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== BACKGROUND INFORMATION ==

I am a rising Junior at The University of Wisconsion. I plan to graduate in May 2015 with Bachelor's Degrees in Mathematics and English Literature after which I am considering graduate study in the field of Mathematics. This summer I am working with [http://www.marquette.edu/mscs/facstaff-factor.shtml Dr. Kim Factor] and [[Nathan Sponberg]] on [[Characteristics of the (i,j)-Step Competition Graphs of Real Food Webs]].

== DESCRIPTION: ==


In the 1960s, Cohen developed competition graphs to model ecosystems. Since then, a lot of theory has been developed on the subject.



Here, we will examine the behavior of the (i,k)-step competition graph of real food webs. Background for the model will be obtained from: 
 1) A bio-math education module developed by Rutger’s University on the topic of food webs
 2) Graph theory texts by Parmenter and Roberts
 3) Material garnered from Cohen’s work in the bio-math field
 4) Mathematics research papers related to competition graphs and (1,2)-step competition graphs


== Weekly Log: ==


'''Week 1:'''

Goals:
* Complete the education module, “The Biology and Mathematics of Food Webs”
* Enter the title, description and milestones/goals into the Wiki 
* Begin working on graph theory exercises to obtain background

Progress:
This week was a shortened week and most of the time on Tuesday was spent on orientation to the program. The goals for the rest of the week were aimed at beginning to form a foundation in Graph Theory and begining to form a clearer idea of how to approch the project. To develop the biological terminolgy that will be used for the food webs, I completed the education module "The Biology and Mathematics of Food Webs". The rest of the week was spent reading graph theory to learn terminology and begining to work through basic examples from "Applied Combinatorics" by Roberts and "Discrete Mathematics with Graph Theory" by Goodaire and Parmenter.

'''Week 2:'''
* Go over basic graph theory material in textbooks, do problems, and have questions ready for meeting with mentor by 11:00 AM Tuesday, 6/4
* Read and discuss “A Characterization of Competition Graphs” 
* Obtain the text Community food webs: Data and theory (Springer-Verlag, New York, 1990) 
* Begin typing background material for the paper into a format you can use in your final paper (LaTex)

Progress:
The main objective of week 2 was to establish a solid foundation in the field of Graph Theory. To do this we continued, early in the week, to work out of the same texts used week one. We completed background reading and worked through problems and proofs given as excersizes in the books. I spent time on Tuesday working with Nate to understand some of the more complicated proofs as well as some of the nuances used in Graph Theory proofs as opposed to more familiar proofs. On Tuesday we also met with Dr. Facor and Kenny to discuss progress thus far, go over questions, and outline the next step. Dr. Factor provided us with some published papers which relate the graph theory concepts we had been working with in the texts to some of the biology we gleaned from the education module. Wednesday was speant reading the papers and working through some of the proofs in the papers and on Thursday, as well as devoting more time to work through the papers, I met with Nate to talk thorough the papers. This week I also aquired "Community Food Webs" by Cohen, Briand and Newman and began to look through the text. The final part of the week was used to begin entering some of the background material from the text books and papers into a LaTex file which will be used when compling the final paper.

'''Week 3:'''
* Read and discuss paper #2 “Competition Graphs of Strongly Connected and Hamiltonian Digraphs”
* Come up with examples of competition graphs of strongly connected digraphs to illustrate the paper
* Find material regarding food webs that answers questions regarding the assumed maximum height of a food web; find examples of real food webs; compare and find the competition graphs of the food webs
* Begin to read paper #3 or “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 web packet, and by creating your own small food web examples.
* Find a way to draw a small food web on the computer and import it into your LaTex draft

Progress:
This week we finished reading the papers and then went back to re-read them and go over the proofs in detail. Part of this step was working with the proofs and definitions, especially the construction proofs, to create examples to illusrate the theorems and proposiions and encourage a better understanding of he ideas presented in the papers. This week time was also speant working to learn how to create digraphs and graphs in LaTeX. We used examples from the initial BioPacket from week one to build examples of food webs as well as their competition and (1,2)-step competition graphs. These examples helped us begin to get a handle on the proporties of digraphs which apply to food webs and the properties of graphs which apply to(1,2)-step competition graphs and in what ways these properties may be related. We met with Dr. Factor to discuss our progress on the examples and papers and talk more about the future direction of the project and possible research topics. I then had a conversation with Nate about what topics might interest each of us. Through out the week I continued to refine my TeX document to reflect new information as well as to elimiante superfluous information. I also learned how to create a bibliography in BibTeX. Time was also spent at the end of the week looking through "Community Food Webs" to supplememnt background and to inform possible research questions.

