SIAM News Blog

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 [2]. 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 (see Figure 1).

Figure 1. Aerial images of flat terrain vegetation patterns in Sudan. 1a. Spot (11.6280, 27.9177). 1b. Labyrinth (11.1024, 27.8228). 1c. Gap patterns (10.7549, 28.5955). Images © Google, DigitalGlobe. Image credit: Karna Gowda.

“The aim here was to find a bifurcation theoretic framework to allow us to investigate the transition sequence emerging in a variety of different conceptual models that had been proposed,” Silber said. Analysis revealed that the gaps\(\rightarrow\)labyrinths\(\rightarrow\)spots pattern was only one of a few different scenarios that could possibly occur in the simple generic setup, and that these patterns occurred when certain conditions are met (a topic explored in their subsequent paper).

Is there a historical precedent to modeling vegetation patterns? Silber mentioned physicist Ehud Meron at Ben-Gurion University of the Negev in Israel as someone who has championed the development of mathematical frameworks for investigating vegetation patterns. She also referred me to her colleague Arjen Doelman, a professor at Leiden University’s Mathematical Institute in the Netherlands and an expert at using mathematical models to predict vegetation patterns.

Doelman described the Klausmeier model [3], a system of advection-diffusion equations used to study banded vegetation. “The Klausmeier model is the oldest model, and it is a simple one. I prefer to think of it as a conceptual or even ‘toy’ model,” he said.

The equations governing the Klausmeier model describe how the dynamics of water (\(W\)) and plants (\(N\)) change by interacting with each other:

\[\frac{\partial W}{\partial T} = A - LW - RWN^2 + V \frac{\partial W}{\partial X}. \tag1 \]

\((1)\) describes dynamics of water supply change over time as a function of rainfall, loss of water due to evaporation, infiltration by plants, and transport of water downhill.

\[\frac{\partial N}{\partial T} = RJWN^2 - MN + D \bigg(\frac{\partial^2}{\partial X^2} + \frac{\partial^2}{\partial Y^2}\bigg) N. \tag2\]

\((2)\) describes the dynamics of plant growth over time as a function of water absorption by plants, distribution of plants over a given area, and loss of vegetation due to animal grazing or lack of water.

Successfully solving the Klausmeier model recapitulates some vegetation patterns on certain terrains, such as sloped surfaces. However, this simple model has a limitation: it cannot be generalized to predict patterns from all kind of terrains, without modifications.

“The local topography of the environment in the Klausmeier model comes in through the spatial derivative (transport) terms, and specifically, the advection term in \((1)\),” Gowda said. “The model in its original form assumes that we are looking at a sloped surface on which the water travels at a constant speed \(V\).” But in places with flat terrain, such as fairy circles of Namibia, advective runoff may be insignificant.

So Silber and her team turned to the Rietkerk model, more realistic for their study. “It distinguishes the differences between ground water and surface water,” Doelman said. Several equations govern the Rietkerk model:

\[\frac{\partial h}{\partial T} = p - I(n)h +D_h \nabla^2 h. \tag3 \]

\((3)\) describes how the dynamics of surface water (\(h\)) change with respect to rainfall, diffusion, and loss of water via infiltration. The infiltration term \(I(n)\) captures the loss of surface water through soil absorption in the presence of vegetation. Infiltration positively influences vegetative growth, which in turn influences infiltration, thus creating a feedback loop.

\[\frac{\partial w}{\partial T} = -vw + I(n)h - \gamma G (w)n + D_w \nabla^2 w. \tag4 \]

\((4)\) describes the dynamics of water (\(w\)). It involves surface water infiltration and loss of water due to evaporation, diffusion, and transpiration by plants.

\[\frac{\partial n}{\partial T} = - \mu n + \: G(w)n + \nabla^2 n. \tag5 \]

Figure 2. Uniform vegetation equilibria of the model by von Hardenberg et al. [5] plotted as a function of precipitation, with patterned solutions shown in insets. Image credit: Karna Gowda.
\((5)\) quantifies the growth of vegetation (\(n\)) as a function of soil water availability, dispersal of plants in a given area, and plant mortality.

