## Metapopulation Modeling and Analysis

### Project Mentors

Professor Richard Rebarber (Department of Mathematics), Professor Brigitte Tenhumberg, and Eric Eager (Graduate Student, Department of Mathematics)

### Project Description

This group learned about modeling demographic stochasticity with Markov models. They also learned about density dependent deterministic models. They constructed models for a metapopulation using both approaches, and compared the predictions for each model, in order to determine the effect of demographic stochasticity on the dynamics of the metapopulation. This involved mathematical analysis and numerical simulation.

The students presented a poster on their research at the Joint Meetings in 2012.

## Discrete Modeling of Biological Systems

### Project Mentors

Professor Alan Veliz Cuba (Department of Mathematics) and Michael Uhrig (Graduate Student, Department of Mathematics)

### Project Description

The students studied a continuous model for Th-cell differentiation based on a previously published discrete model. They produced the most general (as of 2012) result about steady states of continuous of discrete models in biology. Roughly speaking, the results states that if a continuous model is "sigmoidal enough" and has the same "qualitative features" of a discrete model, then there is a one-to-one correspondence between their steady states.

The students produced code to transform a discrete model into a continuous model. The students used the code to create and analyze a new continuous model of Th-cell differentiation, and show that this continuous model does predict the three main types of Th-cells.

The students have presented this work at two conferences.

## Fractional Calculus

### Project Mentors

Professor Allan Peterson (Department of Mathematics) and Michael Holm (Graduate Student, Department of Mathematics)

### Project Description

The students studied the discrete nabla fractional calculus. They defined the appropriate Laplace transformation and proved results for this Laplace transform. Included in these results are the formulas for the Laplace transforms of the fractional nabla differences of a function.

They used their results to verify laws concerning the composition of different order fractional derivatives and prove a generalized power rule. They also showed how their results can be used to solve fractional differential equations. Furthermore, they also defined the nabla fractional Taylor monomials, and found the discrete nabla Laplace transform of the discrete three-parameter nabla Mittag--Leffler function. They used these results to develop important variations of constants formulas for initial value problems for a fractional difference equation.

They presented a poster at the Joint Meetings in 2011.