*I am SO excited for you to read this Monday's #mathematicalmodel post! This week, we go deep into the ocean, and also dive deep into some beautiful mathematics! We'll let Phoebe explain more. She is a research oceanographer with the National Oceanographic and Atmospheric Administration. She studies the North Pacific’s open ocean ecosystem, where her research examines how the ecosystem is impacted by things like fishing and climate change. Today, Phoebe explains how math, partial differential equations in particular, backs the research she does.*

As an undergraduate, I minored in math. I didn’t have an overwhelming interest in math, but a math minor only required one extra math class in addition to those required for my major (meteorology), and I figured it would look good on a resume. Fast forward about a decade and a half. Now, I’m an oceanographer (not a meteorologist) working with ecosystem models that are essentially a series of partial differential equations. Partial differential what? I will explain.

Partial differential equations are used to explain something that’s continuously changing. Ecosystem models are mathematical representations of a food web. Because food webs are always changing as fish grow and eat each other, we use partial differential equations to calculate how many fish of different sizes there are at any given time.

Let’s use the North Pacific Ocean as an example. There are tunas and swordfish and mahi mahi and various other large fish. These fish eat smaller creatures like flying fish and squid. And these guys eat even smaller stuff and so on down through zooplankton and phytoplankton. And at the same time that everyone’s eating each other, they’re also growing, reproducing, and dying. Partial differential equations allow us to calculate all of these different, continuously changing, processes. Here’s what it looks like1:

Translation: the number (*N*) of, say, tuna of a certain size (*w*) at any given time (*t*) is the result of growth (*g*) into that size (again, *w*) minus any tuna that die (*d*). And so on through a range of species and sizes over time. As you might guess, there are more equations for calculating growth (*g*), which happens as fish eat smaller fish, and for calculating reproduction. And there are also equations for calculating death (*d*), which happens when fish are eaten by other fish or by people. There are even equations for determining what size fish (*w*) another fish is going to eat.

Why bother with all this math? Because it’s impossible to count how many fish there actually are in the ocean. Even if you could count them, the number is always changing. These models also allow oceanographers and ecologists to test out different scenarios of what the ecosystem will look like in the future. They help us to understand how changing one piece of the ecosystem affects everything else. Fishing removes large fish from the ecosystem. Climate change has the potential to reduce the amount of phytoplankton supporting the ecosystem. Understanding how these different influences shape the food web can help people figure out things like how many fish can be caught or how fish catch might change in the future.

As an oceanographer, I use math to represent the ocean’s food web. Math helps me estimate how many fish there are of different sizes, what happens when these fish eat each other and are eaten, and what happens when the ocean around them changes. By using mathematical ecosystem models, I can study the ocean in a way that would otherwise be impossible. My research may focus on open ocean food webs, but at its core it’s really all math. Oh, and the one extra math class I had to take in college? By chance, it was partial differential equations.

*1. Scott F, Blanchard JL, Andersen KH, 2014. “Multispecies, trait-based and community size spectrum ecological modelling in R (mizer)”*

*You can follow Phoebe's work on Instagram ( where she posts lots of work updates and explanations!) and Twitter at @PhoebeJefcoats, or contact her at Phoebe.Woodworth-Jefcoats@noaa.gov.*