By Lina Sorg
Figure 1. Blue dye on an ice core illuminates the presence of brine inclusions, which are marked by the darker streaks. Photo courtesy of Anthony Jajeh.
Sea ice is frozen seawater that floats on the ocean’s surface and shapes the corresponding ecosystem in both Arctic and Antarctic regions. Saltwater freezes at lower temperatures than freshwater; as it freezes, the salt accumulates in brine pockets—areas of fluid water with high levels of salinity—that contribute to the overall porosity of sea ice, allow for the flow of seawater, and provide a habitat for a variety of microbes, including algae (see Figure 1).
Algae typically accumulate in brine inclusions at the bottom of sea ice (within the lowest 10 centimeters), as temperatures are slightly warmer there (see Figure 2). These microbes comprise the foundation of the polar marine food web by serving as a food source for nematodes and other organisms, which are consumed by fish that then feed increasingly larger animals — all the way up to polar bears and seals at the top of the food chain (see Figure 3). A dearth of algae hence disrupts nutrient cycles within the entire marine ecosystem. During a minisymposium presentation at the 2024 SIAM Conference on the Life Sciences, which is currently taking place in Portland, Ore., Anthony Jajeh of the University of Utah utilized a dynamical systems model to understand the relationship between algae and nutrient cycles in the sea ice environment.
Jajeh is particularly interested in exopolymeric substances (EPS): gelatinous biofilm coatings that algae secrete to protect themselves from extreme temperatures and other hazards of their harsh environment. EPS blocks some of the fluid flow through the ice, disturbs the inflow and outflow of nutrients, and alters the physics of the ice itself by triggering a biophysical feedback loop. “This changes a lot of different physical properties,” Jajeh said.
Two specific questions drive this research:
- How does EPS affect algal bloom dynamics?
- Does EPS accumulation ever reach levels that inhibit algal growth? I.e., are algae ever victims of their own success?
Figure 2. Algae accumulates at the bottom of an ice core, in brine inclusions where the temperatures are warmer. Photo courtesy of Jody Reimer.
Jajeh and his colleagues modified a classic NP model of phytoplankton blooms [1] to accommodate the inflow of nutrients when water penetrates the ice core, the decrease of nutrients due to algae, and the outflow of nutrients, as well as algae growth and death rates. “What’s really interesting about this is that the solution is just the inflow over the outflow rate,” Jajeh said of the steady state solutions. “There’s no algae in the system.” The lack of algae makes this the trivial solution.
Jajeh then linearized the model, looked for eigenvalues and Hopf bifurcations, and used asymptotic stability analysis to examine other eigenvalues. He found that when the outflow of nutrients is even slightly greater than the inflow (i.e., when \(b>a\)), the trivial solution becomes stable and the algae die out — despite the presence of some nutrients in the system. But when the inflow of nutrients is greater than the outflow (i.e., when \(a>b\)), the trivial solution is unstable and algae persists. “When there’s sufficiently enough nutrients, the algae can use them a lot better,” Jajeh said.
To explore the impact of EPS, Jajeh introduced a three-system ordinary differential equation (ODE) model that explicitly includes EPS. The first part of this coupled system accounts for the density flux of nutrients, the depletion of nutrients due to algae, and the inflow of nutrients; the second part accounts for the growth of algae due to nutrients and the death rate of algae; and the third part accounts for the growth and death rates of EPS. EPS is a function of algae; as it grows, so does the amount of algae (even with a constant death rate).
Figure 3. Poloar bear in Utqiagvik, Alaska. Photo courtesy of Ken Golden.
Bifurcation analysis (via linear stability analysis) reveals surprising behavior in the limit cycle if the trivial solution is unstable. “A spiral solution [to a nontrivial solution] happens when the inflow of nutrients is greater than outflow of nutrients,” Jajeh said. When nutrient inflow is significantly greater than the outflow (i.e., when \(a\) is much bigger than \(b\)), a consistent Hopf bifurcation emerges. When nutrient outflow is greater than the inflow, the algae once again die out.
Jajeh’s study confirms the existence of a biophysical feedback loop: The presence of EPS affects the flux of nutrients within sea ice, and the flux of nutrients affects algae growth and EPS. He also noted that algae never seem to completely deplete the available nutrients, which perhaps suggests that the microbe is not using them as efficiently as it should. In the future, Jajeh hopes to extend the model to include spatial dynamics and specifically examine the dynamics of transient bloom. “A lot of interesting biology happens at the beginning,” he said. “When does the algae bloom happen? How do we get the maximum amount of nutrients and the maximum amount of algae? Stuff like that.”
References
[1] Huppert, A., Blasius, B., Stone, L., & DeAngelis, D. (2002). A model of phytoplankton blooms. Am. Nat., 159(2), 156-171.
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Lina Sorg is the managing editor of SIAM News. |