Modeling Vegetation Patterns in Vulnerable Ecosystems

By Lakshmi Chandrasekaran

On a cold, rainy January morning in a café in downtown Chicago, I met Mary Silber, a leading scientist who applies mathematics to understand repetitive patterns of vegetation, which alternate rhythmically with bands of bare soil. The vegetation that Silber and her team study grows in a part of the world that is quite unlike Chicago – the dry and arid Horn of Africa.

Parts of this region, such as areas in Somalia and Ethiopia, receive very little rain throughout the year. With the world’s population currently at 7 billion and projected to rise to 9.6 billion by 2050, food sustainability—ensuring that we produce enough food for everybody to eat—becomes especially important. It is thus imperative to globally increase the percentage of arable land available for food creation beyond the current 28%. Such an increase involves targeting new areas, such as deserts, which exist in numerous parts of the world.

But surely studying the flora of a region falls under an ecologist’s domain – why would a mathematician possibly be interested in this problem?

Silber, whose career trajectory started with a Ph.D. in physics from the University of California, Berkeley, has never been one to settle for something conventional. She is currently a professor in the Department of Statistics at the University of Chicago and director of a new graduate program called “Committee on Computational and Applied Mathematics.”

Over the past few decades, Silber built her expertise by using dynamical systems to study pattern formation in fluid mechanics. Over time, however, she grew restless and craved newer ventures—real-world problems where she could apply her skills—ultimately shifting her focus to problems relevant to climate change, such as vegetation patterns. 

A bird’s-eye view is necessary to study the vegetative dynamics of any region. “You can only make out the vegetation pattern from the air because of its scale,” Silber said. The instability of the Horn of Africa makes it a challenging and interesting region to study. But what drew Silber’s group to the region in the first place was the beauty of the landscape when viewed from above, whether via modern satellite images or early aerial photographs.

However, the vegetation project was not devoid of challenges. “Equations unknown, parameters unknown, time scales over a century or less, and spatial scales of about hundreds of meters or kilometers,” Silber said, all of which are unlike classical fluid mechanics problems. And, of course, the unpredictability that comes with studying our planet. “Carefully controlled experiments?  No! This is Earth – we don’t repeat things!” Silber exclaimed with a laugh, pointing to the most difficult parameter to control in this problem.

In short, the vegetation problem does not present itself well to testing in controlled, pristine research settings and is prone to much heterogeneity, with a lack of physical monitoring on the ground. Nevertheless, Silber toyed with this challenge, seeking a mathematical workaround for the experimental drawbacks. And in 2012, Karna Gowda, a student from Silber’s ‘Mathematical Modeling in the Earth Sciences’ class at Northwestern University, where she taught until quite recently, expressed interest in working on the project. “Karna has been the main driver behind this work,” Silber said proudly.

Gowda began with the question, “What vegetative pattern sequences can possibly occur when we set up the simplest problem we can think of?” Silber and her colleagues addressed this issue in. They used a system of equations describing the amplitude of Fourier modes on a hexagonal lattice, which permits vegetative patterns that resemble spots, stripes, and gaps...

Read more about the project here