'''Week 4:'''
* Begin to look at properties of the (1,2)-step competition graph (connected other than basal species vertices, complete other than basal species, etc.)  - What must be true about the food web for this to happen?
* Continue to write up findings in LaTeX and create figures to illustrate examples

Progress: 
This week was the true transition from gleaning background information to beginning to work on specific projects. As such I went through each paper and article a final time to ensure I was comfortable with all the concepts, definitions, terminology, and proofs. This was done by reading, re-reading, re-wording, and construction examples. After I was certain I was fully comfortable with all the papers I began to think about my own project. I continuously found myself curious about what type of height resriction a food web with a complete other than basal species (1,2)-step competition graph would have. This led me to decide to look further into the propoerties of the food webs which give rise to complete other than basal species (1,2)-step competition graphs. I began with some basic proofs which helped me to work through and reword some definitions which was helpful in allowing me to look at the food webs in diferent and perhaps more helpful ways. I then, from the reworked definitions, began to look for a height requirement. I then spent my ime coming up with theories and counter examples until I landed on a condition which is neccessary for completeness, but not sufficient. I began to work on some proofs which I hope will become relevant, but still need to go through and ensure they are complete. I also used what I had typed up in TeX to outline a presentation in beamer for the progress report next week. I also took some time to go over with Nate what I was working on, thinking, and hoping to do next, and he shared with me some properties of digraphs which can represent food webs which he dervived from some of the readings. The final project I started this week was playing wih the best way to present defintions and examples in my presentation for next week, my ultimate conclusion that a more visual presentation will be beneficial so I would like to find out how to use different colors on graphs in TeX.  

'''Week 5:'''
* Have a draft 8 – 10 minute talk prepared on what has been learned so far 
* Continue, as time allows, to work on the first bullet point of Week 4
* Present 8 – 10 minute informal talk 

Progress:
The early portion of this week was in large spent on preparing the presentation for the REU group. This allowed me to go through the background and work I had already completed and decide which aspects of the background were relevant and which aspects of the work were interesting. Simultaneously I also worked to refine the conditions and proofs I had worked on Week 4. I have discovered that proof by contradiction is an invaluable tool for this project and therefore took some time to look into the language used in the proofs which allowed me to refine my work. I also continued to look for a sufficient condition for completeness. Assuming limited exposure to graph theory a lot of background needed to be covered in my presentation so I was careful when presenting definitions to make them accessible as well as to use lots of examples which was not only intended to be helpful to my audience but was also helpful to me. By determining which pieces of information were relevant, I got a chance to adjust my perspective and narrow down the amount of information I am dealing with which helped make the problem more manageable. I learned a few more useful tricks in LaTeX, such as how to change the colors of lines and vertices in graphs which has allowed me to more clearly illustrate collections as well as insets and outsets. Then, in determining what was most relevant in my findings, and creating examples to illustrate those findings, I was able to see a new line of inquiry. After completing and practicing my presentation, the new line of inquiry led me to toy with the insets and outsets of vertices. Specifically I have been looking at the consequences of a vertex with a single vertex outset. I spent Friday attempting to find a condition of these outsets which is related to complete (1,2)-step competition graphs but was thus far unable to find them. The final condition I began working with this week was the relation of herbivores to other herbivores, which clearly requires direct competition, but as a result can create interesting questions of insets and outsets.

