A Mathematical Model of Coral Reef Response to Destructive Fishing Practices with Predator-Prey Interactions.
Vanessa Machuca, '18
Advisor, Prof. Karen Ríos-Soto, University of Puerto Rico, Mayagüez
Coral reefs are being degraded by multiple anthropogenic stressors, including excessive and destructive fishing practices. Such activities damage reefs directly, particularly when cyanide and explosives are employed, and deplete reef fishes that keep coral predators and competitors in check. In this work, we focus on the highly problematic corallivore Crown-of-thorns starfish (CoTS), Acanthaster planci, and one of its few known predators, the endangered and overfished Humphead wrasse, Cheilinus undulatus. We built a system of nonlinear ordinary differential equations to model the interactions between coral, wrasse, and CoTS biomasses within the Indonesian province of Raja Ampat. We consider commensalism between wrasse and coral in favor of wrasse, and predator-prey relationships between wrasse and CoTS, and CoTS and coral. We take into account coral damage from illegal, unregulated, and unreported (IUU) fishing and consider constant yield, constant effort, and seasonal wrasse harvesting. Equilibria for the system with and without harvesting are determined, including coexistence equilibria in which all three species persist. We run numerical simulations and conduct sensitivity analyses on key parameters. Through this work, we hope to provide insight on the extent to which the coral reefs of Raja Ampat can hold up to rising fishing pressure as well as describe a model which can be applied to similar ecosystems.
Symmetries of graphs in homology spheres.
Song Yu, '17
Advisor, Prof. Erica Flapan, Pomona College
If we have a frame of a pyramid embedded in the three-dimensional Euclidean space R3, we can use rotations and reflections of R3 to move the vertices of the pyramid around in any manner while keeping its shape. In a similar fashion, we investigate graph symmetries in more general 3-dimensional topological spaces with a focus on homology spheres, an infinite family of spaces based on the space we live in. We will generalize previous findings from ordinary space and see how concepts of rigidity, mirror image symmetry, and linking of rings interact with symmetries of graphs in homology spheres.
Blocking the immune-blockers: a mathematical model of the latest in cancer immunotherapy
Ruby Kim, '17
Advisor, Prof. Ami Radunskaya, Pomona College
Cancer immunotherapy is a relatively new type of treatment that boosts the immune response against cancer cells. Dendritic cell vaccines, one form of immunotherapy, show promising results in ongoing research. A previously developed mathematical model of tumor-immune interactions simulates the current vaccine schedule for prostate cancer and reproduces experimental data showing the effectiveness of dendritic cell vaccines in stunting tumor growth. This model, a system of differential equations, could be used to compute a new vaccine schedule that optimizes treatment outcomes. We update this model using the latest findings in immunotherapy. Programmed Death-1 (PD-1) receptors on effector immune cells, when bound to PD-L1, create a pathway that suppresses effector immune cell activity and prevents autoimmunity. Some cancer cells take advantage of this pathway by using PD-L1 to evade the immune response. Experiments using mouse models show that pathway blocker anti-PD-L1 and ibrutinib (an anti-cancer drug), both independently and when combined, help combat tumor cells. We calibrate the extended model using the two treatments' independent effects, then we simulate their combined effects and achieve a close qualitative fit to real data. This updated model can be used to simulate new treatment strategies.
Characterizing Noise in a Mathematical Model of the Adipogenic Transcriptional Network.
Erin Angelini, '18
Advisor, Prof. Blerta Shtylla, Pomona College
Adipogenesis is the process by which precursor cells develop into mature adipocytes, or fat-storing cells. From a 2012 study by Park et al., we expanded a deterministic model of the transcriptional network of adipogenesis to include a module for adiponectin (AdipoQ) production, an insulin-sensitizing hormone secreted by adipocytes. We analyzed two possible implementations for the adiponectin module to determine if variability within the system parameters alone is sufficient to explain the adipocyte heterogeneity observed in a study by Loo et al. For each model, we first characterized overall susceptibility to noise by calculating the relative local sensitivity of AdipoQ and fat to various parameters. We then simulated the experiment done by Loo et al. with 30% added noise to determine if our system could replicate their data. Our results show that only the model where fat increases the degradation of adiponectin fits the trends observed in the Loo study, indicating that it is more likely than the model where fat decreases the synthesis of adiponectin.
Stability of Agent-Based Models.
