determine how to approach a solution or at least support the conjectures. What more is needed for results? Is it possible to get it before the end of the REU? If not, how can the known information be used to make a satisfying result.
  • Mentor meeting on Monday, 7/13 to discuss the previous issues
  • Go to working lunch on Thursday, 7/16
  • Meet with mentor on Thursday, 7/16
  • Paper outline (brief) based upon last week and this week's meeting due to mentor via email on Friday, 7/17 by 5:00 PM
  • Update Wiki at end of the week or during the week to reflect on what has been accomplished

Week 8

(Mentor is out of town)

  • Begin to finalize some of the results you have; enter theorems, figures, explanations, etc. into LaTeX and create needed figures
  • Initialize your poster draft and email it to mentor by 8 PM Tuesday, 7/28; comments will be back to you by Monday for you to finalize
  • Participate in the luncheon on Thursday, 7/23
  • Update Wiki at end of the week or during the week to reflect on what has been accomplished

Week 9

  • Mentor meeting on Tuesday, 7/28 to discuss finalizing poster, how to change the mini-presentation from week 5 to a formal talk, and pointers on the final paper
  • Have updated poster emailed to mentor by 8 PM Tuesday, 7/28; comments from mentor will be back to you by 10 AM on Wednesday
  • Make any last minute change and submit your electronic version to Dr. Brylow by 12:00 PM on Wednesday, 7/29
  • Finalize any results that are still hanging out
  • Carefully construct a "future research" list that will go into your paper
  • Begin to turn your LaTeX writings into a formal paper.
  • Go to luncheon on Thursday, 7/30
  • Update Wiki at end of the week or during the week to reflect on what has been accomplished

Week 10

  • Draft of formal talk due to mentor by 11:00 AM Monday, 8/3
  • Draft of paper due to mentor by 11:00 AM Monday, 8/3
  • Poster session 1:00 - 3:00 PM on Tuesday, 8/4
  • Formal presentations:
    • Wednesday, 8/5, 10:00 - 2:00, CU 401, lunch provided
    • Thursday, 8/6, 10:00 - 12:00, CU 401
  • Final REU meeting at 11:00 AM on Friday 8/7, CU 401

Daily Log

Week 1

Monday, June 1st

  • Attended orientation staring a 9:00 a.m.
  • Reviewed program expectations and learned what to expect for the next 10 weeks.
  • Started filling out paperwork.
  • Had lunch with my mentor, Dr. Factor. Discussed various topics we are interested in that relate to graph theory.
  • I am planning on doing my project on invasive species in Hawaii and how they affect the eco-system.
  • Dr. Factor mentioned that I may be able to design an educational module based on the topic I am studying to use in a secondary classroom.
  • Orientation in the systems lab with Dr. Brylow. We weren't able to log in to the computers so most of the orientation was put off until Day 2.
  • Took Program Pre-Survey

Tuesday, June 2nd

  • Tour of Marquette Campus
  • Went to Human Resources to fill out more paperwork (W-4, I-9, etc.)
  • Obtained a Marquette ID card.
  • Began researching invasive species in Hawaii.

Wednesday, June 3rd

  • Obtained a Marquette email account granting me access to Marquette computers.
  • Worked with group on BioMath educational module.
  • Participated in Library Orientation. Learned how to use various databases available at Marquette.
  • Performed literature search for information on invasive species in Hawaii.

Thursday, June 4th

  • Met with Dr. Factor to discuss research project and establish milestones and goals.
  • Performed literature search about the invasive Coqui Frog and found several helpful articles.
  • Looked up information about competition graphs.
  • Attended talk by Dr. Factor about how to conduct research and common problems that we may run into.

Friday, June 5th

  • Started researching other invasive species in Hawaii such as the brown tree snake, mongoose, and red-vented bulbul.
  • Read more about competition graphs.
  • Considered which species will be best to use in a competition graph.
  • Started learning how to use LaTeX.
  • Participated in Bad Sci-fi night at the Brylow household.

Week 2

Monday, June 8th

  • Researched competition graphs and made a list of species to discuss with Dr. Factor.
  • Met with Dr. Factor. Discussed research progress and decided to pursue further research with Coqui frogs. Learned about competition graphs and (1,2)-step competition graphs.
  • Started reading "Using Food Webs in Order to Determine Possible Predictors of Primary and Secondary Extinctions" by Kaitlin Ryan.

Tuesday, June 9th

  • Read and took notes on article titled, "Potential predators of an invasive frog (Eleutherodactylus coqui) in Hawaiian forests" by Karen H. Beard and William C. Pitt.
  • Finished reading Kaitlin Ryan's thesis paper on food webs.
  • Practiced drawing various food webs, competition graphs, and dominator trees.
  • Started reading "A Characterization of Competition Graphs" by R.D. Dutton and R.C. Brigham.

