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A Fast-Slow Switching Model of Banded Vegetation Pattern Formation in Drylands

By Punit Gandhi, Sarah Iams, and Mary Silber

Spatial patterns of vegetation in dryland ecosystems were reported in the literature as early as 1950. On very gently sloped terrain, e.g., a  \(\sim0.5\) percent grade, they take the form of bands of vegetative growth that alternate with bare soil. Figure 1 depicts an aerial view of banded vegetation patterns at a site from the Haud region at the border of Ethiopia and Somalia [1]. These patterns occur at a scale that is too large to observe from the ground—with typical band spacing on  the order of a hundred meters—but are unmistakable from air or space.

Figure 1. Aerial photograph of banded vegetation patterns during a U.S. Army Air Force reconnaissance mission in March 1945. The site is located in the Haud region of Africa, at the border of Somalia and Ethiopia [2]. The white arrow in the center of the image indicates the downhill direction where the patterns appear. Photo reproduced in [1].
The dryland ecosystems that support these types of vegetation patterns are characterized by their water deficit. The region in the Horn of Africa where Figure 1 was photographed has an aridity index of approximately 0.07, meaning that it receives roughly  seven percent of the rain that the ecosystem could potentially use. Moreover, that water input is infrequent and largely unpredictable: rainfall data from the Haud (see Figure 2), indicates that a significant fraction of rainfall comes from intense storms that occur every one or two years. Such storms may last just a few hours and deposit several centimeters of rain. A changing climate and increased human pressure make it especially important to understand how the vegetation in these ecosystems responds to environmental changes.

Some of the earliest field studies of banded patterns took place in the Horn of Africa, including the aforementioned Haud region. These studies postulated a connection between the formation of bands and the area’s hydrology. As noted in a recent article by Ehud Meron on SIAM News Online, contemporary models suggest that patterns may increase the resilience of these ecosystems in the face of limited water and potential droughts. Such models are rooted in a nonlinear dynamics perspective on pattern formation, and use systems of reaction-advection-diffusion equations to capture the interactions between plants and available water in their environments.  The simplest models of this type take the qualitative form

\[ \underline{\textrm{Water}}: \frac{\partial W}{\partial T}=  \text{precipitation}-\text{evapotranspiration}(W,B)+\text{transport}(W) \tag {1a}\]

\[ \underline{\textrm{Biomass}}: \frac{\partial B}{\partial T} = \text{growth}{W,B} - \text{mortality}(B) + \text{dispersal}(B). \tag{1b} \]

These types of models predict that increasing aridity (due to decreasing precipitation, for example) cause small-amplitude patterns to emerge from a uniformly-vegetated state that loses stability via a bifurcation. Researchers often formulate such models on the years-to-decades timescale associated with the dynamics of the emergent vegetation patterns, and treat precipitation as a constant year-round input. This “slow drizzle” approximation—combined with the system’s two-field structure—allows for predictions of asymptotic behavior within a framework that is amenable to analysis. See, for example, Arjen Doelman’s plenary lecture at the 2017 SIAM Conference on Applications of Dynamical Systems.

Figure 2. Hourly rainfall data for a site in the Haud region of Africa, near the location of Figure 1. Figure retrieved from [3] and downloaded from [4].

Early observations describe sheet flow in the interband region during rain events, with the rain “arrested by the next vegetation arc down the slope” [5]. The formation of soil crusts in bare soil regions inhibits infiltration of water into the soil, while the presence of root systems in the vegetation bands enhances infiltration. Models of the form \((1)\) heuristically capture this biomass feedback through a nonlinear dependence on biomass in the growth term.1 More detailed models split the water field into surface water and soil water components to explicitly capture this infiltration feedback. These so-called “three-field” models take the form

\[ \underline{\textrm{Surface Water}}: \frac{\partial H}{\partial T} = \text{precipitation}-\text{infiltration}(H,B)+\text{transport}(H) \tag{2a} \]

\[ \underline{\textrm{Soil Water}}: \frac{\partial W}{\partial T} = \text{infiltration}(H,B)-\text{evapotranspiration}(W,B)+\text{transport}(W) \tag{2b}\]

\[ \underline{\textrm{Biomass}}: \frac{\partial B}{\partial T} =\text{growth}(W,B)-\text{mortality}(B)+\text{dispersal}(B). \tag{2c} \]

One challenge associated with three-field models of the form \((2)\) is that the hydrological processes associated with surface water dynamics and soil infiltration occur on a minutes-to-hours timescale, which is much faster than other soil-water processes and biomass dynamics. Despite these fast processes, three-field models are typically formulated at the same years-to-decades timescale as two-field models. This often leads to either an “upscaling” of the parameters associated with the fast processes, or certain unphysical predictions about the system. For example, in the “slow drizzle” approximation with constant year-round rain input, one could artificially reduce the infiltration rate and make a year-round reservoir of surface water that is consistent with observations during rain storms. If a more realistic infiltration rate is taken instead, a three-field model might predict a year-round layer of water with a height that is less than the diameter of a grain of sand.

