# Summary

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 [3] 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).
(II) Modeling competing car manufacturers: The automotive industry is composed of a number of competing manufacturers, which compete to sell more vehicles to customers and maximize revenue for their survival in the market. More often than not, a manufacturer's decision to produce certain car models depends on extensive research into customers' needs, and what the majority of buyers want the most. A customer's decision actually depends on a lot of factors, including price, fuel economy, comfort, cargo space, design, and so on. Thus, the automotive industry may be considered as a multi-species environment with a number of variables for the survival of each species [5]. Interestingly, rivalries are often observed in the automotive industry, even though there are more than two manufacturers. Just by looking at the U.S. market share, GM and Ford take up more than one third of total sales volume. People think about BMW and Mercedes when asked for entry-level luxury models. Similarly, Toyota and Honda come first for the affordable and reliable cars. Therefore, certain segments of the total market may be considered as being dominated by two major species, whose survival depends on a few parameters. For students living in a state whose economy is largely dominated by car companies, it would be interesting to model the behavior of two competing manufacturers with the mathematical tools described above.

# Results and Awards

Our students have presented posters at MSU's Mid-SURE 2014 conference, as well as at the 2014 MI-LSAMP Undergraduate Research Symposium. We are particularly proud of Osvaldo Diaz-Rodriguez and Tai Nguyen for winning the first place prize for best presentation at MI-LSAMP on November 1, 2014. More recently, we are also proud to report that Jamilia Johnson, Cheyenne Peters, and Asia Youngblood won the Award for Most Outstanding Presentation for their talk "Modeling Interactions between Various Cell Populations in a Cancerous System" at the 12th Annual MSU Student Mathematics Conference on April 18, 2015.

# Resulting Publications

• Osvaldo Diaz-Rodriguez, Tai Nguyen, Hyejin Kim, and Tsvetanka Sendova,
Competing dynamics: Analyzing market share in a duopoly (.pdf),
Journal of Statistics & Management Systems, Accepted, 2017.
• Jamilia Johnson, Cheyenne Peters, Asia Youngblood, and Aaron Crump,
Modeling Interactions Between Various Cell Populations in a Cancerous System (.pdf). An edited version appeared in Volume 9 of SIAM Undergraduate Research Online (SIURO).

# Acknowledgment

This program is an MAA activity funded by NSF grants DMS-1156582 and DMS-1359016.

# References

[1] 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.

[2] 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).

[3] 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.

[4] Quanquan Liu and Ziping Liu. Malignancy through cooperation: an evolutionary game theory approach. Cell Proliferation, 45(4):365--377, 2012.

[5] 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.