Wednesday, June 10th

  • Read "Diet of the Invasive Frog, Eleutherodactylus coqui, in Hawaii" by Karen H. Beard.
  • Finished reading Competition Graph paper by Dutton and Brigham.
  • Met with Lenzi to discuss Kaitlin's paper and Competition Graph paper.
  • Started reading LaTeX tutorial.
  • Started reading "Potential Competitive Dietary Overlap Between the Invasive Coqui Frog and the Endemic Hawaiian Hoary Bat on the Island of Hawai'i" by Riley Fehr Bernard

Thursday, June 11th

  • Continued reading Dietary Overlap paper.
  • Printed out useful information about diets of Coqui frogs and Hawaiian Hoary Bats.
  • Lunch with mentors. Dr. Factor gave a short presentation to demonstrate what is expected for our mini-presentations in Week 5
  • Reviewed Characterization of Competition Graphs paper before meeting with Dr. Factor.
  • Met with Dr. Factor and went over Characterization of Competition Graphs
  • Worked with LATEX, learned how to type out equations and math symbols.

Friday, June 12th

  • Learned how to use the Wiki.
  • Updated goals and milestones in Wiki.
  • Typed daily logs into Wiki.
  • Researched different species that may be affected by the growth of the coqui frog population. I found a good amount of articles about ants, rats, and birds in Hawaii.
  • Studied a competition graph theorem and it's proof.
  • Read "Coqui frog invasions change invertebrate communities in Hawaii" by Ryan T. Choi and Karen H. Beard.
  • Skimmed "Biology of the Hawaiian Hawk" by Curtice Rosser Griffin. Gained important information about the prey of the Hawaiian Hawk.

Week 3

Monday, June 15th

  • Updated Wiki and corrected typos.
  • Started typing research into a LaTeX document.
  • Studied theorems from characterization of competition graphs paper.
  • Downloaded GraphTea, a graph theory program that makes drawing graphs on the computer possible.
  • Started reading "The (1,2)-step Graph of a tournament" by Dr. Factor and Sarah K. Merz.
  • Researched types of birds that live in Hawaii and what they eat.

Tuesday, June 16th

  • Continued reading "(1,2)-step Graph of a Tournament" paper.
  • Met with Dr. Factor and discussed "Characterization of Competition Graphs" paper.
  • Attended talk by Dr. Brylow about how to give a research talk and write a research paper.
  • Worked in LaTeX and learned how to create tables.
  • Created a table in LaTeX of different species in Hawaii and their possible prey.
  • Created a food web in Geogebra from the table in LaTeX. The food web was very complex!
  • Created a competition graph in Geogebra using the food web.
  • Learned about scripting in Geogebra.

Wednesday, June 17th

  • Reviewed "Characterization of Competition Graphs" paper.
  • Continued reading "(1,2)-step Competition Graph of a Tournament".
  • Practiced finding (1,2)-step competition graphs and kings.
  • Learned how to include images in LaTeX.
  • Learned how to export GeoGebra files into LaTeX.

Thursday, June 18th

  • Completed literature search on food webs.
  • Attended Responsible Conduct for Research (RCR) Seminar led by Dr. Brylow.
  • Met with Dr. Factor and discussed research progress.
  • Completed part two of RCR online.

Friday, June 19th

  • Looked over "Food Webs, Models and Species Extinctions in a Stochastic Environment" by Patrik Karlsson.
  • Worked with GraphTea and figured out how to import images from GraphTea into LaTex.
  • Downloaded and experimented with a graph theory program called Tulip.
  • Learned how to insert and reference figures in LaTeX.
  • Sketched out plan for educational module.
  • Worked with Bibtex and attempted to create a bibliography for my research.

Week 4

Wednesday, June 24th

  • Finished reading "(1,2)-step Competition Graph of a Tournament".
  • Learned how to import all sources into one place using BibDesk.
  • Started revising food web to create a more accurate model of Hawaii's ecosystem.
  • Started creating slides in LaTeX for mini-presentation.

Thursday, June 25th

  • Continued working on slides in LaTeX.
  • Attended luncheon and listened to talk given by Dr. Spiller.
  • Met with Dr. Factor to discuss progress on mini-presentation.
  • Researched different species of birds in Hawaii and recorded their predators and prey.
  • Sketched out slide content for mini-presentation.

Friday, June 26th

  • Continued working on mini-presentation.
  • Created a list of species to include in food web.
  • Created a food web using Geogebra. The food web was very complex because there were over 25 vertices in the graph.
  • Revised food web and figured out which species can be combined into one vertex.
  • Created a new food web with only 18 vertices.
  • Produced a competition graph and a (1,2)-step competition graph from the food web.
  • Typed up results in LaTeX.

Saturday, June 27th

  • Created slides for mini-presentation
  • Started writing down questions about food webs and their competition graphs.

Week 5

Monday, June 29th

  • Met with Dr. Factor and group to do mock presentations.
  • Started revising slides for Thursday's presentation based on constructive feedback from mentor and group.

