## MCS 358

### Introduction

Describing a mathematical model is often more challenging than presenting a mathematical result without application.  Presenting the proof of a theorem requires precision and clarity, but the truth of the result ultimately depends only on deductive logic.  Describing and analyzing a mathematical model also requires these components, but one also needs to convince the reader that the mathematics accurately describes the phenomenon being modeled.  Thus the writing must be especially clear and convincing.  The reader must understand the implications and limitations of the model and be able to judge the utility and application model and it's analysis.

This document provides some ideas and suggestions that you should keep in mind while preparing your final report.  The Writing Center can provide additional assistance.  Their web site has some helpful documents on grammar rules and other aspects of paper writing.

I have provided a pdf version of the paper Adaptation versus migration in demographically unstable populations (without figures) written by myself and Richard Gomulkiewicz as an example of the ideas I discuss below.

### Organization

The outline below is the basic structure of a modeling paper.  The outline is far from comprehensive; you should add and/or modify the structure to fit your specific needs.

1. Abstract
2. Introduction
3. Model Description
4. Model Analysis
5. Discussion and/or Conclusions
6. References

#### Abstract

The abstract is a brief (about 100 words) synopsis of the paper.  It should concisely describe what you are modeling and what you can conclude from your model.  If appropriate it should also indicate how this result compares to similar models.  The abstract in the example paper describes in very terse terms what is being modeled, the mathematical method of analysis, and the main idea of the result.

#### Introduction

The introduction serves many purposes and is often the most difficult section to write.  I often write this section last (or somewhere near the end) because by then I have a pretty good perspective about what is important and what readers should know about my work.  The introduction should put your work in a historical context.  How does it build from earlier research (either mathematical or in the modeled discipline)?  What is the main idea and how is it related to other results?  The example paper begins by citing the papers on evolutionary biology from which that work builds.  These references are used to reinforce the importance of the work and to put the given work in a historical context.  The introduction concludes by describing the main evolutionary biology results.

#### Model Description

Readers who have gotten this far have already decided your work is relevant.  They have not judged the quality of the work and this is your first opportunity to convince them that you know what you are doing.  This section should contain a detailed description of the phenomenon you are modeling.  Descriptions of relevant experimental or observational work should be presented with references to additional information if appropriate.

The mathematical model should be derived with assumptions and hypotheses clearly noted.  In the example paper the assumptions are embedded naturally in the text and are presented as they are used in the derivation of the model equations.  This is because the use of mathematics in evolutionary biology has a long history and it is assumed that the reader will be sufficiently familiar with the field that no special attention to the assumptions is needed.  In other papers I have written, I have explicitly enumerated my assumptions because I felt that they were sufficiently complex to warrant special attention and additional comment.  In your projects it is probably better to take the latter approach.  Note also that assumptions can be made for a variety of reasons and it is usually important to justify to the reader the reason for the assumption.  Some assumptions are made because the modeled phenomenon is complex and one wishes to limit the scope of the model.  In the example we assume a diploid population (like humans), but many species (notably many plants) are polyploids and this paper does not address these species.  On the other hand, some assumptions are made to simplify the mathematics.  The discrete-breeding assumption, random mating assumptions in the example are of this type.  When in doubt, let the reader know (either here or in the discussion) why a given assumption was made.

#### Model Analysis

The model analysis section (or sections) is where one uses mathematical analysis to describe what the model says about the given phenomenon.  In addition, you may also need to include some mathematical background material if you believe your reader may be unfamiliar with the mathematics being used.  The analysis can take many forms including the traditional Theorem-Proof format of a mathematics paper, computer simulation, statistical analysis, and many other components.  The mathematical analysis should be detailed, but one should also assume that your reader is "working along with you" and can fill in tedious details.  In the example paper we used two sections of analysis, the first is of the Theorem-Proof style, and the second contains some simulations that reinforce the earlier work.  We felt that the techniques used in this paper were sufficiently unusual to require a bit of mathematical background.  Thus we presented a very brief description of these ideas as well as references to sources containing complete descriptions of the methods.  Section 4 contains some numerical simulations.  In this paper this section is used to amplify the results and to point out by way of example some of the more subtle aspects of the theorems of the previous section.

However you choose to present the analysis, the use of figures and examples is essential in conveying your results.  I often read a paper by first scanning the figures and tables and then finding the relevant sections of text.

#### A Note on Figures and Equations

Figures need to be numbered and include a caption.  A caption should be descriptive enough to allow the reader to understand the basic idea without digging into the text.  It should be also short enough to not lose the reader.  This can be a tough task, but it is important.  Details of how the figure was generated should be included in the body of the paper.  Graph axes should be labeled (with units if appropriate and possible).  If more than one graph is shown on a single set of axes then either a legend should be included or a key provided in the caption.

Many times it is better to include equations on separate lines as opposed to imbedded in the text.  Sometimes this is not done if the equation is small (such as x=3).  Use your judgment as to what looks best and what is easiest to read.  Generally speaking, equations are not stand alone objects but should be included in the natural sentence flow.  Equations should generally not be the start of a sentence.  It is often tempting to use the equal sign (=) as a verb.  Don't do it. Sometime this requires some careful sentence crafting to avoid this.  If a displayed equation ends a sentence include the period at the end of the equation.  For examples of all this, see equations 5-8 in the example and the text around them.

#### Discussion and/or Conclusions

The discussion should summarize your results and put them into context with similar work.  It should also point out any limitations that may exist and possibly point the direction to further work.  You can be more detailed in your summary because the reader has presumably read the rest of the paper (remember this is not necessarily true, but the level of detail in your discussion should assume that).  It is here where you will probably need to cite other research on similar topics.  Note that in the example we begin with a paragraph that summarizes the results.  The next 5 paragraphs we discuss other papers and how this work is either related to them or differs from them.  This is extremely important, academic honesty requires us to give credit where credit is due.  This not only includes providing references to books and papers we looked at, but also honestly describing how other authors dealt with similar phenomena and how we did or did not borrow from their ideas.

#### References

In any research paper references are essential.  They indicate where you got many of your ideas and data and they point others to related work.  Not much to say here except follow some standard format.  Also remember to cite web sites if appropriate.  I cited by name and year (because that is what the journal wanted) but citing by number is fine as well.  You probably only need to cite references directly related to your work, either mathematically or otherwise.  If you have to look something up in a Calculus book, you probably don't need to cite it.  When in doubt cite.  Also note that it is sometimes necessary to cite people who assisted you in some way.  We have done this in this paper.  If you get help from someone other than the instructor (either personally or via email) you probably should cite them as a personal communication.