'''Week 6:'''
* Continue working on the first bullet point and LaTeX updates

Progress:(4th of July Week - no work Thursday)
This week was a continuation of the past week and my interest in the insets and outsets of specific vertices. I was able to determine that the only type of vertex which may have an outset of size one is an herbivore. This condition, while also necessary for a complete (1,2)-step competition graph, is also not sufficient. But, because of my claim last week, that all herbivores must compete with other herbivores, this claim means that if there is an herbivore with outset one, then all other herbivores must have that same basal species in their outset for the (1,2)-step competition graph to be complete. I have then considered that this makes all other basal species superfluous to the (1,2)-step competition graph but have not yet proven or disproven this. I instead became interested in under what conditions a vertex may have an inset of one. After finding lots of possible conditions, followed by even more counterexamples, I think that the outset line of inquiry is more revealing than the question of insets. Although I would still like to explore if I can perhaps create sufficiency by combining the inset and outset question by placing a minimum on the outset of a vertex then adjusting the inset. Other than these conditions, the main problem I faced this week was in the wording of my claims preventing me from proving them with any sort of elegance or accuracy, so although I could see the proof, I was unable to write in until rewording my claim which ended up posing a greater challenge than anticipated and results which are still being adjusted and readjusted as new pieces of the puzzle become clear. A future goal which has stemmed from my work this week, is to create a condition which combines some of these smaller findings. This week also included an introduction to making a poster, an undertaking which I have as of yet not attempted therefore an introduction to the basics was helpful, as well as the chance to see some good and bad examples.  

'''Week 7:'''
* Continue according to progress during Week 6 - a solid end goal should be clear at this point in time
* Paper outline (brief) due to mentor 

Progress:
This week I began working on a new approach to the problem at hand. I started working the opposite direction, so instead of stating necessary properties of Digraphs with complete on all non basal vertices (1,2)-step competition graphs and focusing more on forbidden subgraphs and properties, I looked at complete on all non basal vertices (1,2)-step competition graphs and started to draw the possible minimal digraphs which correspond to each of the (1,2) step completion graphs. The goal of this exercise was to look for patterns in the graphs which give rise to (1,2)-step competition graphs which are complete on all non basal vertices. Although there are some patterns, I have not yet been able to find a consistent condition sufficient for completeness. I also spent time this week working with an extension of the condition from last week the only type of vertex which may have an outset of size one is an herbivore. I was able to extend this condition to say that a vertex with a path length of two to a basal species must have a second path of length no greater than two to the same basal vertex to ensure (1,2)-step competition with the intermediary vertex. Although this condition solved some of the problems I was running into, the counter examples which show that this condition is not sufficient run into the problem of two top predators being non-competing. This led me to look somewhat into the number of species at each tropic level but that line of inquiry has yet to yield anything of value. The discussion this Thursday was about graduate school. I really appreciated the information and found it al to be very helpful. I still however find the idea of graduate school intimidating and am forced to think of which fast food joint is the best career option. I am currently leaning towards Panera because bread bowls.   

'''Week 8:'''	
* Begin to finalize some of the results and enter theorems, etc. into LaTeX and create needed figures
* Initialize  poster draft and email it to mentor

Progress:
This week I worked mainly on trying to finding a condition that proved to be sufficient to imply that a digraph has a (1,2)-step competition graph which is complete on all non basal vertices. Where I have been running into problems is in cases where two top vertices do not compete or (1,2)-compete and therefore are not connected in the (1,2)-step competition graph which in turn leaves it incomplete. There are multiple ways to remedy this problem, two top predators may directly compete by preying on the same herbivore or omnivore, they may also however (1,2)-compete if one is an omnivore which eats a basal species which is a path of 2 away from the other, or one eats an species which eats a species which is eaten by the other. It is therefore inaccurate to say all top predators must directly compete. The solution began to appear to be a condition on the size of each tropic level. It began to appear that if the number of top predators was greater than half the number of intermediary species, the conditions I have already presented would be sufficient to imply completeness and that if the number of top predators was less than or equal to half the number of intermediary species I could coerce the digraph into having a (1,2)-step competition graph complete on all non basal vertices by placing a lower bound on the size of the outset of the top predators. As indicated above however, there are ways in which omnivores can create (1,2)-competition so this condition is not necessary although it appears to be sufficient. I also began to look at poster outlines and examples and to plan out what information I wanted on my poster. I decided to save the conditions I was working on this week for the paper as the thoughts are not yet fully formed.     

'''Week 9:'''   (Mentor out of town)
* Have updated poster emailed to mentor 
* Make any changes to the poster on Tuesday; submit the electronic version 
* Finalize any results
* Carefully construct a “future research” list that will go into your paper
* Begin to turn  LaTeX writings into a formal paper (based on papers read regarding competition and (i,k)-step competition) 

'''Week 10:'''
* Draft of formal talk 
* Draft of paper 
* Poster session 1
* Formal presentations 
* Paper due electronically