Tuesday, June 30th

  • Rewrote definitions for presentation.
  • Finished slides for presentation.
  • Created works cited slide for presentation and cited all sources.

Wednesday, July 1st

  • Read part of "Interdisciplinary Curriculum: Design and Implementation" by Heidi Hayes Jacobs.
  • Began writing an outline for education module.
  • Created new food webs and competition graphs that do not include Brown Tree Snake.
  • Worked with group to revise presentations.

Thursday, July 2nd

  • Gave mini-presentation about how project is going so far.
  • Listened to other students give presentations and filled out evaluations.

Friday, July 3rd

  • Researched information on weighted competition graphs and weighted food webs.
  • Sketched a weighted food web and considered how to create a competition graph.

Week 6

Monday, July 6th

  • Continued exploring weighted competition graphs
  • Met with Dr. Factor and discussed weighted food webs and educational module.
  • Explored how to create a competition graph from a weighted food web.
  • Considered the implications of (1,2)-step competition graphs of food webs.
  • Attempted to create weighted (1,2)-step competition graphs.
  • Worked on outline of educational module.

Tuesday, July 7th

  • Worked on definitions of new graphs.
  • Completed the last phase of RCR training.
  • Started reading "Food Web Patterns and Their Consequences" by Stuart L. Primm, John H. Lawton, and Joel E. Cohen.

Wednesday, July 8th

  • Wrote definitions of new graphs in LaTeX.
  • Revised definitions and came up with some conjectures.
  • Researched ecology curriculum for educational module.

Thursday, July 9th

  • Finished reading "Food Web Patterns and Their consequences".
  • Researched Trophic Cascades in food webs.
  • Learned how to draw curved arcs in Geogebra.
  • Attended luncheon with peers and mentors.
  • Met with Dr. Factor to discuss findings.

Friday, July 10th

  • Worked on proofs for conjectures.
  • Revised definitions.

Saturday, July 11th

  • Revised proofs and came up with counterexamples for some conjectures.
  • Constructed weighted graphs in GeoGebra.
  • Started learning how to draw graphs using LaTeX.

Week 7

Monday, July 13th

  • Constructed diagrams in LaTeX.
  • Discussed progress with Dr. Factor via telephone.
  • Continued revising definitions and proving conjectures.

Tuesday, July 14th

  • Continued to write proofs for conjectures.
  • Constructed additional diagrams to supplement proofs.
  • Created example for finding benefit weight.

Wednesday, July 15th

  • Revised the definition of weighted benefit digraphs.
  • Tested various types of digraphs and observed the resulting benefit digraphs.
  • Began working with weighted vertices.

Thursday, July 16th

  • Meeting with Dr. Factor to discuss progress.
  • Attended luncheon and listened to Dr. Factor give talk about poster presentations.
  • Continued working with weighted vertices in food webs.

Week 8

Monday, July 20th

  • Revised definition of vertex weight in food webs.
  • Created weighted food web in Geogebra.
  • Calculated the vertex of species in Hawaiian food web.

Tuesday, July 21st

  • Created weighted competition graph in Geogebra.

Wednesday, July 22nd

  • Constructed weighted benefit digraph in Geogebra.
  • Found counter example for one of my theorems.
  • Revisited the definition of a weighted benefit digraph.
  • Started working on draft of poster.

Thursday, July 23rd

  • Began gathering findings and putting together draft of research paper.
  • Participated in luncheon and listened to talk by Dr. Brylow.
  • Continued working on poster.

Friday, July 24th

  • Explored different types of posters online and read guides on creating an effective research poster.
  • Decided on which style I wanted for my poster and continued to fill the poster with content.

Week 9

Monday, July 27th

  • Transferred definitions from LaTeX to PowerPoint for poster.
  • Wrote Introduction and Conclusion sections for poster.

Tuesday, July 28th

  • Finished up contents of poster and started working on poster design.
  • Sent draft of poster to Dr. Factor for comments and feedback.

Wednesday, July 29th

  • Applied finishing touches to poster and submitted poster to Dr. Brylow for printing.
  • Meeting with Dr. Factor to discuss progress on paper and final presentation.
  • Prepared findings in LaTeX for use in research paper.

Thursday, July 30th

  • Created abstract for formal presentation.
  • Participated in weekly luncheon with peers.
  • Brief meeting with Dr. Factor to discuss abstract for presentation.
  • Began creating slides for final presentation.

Friday, July 31st

  • Created figures for presentation.
  • Added animations to presentations.

Week 10

Monday, August 3rd

  • Met with group to rehearse formal presentations.

Tuesday, August 4th

  • Participated in poster presentations.

Wednesday, August 5th

  • Attended formal presentations.
  • Delivered 15 minute formal presentation.

Thursday, August 6th

  • Attended formal presentations.

Friday, August 7th

  • Attended final meeting of REU.
  • Completed post-REU survey.
  • Revised research paper and submitted final draft.