A Fast-slow Switching Model

An alternative approach is a fast-slow switching framework [4] that captures the natural timescales of important processes, particularly fast hydrological processes and slow biomass dynamics. An advantage of this fast-slow framework is that one can choose the parameters based on the appropriate values for the associated process. This reduces the required number of “fitting” parameters that ensure that the emergent pattern characteristics in the model match observations. The model is considered on a one-dimensional spatial domain with uphill along the positive \(x\)-direction.

Rainfall initiates fast hydrological processes associated with overland waterflow and surface infiltration into the soil. These processes occur on timescales of minutes to hours. Evapotranspiration and plant dynamics transpire on much longer timescales of weeks to months, which we assume can be neglected during rain events. This separation of scales suggests a “fast system” that captures the hydrology on the fast timescale \(t\) (associated with rain events), under the assumption of a fixed biomass distribution. In dimensionless form, the fast system is given by 

\[ \underline{\textrm{Surface water}}: \frac{\partial h}{\partial t} = \underbrace{p(t)}_\text{precipitation}  - \underbrace{\iota(b,s,h)}_\text{infiltration} + \underbrace{\frac{\partial}{\partial x}  \bigg(\nu(b,h) h\bigg)}_\text{transport} \tag{3a} \]

\[ \underline{\textrm{Soil moisture}}: \frac{\partial s}{\partial t} = \alpha \underbrace{ \iota(b,s,h)}_\text{infiltration}, \tag{3b} \]

where the infiltration rate \(\iota(b,s,h)\) of surface water into the soil and the surface water advection rate \(\nu(b,h)\) are given by

\[ \iota(b,s,h)=\left(\frac{b+qf}{b+q}\right)\left(\frac{h}{h+1}\right)(1-s)^\beta \quad \text{and} \quad \nu(b,h)=\frac{h^{\delta-1}}{1+\eta b}. \]

During rain events, the surface water height \(h\) increases via a time-dependent precipitation input \(p(t)\), and surface water flows with advection rate \(\nu(b,h)\). The surface water infiltrates the soil and increases soil moisture with a rate determined by \(\alpha \iota(b,s,h)\). The parameter \(f\) captures an infiltration contrast between vegetated regions and bare soil. Infiltration-inhibiting crust forms on bare soil, and vegetation enhances infiltration in regions where biomass \(b>q\), which results from increased macroporosity associated with roots and heightened biological activity. The exponent \(\beta>1\) captures the slowing infiltration rate as soil moisture approaches saturation (\(s\to 1\)). The model for the downhill surface water speed is based on empirical relationships observed for open channel flow, and also captures a feedback indicating that the presence of biomass slows surface water flow by effectively increasing surface roughness. The empirical evidence suggests that \(\delta=5/3\), although we often use \(\delta=1\) to speed up simulations. 

Once the surface water has infiltrated the soil, the fast-slow model switches to a “slow system” that captures the interactions between soil moisture and biomass on a timescale \(\tau\) associated with biomass growth: 

\[ \underline{\textrm{Soil moisture}}: \frac{\partial s}{\partial \tau} = -  \underbrace{\left( \sigma s + \gamma s b  \right) }_\text{evapotranpiration} \tag{4a} \]

\[ \underline{\textrm{Biomass}}: \frac{\partial b}{\partial \tau} = \underbrace{s b (1-b)}_\text{growth}  - \underbrace {\mu b}_\text{mortality} + \underbrace{\delta_b \frac{\partial^2 b}{\partial x^2}}_\text{dispersal}. \tag{4b} \]

The soil moisture is lost by evaporation (with rate \(\sigma\)) and transpiration (with rate characterized by \(\gamma\)). The biomass growth is linear in \(s\) and logistic in \(b\). It is lost with mortality rate \(\mu\), and seed dispersal is modeled by diffusion. 

Figure 3. Range of biomass values for uniform vegetation (green dotted lines) and patterned states (red/blue bars) as a function of mean annual precipitation (MAP). As MAP decreases, the uniform vegetation state loses stability and patterns emerge. The band widths decrease and bands are lost for very low MAP values until a single band remains on the domain. This single-band state eventually collapses to a bare soil state. When starting with the single band state and increasing MAP, the state persists past the point where the uniform state loses stability. The width of the single band grows with increasing MAP until the band covers the entire domain and the simulation reaches a uniform vegetation state. More details are available in [4].

Profiles with increasing/decreasing mean annual precipitation (MAP). Video courtesy of Punit Gandhi.

