NREUP at Michigan State University
May 12, 2014 - June 20, 2014
NREUP Student Participants: Aaron Crump (Wayne State), Osvaldo Diaz-Rodriguez (MSU), Jamilia Johnson (MSU), Cheyenne Peters (MSU), and Asia Youngblood (MSU).
SROP Guest Student Participant: Tai Nguyen (MSU).
Faculty Mentors: Hyejin Kim, Tsvetanka Sendova, and Mark Iwen.
Mathematical models are powerful tools for understanding and exploring the meaning and features of dynamical systems, and they have been extensively used in various fields such as biology, physics, ecology, finance, economics and many others. Mathematical models are usually categorized into two groups - deterministic and stochastic. Since noise can play a significant role in the dynamics of some systems, stochastic dynamics can sometimes provide additional insight into real world applications. In the proposed project students will model and simulate the competing dynamics of systems in the areas of biology and economics.
Projects(I) Cancer Modeling: This project will focus on the modeling of cancer, starting with deterministic systems of ordinary differential equations, and then incorporating stochasticity effects to account for natural birth and death fluctuations as well as the migration of malignant tumor cells. Student participants will compare the predictions of the deterministic and stochastic models using both asymptotic analysis and numerical solutions of each system. Basing their work on cancer models like [4, 1] and recent studies by Lin, Kim and Doering [2, 3], students will draw parallels between evolutionary biology and modeling cancer. This will allow them to look for conditions on the model parameters which help desirable ensure evolutionarily stable states (ESS) -- e.g., healthy states. In particular, questions of interest will include the following:
- Under what conditions on the parameters the ESS includes only stromal (healthy native) cells, or only stromal and benign tumor cells?
- How do the predictions of the stochastic and deterministic models differ? In  Lin et al have shown that for a model involving two competing species with identical birth and death rates, who only differ in their migration propensity deterministic models predict that the species with lower migration propensity always wins, but on the other hand, stochasticity enhances the chances of survival of the species with higher migration propensity. It would be of interest to see if this would translate to a model involving three species (stromal cells, benign tumor cells, and malignant tumor cells).
This program is an MAA activity funded by NSF grants DMS-1156582 and DMS-1359016.
References David Basanta, Jacob G. Scott, Mayer N. Fishman, Gustavo Ayala, Simon W. Hayward, and Alexander R.A. Anderson. Investigating prostate cancer tumour-stroma interactions: clinical and biological insights from an evolutionary game, British Journal of Cancer, 106(1):174--181, 2011.
 Yen Ting Lin, Hyejin Kim, and Charles Doering. Demographic stochasticity and the evolution of dispersion I. Homogeneous environments. Journal of Mathematical Biology, doi:10.1007/s00285-014-0776-9 (online published).
 Yen Ting Lin, Hyejin Kim, and Charles Doering. Demographic stochasticity and the evolution of dispersion II. Heterogeneous environments. Journal of Mathematical Biology, Print ISSN 0303-6812.
 Quanquan Liu and Ziping Liu. Malignancy through cooperation: an evolutionary game theory approach. Cell Proliferation, 45(4):365--377, 2012.
 V. L. Beresnev and V. I. Suslov. A Mathematical Model of Market Competition. Journal of Applied and Industrial Mathematics, Vol. 4. No. 2, pp 147-157, 2010.