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Research Plans

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The general theme of my mathematical research is using numerical modeling to solve problems that are intractable from a purely theoretical standpoint. I currently work on projects that lend themselves to collaborations with undergraduates. I also develop projects specifically designed to draw otherwise uninterested undergraduates in to research. I complement this work by investigating best practices in undergraduate education especially as it relates to preparing them for research and project work.

I originally studied equations at the intersection of differential geometry and mathematical physics. Numerical techniques must be chosen carefully. Other researchers used Runge-Kutta methods with higher-order accuracy, but these broke the symmetry of the equations calling all numerical results into question. I used stable but less accurate methods known to preserve the symmetries in the original problem and small step sizes to get highly accurate results (Linhart[2002]).

I spent several years working in a variety of mathematical and statistical software positions, adding knowledge of statistics to my classical mathematics training and broadening my potential areas for research and my teaching flexibility. During this time I developed another interest in fast and accurate algorithms; I published algorithms and code to accurately compute the logarithm of the normal distribution both in double-precision arithmetic and to any desired precision (Linhart[2008]).

My current research is in mathematical ecology with phytoplankton species, where systems of differential or stochastic equations are used to study the competition and coexistence of species. The dynamics of these equations are very interesting; in some cases you see chaotic behavior (Huisman[2001]), and with seemingly minor changes to the system, the behavior becomes predictable (Roelke[2003]).

Phytoplankton are unicellular organisms that photosynthesize. They depend on three main nutrients, nitrogen, phosphorus, and silicon. Phytoplankton are present in all water environments. Chemostat experiments for growth and competition in the lab have verified this differential equation system governs species interaction (Tilman[1982], Sommer[1989]), making them ideal for numerical experiments to explore questions of competition and coexistence

(1)   \begin{align*} \dfrac{dN_i}{dt} &= \mu_i N_i - \nu N_i \\ \dfrac{dS_j}{dt} &= \nu (\tilde{S}_j -S_j) - \sum_{i = 1}^n Q_{ij} \mu_i N_i \label{nutrient_eq}\end{array}\end{align*}

These equations are linked by an equation for resource dependent growth. Below is the Monod equation, an essential nutrient model, where species need a minimal amount of each nutrient in order to survive, and nutrients are used proportionally:

(2)   \begin{equation*}\mu_i = \tilde{\mu}_i \min_j\left\{\dfrac{S_j}{S_j + \kappa_{ij}}\right\}\end{equation*}

The three equation system governs species and nutrient interaction, competition and coexistence. Here N_i is the population density of species i, S_j the concentration of resource (substrate) j, \tilde{\mu} the maximum growth rate per capita for species i, \mu_i the instantaneous (resource dependent) growth rate per capita for species i, \nu the flushing rate, and Q_{ij} a constant rate for species i to consume resource j. The simplicity of this equation system, especially when restricted to only a few species, makes it ideal for introducing undergraduates to mathematical modeling and research.

Tilman[1982] argues persuasively based on these equations that, with three nutrients available, at most three species may coexist, but this result contradicts the diversity of species in our ecosystem. Explaining the physical and biological processes that maintain diversity, including finding mathematical models that show how diverse ecosystems develop and interact, are overarching questions.

My current work includes identifying mechanisms that allow two species to coexist in nature, even though chemostat experiments and initial numerical models indicate one should out-compete and exclude the other species. Future work will include addressing the question by which mechanisms do meta-communities (Hanski[2004]), e.g. lakes connected by rivers, promote diversity in the ecosystem. This involves much more complicated interconnected systems of differential equations. Another interesting question is what mathematically explains why adding species to an ecosystem, i.e., increasing species diversity, prolongs species diversity, i.e., the time it takes for the ecosystem to reach equilibrium where all but a few species are excluded. A third direction for exploration is adding stochasticity via the Gillespie algorithm and to seeing if the results of such an addition match laboratory experiments better than the original equations.

Inspired by an undergraduate research project I created on the Zombie Apocalypse based on Munz[2009] that asks students to question the paper’s assumptions and look for scenarios in which humans are more likely to survive, I intend to create a similar paper and concomitant projects based on the idea of parasitoid models. A parasitoid, unlike a parasite, eventually kills its host. The most famous example of a parasitoid is the alien from the movie Alien, which, like the zombies, will capture student interest. Models based on topics in popular culture are a good way to bring mathematical research to undergraduates, especially those not normally drawn enthusiastically to mathematics research. My students have won several awards for their work on the zombie project.

The final pillar of my scholarly research is work on best practices in teaching mathematics. For the most part, mathematics is taught in a lecture/exam format with few changes from how courses were taught nearly a thousand years ago, but these practices are not optimal today. Effective writing and communication are two skills at the head of every list of what employers and graduate schools are looking for, and I have work in review (Linhart[2013]) on writing in my mathematical modeling course. Since effective writing and communication is so important, I have added it to all of my classes with 50 students or fewer. I have noticed that reflective writing, which is not at all typical for mathematics classes, has a lasting impact on what students learn. I would like to investigate two interesting questions that result from this observation: 1) What type of reflective assignments have the greatest impact? 2) Why is it that these assignments have such impact?

Programming is another fundamental skill desired by employers and useful in graduate work, yet often programming stymies modern students. I have had a great deal of success teaching this in my classes, and this would also be a rich subject for exploration and discourse.

Writing, reflective work, and programming are the fundamental skills that allow students to succeed at undergraduate research projects, especially the ones I have available; thus my education research complements my more traditional mathematical research.

Bibliography

Ilkka Hanski and Oscar E. Gaggiotti, editors. Ecology, Genetics and Evolution of Metapopulations. Elsevier Academic Press, 2004.

Jef Huisman and Franz J. Weissing. Fundamental unpredictiability in Multispecies competition. The American Naturalist, 157(5):488–494, 2001.

J. M. Linhart. Algorithm 885: Computing the logarithm of the normal distribution. ACM Transactions on Mathematical Software, 35, 2008. Article 20 (10 pages).

J. M. Linhart. 2013. Teaching Writing in a mathematical modeling course. Accepted to PRIMUS.
URL:
http://www.jeanmarielinhart.info/wp-content/uploads/2014/01/linhart_primus_20131.pdf

J. M. Linhart and L. A. Sadun. fast and slow blowup in the S^2 \sigma model and $(4+1)-dimensional Yang Mills model. Nonlinearity, 15:219–238. 2002.
URL: http://www.jeanmarielinhart.info/wp-content/uploads/2014/01/blowup.pdf

P. Munz, I. Hudea, J. Imad, and R. J. Smith? When zombies attack! Mathematical modeling of an outbreak of zombie infection. In J. M. Tchuenche and C. Chiyaka, editors, Infectious Disease Modelling Research Progress, pages 133-150. July 2009. ISBN 978-1-60741-347-9.
URL: http://mysite.science.uottawa.ca/rsmith43/Zombies.pdf

Daniel Roelke, Sarah Augustine, and Yesim Buyukates. Fundamental predictability in multispecies competition: the influence of large disturbance. The American Naturalist, 162(5):615–623, 2003. Plus an on-line 6 page appendix.

Ulrich Sommer, editor. Plankton Ecology: Succession in Plankton Communities. Springer-Verlag, 1989. Chapter 3.

David Tilman. Resource Competition and Community Structure. Princeton University Press, 1982.

Another of my publications

M. Allmaras et. al. 2013. Estimating Parameters in physical models through Bayesian Inversion: a complete example. SIAM Review 55(1):149-167
URL: http://www.jeanmarielinhart.info/wp-content/uploads/2014/01/2010-sirev.pdf