Model simulations indicate that patterns form with characteristics—such as band spacing and upslope migration speed—that are within an order of magnitude of observation. This is with parameter values that are appropriate for the associated processes, and without fine-tuning. Figure 3 illustrates the patterns’ dependence on the mean annual precipitation (MAP), assuming two equal six-hour rainstorms each year that are spaced six months apart. The animation illustrates that the width of the vegetation bands depends strongly on MAP, with lower MAP leading to narrower bands at a given band spacing. It also depicts the strong history dependence of the patterns; decreasing precipitation leads to band narrowing but also band loss, while increasing precipitation leads to band widening without the insertion of new bands. This is consistent with C.F. Hemming’s 1965 predication that increasing rainfall might widen individual vegetation arcs, while decreasing rainfall might reduce the number of arcs [5].

Where is the Soil Moisture?

One open question concerns how the spatial distribution of water within vegetation patterns changes over time. Soil moisture might be higher within the vegetated region or the bare interband region, depending on the length of time since the last rainfall. In the vegetated region, infiltration is higher and surface water movement is slower, which results in increased input. However, soil moisture is lost at a faster rate due to transpiration, thus causing increased output. The competition between soil moisture increase (which happens on a minutes-to-hours timescale) and soil moisture loss (which happens on a days-to-months timescale), suggests that appropriately-resolved time series data of soil moisture distribution could be useful for building and testing models.

Figure 4. Rainfall data in 1985 at a site in Mexico’s Chihuahuan Desert exhibits banded vegetation patterns (top). The dashed line indicates measurements of soil water content in the vegetation band, and the dotted line indicates the in-between bare soil region (bottom). Figure adapted from [7].

One field study does provide a time series of soil water content for a vegetation band and for the bare soil region at a site in Mexico’s Chihauahuan Desert [7]. After rain events, the soil water content within the band increases dramatically relative to that of the bare soil (see Figure 4). Following extended periods without rain, the soil water content within the vegetation band becomes comparable to—or is even slightly below—the soil moisture of the bare ground. The predictions of the fast-slow model are qualitatively consistent with these observations. While further comprehensive field studies are necessary to determine whether Figure 4’s observations hold more broadly, model predictions of soil moisture distribution vary widely, and many are qualitatively inconsistent with these results.

Slow-timescale conceptual reaction-advection-diffusion models of the forms \((1)\) and \((2)\) do not resolve individual rain events and miss details of soil moisture on shorter timescales, meaning that researchers interpret resulting predictions as those of the annually-averaged system. Two-field models of the form \((1)\)—which lump soil  moisture and surface water into a single “water” field—indicate that the highest concentration of water exists in the regions of bare soil between the vegetation bands [6]. This disagrees with the measurement averages in Figure. 4. Three-field models of the form \((2)\) separately track the dynamics of surface and soil water and predict that soil moisture (on average, i.e., for typical parameter choices), will peak within the vegetation band [8, 9]. In these models, the average surface water height peaks in the bare soil region, which is in contrast to the fast-slow model’s predictions. In addition, the relative difference between bare soil and vegetation band is often not very strong for both water profiles in these three-field models. Figure 5 illustrates the types of possible prediction differences between the three model classes. 

Figure 5. Possible predictions for relative spatial distribution of water (W and H/s) and biomass (B) fields in conceptual models. For the fast-slow model, the solid line marks the time-averaged profiles while the dashed lines show pointwise minimum and maximum.

Interpreting Two-field Models Using a Fast-slow Framework

The fast-slow switching model in \((3)\)-\((4)\) produces physically reasonable predictions in the absence of parameter tuning. It provides a testbed for exploring the role of different feedbacks in the model and identifying the impact of various precipitation regimes on patterns. This framework may also provide a path towards a deeper understanding of how to interpret predictions of two-field, reaction-advection-diffusion models of the form \((1)\), which are well-suited for mathematical analysis but often dismissed as too oversimplified. One can interpret the fast-slow model as the fast system \((3)\) that provides a kick to the soil moisture in the two-field slow system \((4)\) every time it rains. The time-averaged evolution of the fast-slow model may therefore provide some hints about how to appropriately capture key fast feedbacks—such as infiltration and water transport—when formulating a consistent two-field model on the slow timescale.

1 The presence of biomass increases infiltration rate which, in turn increases the biomass growth rate: hence a \(B^2\) dependence of growth [6].

Punit Gandhi presented this work during a minisymposium presentation at the 2019 SIAM Conference on Applications of Dynamical Systems, which took place in May 2019 in Snowbird, Utah.

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Punit Gandhi is an assistant professor in the Department of Mathematics and Applied Mathematics at Virginia Commonwealth University.  
Sarah Iams is Associate Director of Undergraduate Studies and a lecturer in applied mathematics in the Paulson School of Engineering and Applied Sciences at Harvard University.  
Mary Silber is a professor in the Department of Statistics at the University of Chicago and director of the Committee on Computational and Applied Mathematics, an interdisciplinary Ph.D. program. She is a SIAM Fellow. 
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