I am in my fifth and final year of my PhD program at Northwestern’s Department of Engineering Sciences and Applied Mathematics, where I work with Danny Abrams to develop mathematical models of social and biological systems. This fall, I will be a Postdoctoral Research Fellow at University of Michigan’s Center for the Study of Complex Systems (CSCS). I will teach courses in agent-based modeling and develop models to better understand income inequality and segregation.

The central limit theorem states that under certain circumstances the distribution of the sample mean tends towards a normal distribution [1]. The elegance of this theorem leads many to assume properties in nature to take on a normal distribution. There are many instances where this assumption is justified or--at the very worse--harmless. But, there are many instances where this assumption is not justified, such as animal ornaments or Atlantic salmon body mass--these distributions are bimodal [2, 3].

My thesis work [4, 5, 6] at Northwestern has been studying social and biological systems that are inherently bimodal (or multimodal) and developing a model to explain why they split into multiple groups. I have investigated three domains where this splitting occurs: coupled oscillators, advertising competition, and sex cell size dimorphism (anisogamy).

The Kuramoto model [7] is a coupled oscillator model where the oscillators move around a circular domain due to two factors: their natural frequency and their interactions with other oscillators. The oscillators can attract each other--leading to oscillators collapsing to a unimodal phase-locked state--or repel each other--leading to a state where oscillators drift around the circle incoherently. Specifically, there is a sine coupling between the oscillators that determines the strength of attraction or repulsion.

In my work, I take the paradigmatic Kuramoto model and replace the sine coupling function with a generic coupling function to see what features of the coupling function are associated with the generation of a multimodal phase distribution when oscillators repel each other. Assuming the coupling function has multiple fixed points with at least one at zero, I find that the slope near zero must be less than the slope at one of the other fixed points for multimodality to arise. This means that oscillators must “dislike” oscillators far away from them more than they dislike oscillators near them. This result may give a helpful explanation for the development of multimodal distributions for periodic variables.

Advertising is a ubiquitous feature of life in modern times, with total U.S. advertising expenditures hovering around 1 percent of GDP [8]. Through advertising many brands have become mainstays and even becoming synonymous with the product itself (ex: Kleenex) while others have become much lesser-known, generic brands. I develop a model to explain this splitting into these two groups. The main feature of this model is that companies can spend advertising to increase their demand, but they have to spend more than the average company to get this benefit--cutting through the noise if you will. This feature leads to a “go big or go home” attitude where companies choose to invest heavily in advertising or invest the minimal amount. This gives an explanation for the coexistence between name-brand and generic-brand companies.

It is often assumed that organisms came from simple organisms. When these organisms reproduced sexually, they reproduced with sex cells that were similar in size. Today, many complex multicellular organisms reproduce with sex cells are vastly different in size (e.g. the egg cell and the sperm cell). This size difference between the two sex cells is called anisogamy. Part of my thesis work is building on previous work modeling this evolution from similar sized sex cells to the size dimorphism between the sex cells we see today.

The model is based on two components. I take an individual’s fitness, the likelihood of transmitting your genes to the next generation, as the product between the number of gametes one can produce and the fitness of the gametes produced. Here, gamete number decreases with size of the gamete and gamete fitness increases with the size of the gamete. I show that this leads to a balancing act where organisms choose whether they maximize their fitness by having many chances to mate (large number of small gametes), or by ensuring they have healthy, strong gametes (small number of large gametes). This corresponds to the current biological paradigm we know now with sperm cells being small and numerous and egg cells being large and small in number.

This work was supported by NSF. Additionally, I would like to give a shout out to everyone who has supported me: my parents, my siblings, my wife, my teachers, and my advisor, Danny Abrams. I hope to be an inspiration and to give to others the support I received throughout my life.

 

1. Montgomery, D. C., & Runger, G. C. (2010). Applied statistics and probability for engineers (pp. 214). John Wiley & Sons.
2. Emlen, D. J. (1996). Artificial selection on horn length‐body size allometry in the horned beetle Onthophagus acuminatus (Coleoptera: Scarabaeidae). Evolution, 50(3), 1219-1230.
3. Damsgård, B., Evensen, T. H., Øverli, Ø., Gorissen, M., Ebbesson, L. O., Ray, S., & Höglund, E. (2019). Data from: Proactive avoidance behaviour and pace-of-life syndrome in Atlantic salmon. Dryad Digital Repository.
4. Johnson, J. D., & Abrams, D. M. (2019). A coupled oscillator model for the origin of bimodality and multimodality. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(7), 073120.
5. J. D., & Abrams, D.M, in preparation. A Minimal Mathematical Model for Free Market Competition Through Advertising (Science Format)
6. White, N., Kangabire,  A., Johnson, J. D., & Abrams, D.M, in preparation. A Mathematical Model of Anisogamy
7. Kuramoto, Y. (1975). International symposium on mathematical problems in theoretical physics. Lecture notes in Physics, 30, 420.
8. Guttmann, A. (2020, February 25). Advertising spending in the United States from 2010 to 2021. Retrieved April 5, 2020, from https://www.statista.com/statistics/236958/advertising-spending-in-the-us/