The math of blue whales’ migration patterns

Blue whales, the largest creatures ever to roam the Earth, are highly migratory animals. Each summer, they travel northward along the California coast to forage krill. Then in the fall, they return to the southern breeding grounds where they pass the winter. Sadly, blue whales are endangered, facing the threats of fishing gear, ship strikes, and climate change. To mitigate these threats, researchers must understand the factors that influence the whales’ migration patterns.

Today at the AMS Special Session on Agent-Based Dynamics and Self-Organization in Biology, Stephanie Dodson of the University of California, Davis, gave a talk about her ongoing work to use agent-based models to study blue whale migrations. Agent-based models computationally simulate the actions and interactions of individuals in an attempt to uncover the large-scale dynamics of a community. 

Dodson considered a state-switching model where each whale chooses either a “transit” state (moving long distances with few turns) or a “forage” state (moving short distances with many turns) according to the current krill density and sea surface temperature at the whale’s location. The oceanic data used in the simulations came from the Regional Ocean Modeling System (ROMS) and the North Pacific Ecosystem Model for Understanding Regional Oceanography (NEMURO), covering the 2000-2010 migration seasons with a 3-kilometer spatial resolution and a daily temporal resolution.

Graphs explaining the transit-forage state-switching model used by Dodson.

Blue whales are more likely to forage in waters with low sea surface temperatures (SST) and high krill densities. The distributions of step lengths and turning angles used in Dodson’s model came from data from tagged whales.

Dodson initialized the simulations with the whales entering the region of interest from the south between May 1 and June 1. She showed the audience an animation of one simulated season, which clearly reproduced the broad-scale northward migration pattern. She explained that her model successfully captures differences from year to year but fails to show any southward migration in the fall. That’s because the southward migration is likely driven by additional factors beyond just sea surface temperature and krill density. (Her current project is investigating the role of social calls in the southward migration.)

Satellite data of the ocean often has gaps in time or space, forcing researchers to use lower-resolution data than they would prefer. To improve her model’s sophistication, Dodson compared its performance on the “gold standard” 3 km, 1 day data to its performance on 9 km, 1 day data and 3 km, 8 day data. She found that coarse spatial data caused the simulated whales to form clumps, which she fixed by lengthening the simulation’s time steps to make the average step distance comparable to the spatial resolution. On the other hand, coarse temporal data caused the simulated whales to stay in one state (transit or forage) for too long. The best way to address this issue, she explained, was to add in whatever higher-resolution temporal data was available, even if it had spatial gaps. 

If you want to read more, Dodson’s work, which she started as a graduate student with collaborators at NOAA, appeared last year in Ecological Modeling. It’s just one example of how math can illuminate animal migrations and inform conservation efforts.

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