PDE Dynamics of Multiscale Systems via a GSPT Approach - A Study of Ecosystem Models

• Applicant: Jan-Eric Sulzbach, Ph.D.
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 571660837

This research proposal focuses on understanding how complex patterns form in nature - specifically, in dryland ecosystems like deserts and semi-arid regions. These areas often show beautiful but mysterious patterns of vegetation, such as stripes and spots, which scientists now believe result from interactions between plants, surface- and groundwater, and environmental factors like rainfall. The study builds on the mathematical model of reaction-diffusion systems. These are systems that simulate how substances spread out and interact. In the 1950s, famous scientist Alan Turing showed how such models can create patterns naturally, just by having substances that move (diffuse) at different speeds. Since then, these models have been used successfully to model everything from animal skin patterns to plant growth. One well-known model from ecology that the results of this study are applied to is the Klausmeier model, which simulates how water and plants interact on flat or sloped terrain. It includes factors like rainfall, plant growth, downhill water flow, and evaporation. This model helps scientists explore how climate changes might lead to deserts spreading or ecosystems collapsing. But here’s the challenge: while these models work in simplified cases, real-world systems are far more complex. For example, rainfall isn’t constant - it changes over time and across locations. Also, the dynamic behavior of most models can only be analyzed in one-dimensional space (like a line), but nature is three-dimensional and highly dynamic. To address this, this research plans to extend a powerful mathematical tool called the Geometric Singular Perturbation Theory (GSPT). Traditionally, GSPT helps analyze systems that evolve at very different speeds, but it mostly applies to simpler, finite cases. This research wants to expand GSPT to work with full-scale, infinite-dimensional systems like those described by partial differential equations. The project is structured around three main goals, each emphasizing both the advancement of theoretical foundations, and their application to dryland ecosystem models as an important case study. The goals are: Upgrading the mathematical toolbox, understanding transitions (bifurcations) and exploring seasonal and spatial changes. Overall, this project is about creating better mathematical tools to study how natural systems behave under stress — something especially important with the challenges we face due to climate change. If successful, the methods developed could apply not only in ecology but also in biology, chemistry, and physics—wherever complex systems evolve in space and time.

• DFG Programme: WBP Fellowship
• International Connection: Netherlands
• Host: Professor Dr. Arjen Doelman