Starting with a uniform vegetative cover, Gowda and colleagues examined when the vegetative patterns transition to patches with decreasing rainfall. Mathematically, these transitions occur between the lower and upper Turing points (see Figure 2).

These results, published last year in [1], use numerical simulations that match qualitatively with the analytical predictions in [2]. Key highlights of this work include understanding the transitions in patterns as a function of change, both in the rate of infiltration and its interaction with the amount of available vegetation (see Figure 3).

But the Rietkerk model has limitations too. “The models we used are idealized, since different types of vegetation are thrown into a single biomass variable,” Gowda said. “For instance, if a drought hits a certain type of vegetation more than others, how is that going to affect the pattern? That is hard to predict.”

Rather than predict how vegetation patterns develop in arid regions, this model could instead predict the increasing desertification in drought-prone areas. “The final ‘catastrophe’ of desertification is preceded by ‘mini-catastrophies’ in which the pattern undergoes significant changes, say, half of the stripe patterns disappear at a very fast time scale,” Doelman said. “These are model predictions, and we’re presently working with ecologists to validate this.”

Figure 3. Map of pattern sequences observed via numerical simulation in a two-parameter space of the model by Rietkerk et al. [4]. The parameter f controls differential infiltration between bare and vegetated areas, and Dh is the surface water diffusion parameter. A “spots only” sequence occurs in a thin red region of the parameter space predicted by analytical theory, while analogs of the “standard” gaps→labyrinths→spots occur in most simulations elsewhere. Example simulations are shown for three distinct parameter sets circled in the left panel, corresponding to f = 0.2 and Dh = 0.6, 1.0, and 4.0. Image credit: Karna Gowda.

To help constrain the model and determine what satellite images could actually measure and quantify, Gowda has been studying the available literature on these vegetation patterns from the past 60 years. He is currently looking at British aerial survey data of dry lands in Somalia, collected during World War II. “Our goal is to try and construct some record of dynamics,” he said.

Another aspect of the problem involves analyzing the role of terrain-topography in influencing patterned vegetation. “We could see that the shallow topography was playing a role, and we want to bring together mathematical modelers in this area with eco-hydrologists for an exchange of ideas,” Silber said. “This meeting of the minds is what we hope will happen during the minisymposium that Sarah Iams and Punit Gandhi have organized for the SIAM Conference on Applications of Dynamical Systems (DS17), to be held in Snowbird, UT, in May 2017.”

With both Silber and Doelman’s groups as key players in the mathematical study of vegetation in arid ecosystems, it should be very exciting to hear their sessions, as well as plenary lectures by Silber and Doelman, at DS17. Check out more details on the conference and register!

[1] Gowda, K., Chen, Y., Iams, S., & Silber, M. (2016). Assessing the robustness of spatial pattern sequences in a dryland vegetation model. Proc. R. Soc. A, 472(2187), 20150893.
[2] Gowda, K., Riecke, H., & Silber, M. (2014). Transitions between patterned states in vegetation models for semiarid ecosystems. Phys. Rev. E, 89(2), 022701.
[3] Klausmeier, C.A. (1999). Regular and Irregular Patterns in Semiarid Vegetation. Science, 284(5421), 1826-1828.
[4] Rietkerk. M., Boerlijst, M.C., van Langevelde, F., Hillerislambers R., de Koppel, J.v., Kumar, L., Roos, A.M. (2002, October.) Self‐Organization of Vegetation in Arid Ecosystems. The American Naturalist, 160(4), 524-530. 
[5] von Hardenberg, J., Meron, E., Shachak, M., & Zarmi, Y. (2001, Oct. 18). Diversity of Vegetation Patterns and Desertification. Phys Rev Lett., 87(19), 198101.

Lakshmi Chandrasekaran received her Ph.D. in mathematical sciences from the New Jersey Institute of Technology. She is currently pursuing her masters in science journalism at Northwestern University, and is a freelance science writer whose work has appeared in several outlets. She can be reached on Twitter at @science_eye. 
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