Qing Fan, '18
Advisor, Prof. Björn Sandstede, Brown University
Situations in which interactions of individual agents form macro-level patterns are ubiquitous, and agent-based modeling provides a natural approach to understanding them. This paper is motivated by an agent-based model of pigment cells forming stripes on zebrafish, a tropical fish studied for medical research. We analyze the stability of agent-based models similar to the zebrafish model, in which deterministic and stochastic processes occur on the same timescale. However, it is difficult to analyze agent-based models with randomness. We therefore examine these dynamics in the framework of piecewise-deterministic Markov processes (PDMPs), which consist of continuous-time deterministic flow (described by ordinary differential equations) punctuated by random jumps in position. We use results from previous literature to specify sufficient conditions for the stability of general agent-based models with setups similar to the zebrafish system, and provide examples to illustrate the use of these conditions. We also simulate a one-dimensional toy model of zebrafish cells to numerically explore the effects of deterministic and stochastic processes on the long-term stability of the model.
Automatic Locally Adaptive Regression Smoothing.
Xuanchi Liu, '17
Advisor, Prof. Gabriel Chandler, Pomona College
Our goal is to fit a smooth regression curve to data, under the assumption that the target is smooth. Traditional regressions techniques use cubic splines; however, they suffer from having to choose the number and location of the knots, whose choices vary the result significantly. We propose a non-parametric spline regression method that is based on linear splines, adapts the smoothing parameter automatically, and is unlikely to overfit the data. The parameters involved in this algorithm are not data specific and do not affect the robustness of the results.
Computational perspectives on coral bleaching.
Adam He, '17
Advisor, Prof. Daniel Martínez, Pomona College
Worldwide, coral reefs are being threatened by coral bleaching, a potentially lethal stress response. Despite much research on bleaching in scleractinian (reef-building) corals and the model organism Aiptasia, our understanding of bleaching in octocorals (soft corals and sea fans) is limited. To resolve this, we have sequenced and assembled transcriptomes of the octo-coral Symposium sp. and its endosymbionts at key time points in the bleaching process, and are analyzing changes in gene expression patterns. We discuss various computational methods relevant to the study of large biological datasets and connections with previous results on the molecular and cellular biology of bleaching in octocoral.
Defining Humanistic Mathematics: The HMNJ Archives.
Nurullah Goren, '18 & Tiffany Zhu, '17
Advisor, Prof. Gizem Karaali, Pomona College
The Humanistic Mathematics Network Journal (HMNJ) was an eclectic publication which ran from 1992 to 2004. The HMNJ was dedicated to developing a pedagogical program around the phrase humanistic mathematics. However, the exact meaning of "humanistic mathematics" was never sufficiently and consistently defined throughout the journal’s lifespan. Our goal in this project was to determine the meaning of this phrase, at least as it was understood by the editors and the contributors of the journal, by reviewing the corpus composed of all issues of the HMNJ. Our work implies that humanistic mathematics is a philosophy which elevates the sociocultural, ideological, and educational aspects of mathematics to the same level of importance as the mathematics themselves. This philosophy has both descriptivist and prescriptivist halves. The descriptivist half inquires about the nature of mathematics and its relationship to other fields, arguing in particular that mathematics is a part of the humanities. The prescriptivist half takes a more top-down interpretation of the term humanistic mathematics, arguing that mathematics education should be treated as cultural exchange, and that mathematics should be taught in the project- and discussion-based way that humanities are taught, in contrast to the rote-memorization that plagues mathematics education.
An Upper Bound on the Number of Integer Roots of Certain Sum-Product-Sparse Polynomials
Kayla Cummings, '18
Advisor, Prof. J. Maurice Rojas, Texas A&M University
An SPS-polynomial is a polynomial expressible as a sum of products of sparse univariate polynomials. SPS-polynomials are closely related to depth-4 arithmetic circuits (of recent interest in complexity theory), and Koiran has shown earlier that new lower bounds for the complexity of the permanent hold if SPS-polynomials of low complexity have few integer roots. Some effort has been made toward bounding the number of real roots of SPS-polynomials, but bounding the number of integer roots still appears out of reach. Bounding p-adic valuations of the integer roots is a potentially promising, alternative approach that has yet to be explored.
Let ordp (ck) be the p-adic valuation of the coefficient corresponding to the kth-degree term of an SPS-polynomial f. Then define the p-adic Newton polygon of f to be the convex hull of all ordered pairs (k, ordp (ck)). Due to work of Hensel and Dumas, the number of edges in the lower hull of the p-adic Newton polygon is equal to the number of distinct p-adic valuations of the integer roots. Using p-adic Newton polygons, we show that an upper bound for the number of p-adic valuations, in line with Koiran's conjectures, can be proven for a particular family of SPS-polynomials. We then point out a larger family of SPS polynomials where p-adic methods may be more tractable than real analytic methods.