# Student Feature - Yuxin Chen

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I am a fourth year PhD student in Applied Mathematics at Northwestern University, advised by Mary Silber who is currently a professor at Department Of Statistics at University of Chicago. I obtained a B.S. Honors degree in Mathematics from Dalhousie University in Canada, advised by Theodore Kolokolnikov.

In my undergraduate years, I have worked on several research projects. One of them involved modeling predator and prey swarm dynamics. In the study, my undergraduate adviser Theodore Kolokolnikov and I simulated how the predator attacks swarming prey by testing different interacting potentials and analyzed the stability of certain patterns that the system presents by using linear stability analysis [1]. A nonlinear mathematical model was set up and captured the essential features of predator-prey dynamics, including predator confusion, predator attack, and prey escape. Additionally, the shape of the swarms and conditions under which these dynamics occurred were derived via perturbation method. In this experience, I was fascinated by how natural organisms behaved collectively, ranging from bacterial colonies to locust swarms to schools of fish. It also roused my interest in doing research in applied mathematics.

At Northwestern University, I continue investigating natural phenomena via mathematical methods. Working with Mary Silber, Karna Gowda, and Sarah Iams, I started my first project on studying pattern sequences of dry-land vegetation models via bifurcation analysis [2]. Vegetation patterns have been observed in semi-arid or arid areas. Theoretical ecologists have built partial differential equation vegetation models, where a particular sequence of patterns, “gaps -> labyrinth -> spots”, occurs as precipitation decreases. It has been suggested that vegetation patterns may serve as early-warning signs of desertification. Looking at the sign of quadratic coefficient of the hexagonal lattice amplitude equation that was derived, we tried to identify the standard pattern sequence as parameter, which is precipitation in this case, varies in the model developed by Rietkerk et al. [3].

Currently I am interested in the problem of noise-induced transitions between alternative stable periodic orbits of a periodically forced system and hopes to answer the question if there is a preferred phase of the transitions. I have been an active member in Mathematics and Climate Research Network (MCRN) and with collaborators in the Long-term Ecological Research focus group from this network, I am looking at some stochastic ecological models by connecting with time series data collected at Konza ecological sites in Kansas. This idea originated when I was a TA at a mathematics and climate graduate students summer program in 2016, where I mentored a group of junior graduate students. Through this program, I have developed the interests in parameter estimation and validation of conceptual models using real data. In addition, I have learned about how to lead a research group from fellow TAs and senior faculty members. I have benefited a lot from communicating with people in related research areas, especially through MCRN, where I am exposed to a wide variety of exciting research topics in application to ecology and climate sciences and have the chance to build the connection with ecologists, which is crucial for studying and validating ecological models.

Originally coming from China, completing the undergraduate study in Canada, and currently studying in graduate school in the US, I have always been seeking new environment and trying to move out of comfort zones. During my time in Canada and the US, my research advisers, Theodore Kolokolnikov and Mary Silber respectively, have helped me tremendously with proposing new research ideas and teaching me how to do research. They have been generously providing me with network opportunities, which have also helped me jump the language barrier.

Reference:
[1] Chen, Y., & Kolokolnikov, T. (2014). A minimal model of predator–swarm interactions. Journal of The Royal Society Interface, 11(94), 20131208.
[2] Gowda, K., Chen, Y., Iams, S., & Silber, M. (2016, March). Assessing the robustness of spatial pattern sequences in a dryland vegetation model. In Proc. R. Soc. A (Vol. 472, No. 2187, p. 20150893). The Royal Society.
[3] Rietkerk, M., Boerlijst, M. C., van Langevelde, F., HilleRisLambers, R., de Koppel, J. V., Kumar, L., ... & de Roos, A. M. (2002). Self-organization of vegetation in arid ecosystems. The American Naturalist, 160(4), 524-530.
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