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Mathematical Models Predict Effectiveness of Mitigation Strategies for Invasive Watermilfoil

By Jillian Kunze

Figure 1. A large growth of invasive watermilfoil beneath the surface of a lake.
The Eurasian watermilfoil is one of the most invasive aquatic plants in the entire U.S. (see Figure 1). Though it is native to Europe and Asia, this plant has become prevalent throughout the U.S. as well as many Canadian provinces, and it is found in over 50 lakes and ponds in upstate New York’s Adirondack Mountains alone. Diana White of Clarkson University was inspired to take a mathematical approach to reducing watermilfoil’s spread by working with her local community in upstate New York, where the invasive watermilfoil is prevalent.

“Watermilfoil outcompetes native plants, grows very quickly, and forms monocultures,” White said. “It typically outcompetes everything else.” It is also able to reproduce by fragmentation, meaning that fragments of the plant can grow into new plants. And since watermilfoil is very versatile, it can grow in conditions with too many or too few nutrients for other kinds of flora. Though it prefers to be in water that is three to 13 feet deep, watermilfoil can survive up to 33 feet beneath the surface if the water is clear enough. 

Current control practices for reducing invasive watermilfoil include herbicides and mechanical harvesting, the latter of which can cause fragmentation and further growth of the watermilfoil. White instead focused on three different techniques for controlling watermilfoil: introducing milfoil weevils to an affected lake, placing benthic mats to block the plant from the sun, and harvesting watermilfoil by hand.

During a minisymposium presentation at the SIAM Annual Meeting, which took place virtually last week, White discussed what mathematical modeling can reveal about sustainable approaches for controlling the invasive watermilfoil. She described two models that explore different control strategies, the first of which uses ordinary and partial differential equations to predict the growth and spread of watermilfoil. This project was funded by the New York State Department of Environmental Science and done in collaboration with the Norwood Lake Association and Clarkson University professors Michael Twiss and Jonathan Martin, as well as undergraduate and master’s students Jillian Neaves, Noah Bohl, and Isabel Dengos. 

The model incorporates a reaction-diffusion framework to measure the spread of watermilfoil’s stolons, or stems that grow and spread beneath the lakebed to propagate the plant. The “reaction” component represents the growth of the stolons, and “diffusion” represents their spread. Layered on top of that is another framework that represents the growth and decay of the biomass above the lakebed. 

To parametrize this model specifically for watermilfoil, White and her collaborators collected data from the nearby Norwood Lake. Students placed hula hoops filled with sand on the lakebed and secured them with tent pegs, then pulled out the biomass from that hooped region (see Figure 2). They then dried the watermilfoil and found its dry weight. “We’ve been doing that every week this summer, so we can fit that model and try to parametrize it for Norwood Lake,” White said. The students also revisit and take pictures of the hula-hooped areas each week to investigate the rate of regrowth.

Figure 2. The process of data collection to parametrize the watermilfoil model. Students drop hula hoops weighed down with sand on the bottom of Norwood Lake and secure them with tent pegs, then pull out of the biomass out of the hooped region. Later, they return to measure the extent that the watermilfoil has regrown.

White presented a few example simulations using this model, including a simulation of how pulling out watermilfoil by hand effects the biomass in an area over time. Since it is not possible to totally remove all the stolons from an area of lakebed in real life, an interesting exercise would be to leave five to 10 percent of the stolons in a patch that was otherwise hand-pulled and observe the change in results. There are a number of additional questions that White is interested in investigating with this model, such as the effect of pulling out stolons several time during a season and at different locations, or the impact of laying down benthic mats over an area.

The second portion of White’s presentation focused on a model for control strategies that involve the milfoil weevil, a variety of beetle that only targets watermilfoil. This project was funded by a New York Sea Grant, and is in collaboration with the Indian River Lake Conservancy and Indian River High School students and lead environmental science teacher Andrea Inserra. Clarkson University undergraduates Nicolas Bos-Lad and Nathalie Barrios and recent graduate Thibaud Antoniou are also contributing to the project.

Adult weevils lay their eggs in the meristem of the watermilfoil; when they hatch, the larvae and pupa feed on tissue in the stem and halt its growth. This can reduce the biomass and even kill watermilfoil plants. Weevils also spend their winters in leaf litter, so they do best in lakes near forested areas where leaves gather near the water’s edge. “We’re using a machine learning approach to predict whether weevils, which are milfoil specialists, will be productive at reducing Eurasian watermilfoil,” White said.

White and her collaborators considered real-world data that resulted from a company which added weevils to ponds in a very haphazard way; they were inconsistent in when they performed the weevil augmentation, how many times they did it, and how many weevils they added each time. White wanted to know what factors led the weevil augmentations to sometimes succeed and sometimes fail at reducing watermilfoil in lakes. “We compiled metadata from studies and used machine learning techniques to determine the lake conditions under which the weevil augmentation is successful,” she said. The machine learning algorithm’s goal was to use data on lakes infested with Eurasian watermilfoil to find the probability of success for different augmentation strategies.

Figure 3. Predictions from the machine learning model for weevil augmentation strategies at different lakes.
The researchers trained the machine learning model based on sets of predictors — qualities of lakes for which they knew whether the weevil augmentation strategy succeeded or failed. They then ranked these predictors and removed those that were ranked very low; the most important parameters for predicting weevil success were found to be lake depth, lake nutrients, lake turbidity, lake temperature, lake size, total shore length, presence of weevil habitat, and the augmentation strategy (how many weevils were added, and how often). 

They then validated the model using a subset of predictors where success or failure was known, but that were not used in training. The model had a precision of about 86 percent — i.e., anyone who inputs data into the model can predict with 86 percent accuracy whether weevil augmentation would be appropriate for a given lake. For example, Figure 3 shows a table with the probabilities of success for different augmentation strategies in several lakes; the machine learning model predicts that two augmentations of weevils will always be successful at those locations.

White concluded her presentation by listing a number of ways that interested individuals can help to reduce invasive watermilfoil in lakes. Residents near lakes can participate in small-scale pulls, and boat owners can clean any watermilfoil off their boats by hand or by pressure washing. And leaving leaf litter by the side of lakes can provide habitat for weevils, helping the beetles to eat away at the invasive watermilfoil. 

  Jillian Kunze is the associate editor of SIAM News