Mathematical Modeling to Forecast Chlorophyll Levels and Algae Blooms

I worked on mathematical models for forecasting chlorophyll concentration and finding a condition for blooming. Mathematical models help with understanding important processes that govern chlorophyll concentration.  These models are analyzed by considering all relevant variables on the same time and length scale to determine the relative impact of each physical process. This normalization yields non-dimensional parameters that assist in the interpretation of analytic and computational results. We analyzed data from past REU students and the DNR to use in our models.We modeled chlorophyll growth because there is a direct relationship between chlorophyll and algae counts because algae produces chlorophyll as a by-product of photosynthesis and chlorophyll is more easily measured.

The forecasting model predicts how long after a large flushing event it takes for the lake to turn green and smelly if there’s no rain. We used a logistic growth model to capture longer term algae growth and a carrying capacity was estimated by looking at how much the algae can grow if there’s no nutrients for them to consume. We ran 100 simulations varying the carrying capacity and then the growth rate. We found the most variation when the carrying capacity was varied but in all simulations we found that it takes about 3 weeks to hit the carrying capacity level, which corresponds well with data collected from past REUs and the DNR.

The bloom condition was found by solving an equation that describes chlorophyll growth over time based on its growth and movement. The bloom condition was found by solving the equation analytically and was found to rely on Secchi depth and the ratio between growth rate and turbulence. The equation was also solved numerically using computer programming and found very similar results to the analytic solution.  

 Both of these models can take proposed solutions to the algae problem and test how effective they would be at preventing blooms. Both models advocate the solution of increasing turbulence and flow in the lake and the bloom condition models also suggests lowering the algae growth by limiting phosphorus and nitrogen to help solve this problem. 

Integrating Math and Life

As a math major, I find most of my time consumed by stressing out over problem sets and computer programming. However, when I do get a chance to relax I love to read, paint, and golf. None of these hobbies on the surface seem to have anything to remotely do with math but they all deal in some way with problem solving or detail work. A common complaint in math classes is that people never feel like they’re going to use some abstract formula in their daily lives. And while I might not use the Pythagorean Theorem every day, the problem solving skills and detail oriented mindset I gain help me in everyday activities and my hobbies; just as those skills I learn from hobbies also help me with learning math.  

                I love to read popular math books that talk about beautiful abstract math theories that I don’t get to cover in my numerical computing classes. This gives me a more holistic view of math and also helps me to think of numerical computing in a fuller view and not just let the computer do all the thinking for me. I also love to read fictional short stories which allows me look at situations from different angles and tap into my creative side. When I paint I get to look at the full picture of what I want and then try to break it into smaller, manageable chunks so I don’t get overwhelmed in the long process. In addition, I also have to be very patient when dealing with smaller details. When I golf I’m always looking for the best place to position my golf shots and details like wind speed, grass type, and temperature that could possibly change the outcome of the shot. By combining my love for reading, painting, golfing, and math I’m able to pay attention to detail and gain critical thinking skills which helps inspire me to always be creative and learn as much as possible.  

A Model Summer

This summer, I’m going to be working on developing mathematical models for the lake. First we’re going to develop a hydrodynamic model to tell us about the temperature and turbulence of the lake and then use that information to see how that affects the growth of the algae blooms on the lake. Math modeling is really exciting because it lets us change inputs in the model and run experiments varying temperature and turbulence so scientists don’t have to run costly and time consuming experiments. Then once we know how much algae is predicted we can use that information to see how much chlorophyll and pollution is in the water. In addition, we can see how much phosphorus contributes to algae growth.  

                I’m excited to learn more about the biology in this process and to see how sustainability efforts are affecting the algae growth. I’m also excited to see the differences in the predictions of algae blooms during several hot summers or several cooler summers in addition to decreased pollution of the lakes. Hopefully, by understanding the biology and physics behind algae growth we’ll be able to suggest other policies to implement to help make the lakes cleaner. 

Both the work and civic environment have exceeded my expectations. As a mathematician, it’s exciting to work on projects that have the potential to help out a whole community of people and not just focus on solving abstract equations in a white walled room. It’s extremely helpful and interesting to work with others from different disciplines to get insight on other aspects of the problem I might never have considered. It’s also really inspiring to see the whole community support this project especially since everyone is interested in the research that could potentially help the community. 

I come from a midsized city, but adjusting to this town has been really easy since everyone has given us such a welcoming reception. The town and surrounding area is very beautiful and I’m excited for the chance to get outside and explore the trails, which I don’t really get a chance to do in my hometown. My mentor and peer collaborator have also been friendly while also pushing me to be a better mathematician and I can’t wait to see what the rest of the summer holds!  

-Mackenzie Jones