Collaborative Research: The mathematics of animal migration

This project will develop mathematical models that will aid in the understanding of animal migration. Migration is a widespread phenomenon that occurs seasonally as animals shift their locations in response to changing conditions. Oftentimes these changes involve spatial variation in resources that serve as cues for animals to track, resulting in wave-like population expansions. This research will use a series of novel mathematical modeling approaches to explore such seasonal, wave-like migratory dynamics, with a specific focus on understanding how the quality and quantity of resources interact to shape the pace and pattern of migration for varied theoretical scenarios. In addition, a pre-existing dataset of GPS tracking data for the critically endangered scimitar-horned oryx (Oryx dammah) will be analyzed to characterize when, where, and how well the animals track seasonal changes in resource availability in a resource-poor landscape. The project will support the training of undergraduate and graduate students who are developing skills and knowledge at the interface of mathematics and biology. Consumer tracking of transient resources occurs worldwide in a wide range of systems and taxa. The 'green wave surfing' hypothesis is a recent conceptual advance in understanding such resource tracking that is now widely discussed with regard to seasonal migrations of ungulates, birds, and marine species. According to this hypothesis, migrating consumer species living in seasonal systems should closely track the progression of the highly nutritious plant green-up wave that moves across the landscape as the growing season begins. Empirical data demonstrates that such tracking does occur for some individuals, populations, and species; however, 'surfing the green wave' is not universal, and instead some taxa either jump ahead of the green wave or lag behind it as it seasonally translates in space. The project will develop hybrid dynamical system models involving reaction-advection-diffusion equations with reaction and diffusion coefficients and growth governed by the quantity and quality of the resource green-up wave. Model variants including Allee effects, shifting habitats, and population structure will bring added biological realism. Research will address the impacts of sex- and age-specific migratory behaviors, predation, and mating success on migratory dynamics. Methods from differential equations, integral equations, and dynamical systems will be employed to identify conditions under which populations can persist in the long run. Existence of equilibrium solutions, traveling wave solutions, and oscillating solutions in time and density will be established to understand how 'surfing the green wave' promotes population growth and develops spatiotemporal patterns in population persistence on bounded domains. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.