By Ehud Meron
Instabilities are generally classified as events that one should avoid. This is not the case in drylands, for which model studies predict that instabilities can reverse desertification processes and increase ecosystem resilience to drought. These positive outcomes are consequences of plant populations’ pattern-forming instabilities that result in spatial self-organization and the formation of nearly periodic vegetation patterns.
Two general types of pattern-forming instabilities are evident [1]. The first type corresponds to nonuniform instabilities of uniform states, where spatially-periodic modes with particular wavenumber \(k_0\) begin to grow while an instability threshold is traversed. When the growth is monotonic in time, stationary asymptotic patterns generally form. In isotropic systems, the first pattern that typically appears is a hexagonal pattern: a result of the simultaneous growth of three modes with wave-vectors \(\boldsymbol{k}_i (|\boldsymbol{k}_i|=k_0)\)—oriented 1200 apart—that resonate with one another \((\boldsymbol{k}_1+\boldsymbol{k}_2+\boldsymbol{k}_3=0)\). When the growth is oscillatory, traveling or standing waves form. The second kind of instability corresponds to instabilities of localized structures, such as fronts that separate spatial domains of different system states. Unlike the former instabilities that simultaneously induce pattern formation throughout the whole system, instabilities of fronts develop locally and yield growth patterns that gradually expand to the entire system.
Vegetation patterns with clear periodicity are present in many dryland ecosystems. Figures 1a and 1b depict a nearly hexagonal gap pattern in a grassland in northwestern Australia [2]. The pattern consists of circular bare-soil gaps, each of which is surrounded by about six other gaps at nearly equal distances. One can explain the emergence of such patterns in terms of positive feedback loops between local vegetation growth and water transport toward the growth location (see Figure 2) [1]. Growing vegetation draws water from its surroundings that helps further its growth but also inhibits growth in the surrounding areas where water content is depleted; this favors the formation of vegetation patterns [3]. Many other examples exist, including vegetation bands or stripes on hill slopes that are perpendicularly oriented to the slope direction (see Figure 1c). These stripes often migrate uphill; plants at the top of the band receive runoff water and grow, while plants at the bottom lose runoff and die [4].
Figure 1. Vegetation patterns in northwestern Australia. 1a. A pattern of circular gaps in a grassland of nearly hexagonal geometry. 1b. A single gap, four meters in diameter on average. 1c. A nearly-periodic stripe pattern of acacia trees on a gentle slope. Images courtesy of Kevin Sanders and Stefan Getzin.
Can we move beyond these heuristic explanations of vegetation pattern formation and relate observed patterns to pattern-forming instabilities? Unfortunately, controlled laboratory experiments to identify instabilities—as researchers do in the context of other pattern formation, such as fluid dynamics, chemical reactions, and nonlinear optics [5]—are difficult (if not impossible) to carry out because of vegetation growth’s long time scales. However, studies of nonlinear partial differential equation models that capture positive feedback loops—such as those in Figure 2—have been highly instrumental in explaining observed patterns. For example, researchers have used a model consisting of a biomass variable—which describes the plant mass above ground level—and two water variables—which describe soil moisture and overland water flow—to account for the hexagonal grass pattern in Figure 1a, in terms of a nonuniform stationary instability of uniform vegetation as the precipitation rate decreases below a critical value [1-2]. Scientists have also used a simpler model to account for banded vegetation migrating uphill in terms of a nonuniform oscillatory instability of uniform vegetation [6], and for the existence of a family of periodic patterns with decreasing wavenumbers as precipitation rate decreases [7]. One can obtain information about instability thresholds and the types of modes that grow beyond instability points by using linear stability analysis. Global information, i.e., away from instability points, is accessible via numerical continuation methods. That information is often summarized in a bifurcation diagram (see Figure 3).
Figure 2. A general positive feedback loop that drives vegetation pattern formation. An incidental location of denser vegetation draws more water from its surroundings through various water-transport mechanisms, including overland water flow, water conduction by laterally-spread roots, and soil-water diffusion. The enhancement of water transport as vegetation grows denser is related to the development of infiltration contrast (higher in denser vegetation) and the positive correlation between shoot growth and root growth. The extra water contribution that dense vegetation receives helps it grow even denser and draw increasingly more water from surrounding areas, making the vegetation in those areas sparser and favoring the formation of vegetation patterns. Further details are available in [1].
Observations of vegetation growth patterns induced by front instabilities (the second type of pattern-forming instability) have not yet been reported, but model studies predict their feasibility [8]. High evaporation rates in bare soil, along with significantly reduced evaporation in vegetation patches, may stabilize the bare-soil state—i.e., a state devoid of vegetation—at relatively high precipitation rates where uniform vegetation is also a stable state. The bifurcation diagram in Figure 3 indicates this occurrence. In the bistability precipitation range \(P_T<P<P_0,\) where \(P_T\) denotes the instability threshold of uniform vegetation to periodic patterns and \(P_0\) denotes the bare-soil instability, front solutions—bi-asymptotic to the bare-soil and uniform-vegetation states—exist. These fronts can experience a transverse front instability, where small front-line undulations begin to grow and lead to growth patterns (see Figure 4). The instability sets in when water transport to an incidental vegetation bulge is fast enough to enhance further vegetation growth while inhibiting growth on both sides of the bulge [8].
Figure 3. A partial bifurcation diagram obtained by numerical continuation of model solutions in one space dimension [14]. The vertical axis is the spatial biomass average, and the horizontal axis represents precipitation rate. Solid lines represent stable solutions, and dashed lines represent unstable solutions. Periodic pattern solutions (green line), of which only one is shown, extend to lower precipitation ranges in comparison to the uniform vegetation solution. Under conditions of high evaporation rates in bare soil, the bare soil state may remain stable at high precipitation values, thus forming a bistability range with uniform vegetation.
Vegetation patterning is an intriguing phenomenon that calls for further study, particularly when considering more complex aspects of real ecosystems. However, a different kind of question related to the function of dryland ecosystems in variable climate and under human pressure is also pressing. The bifurcation diagram in Figure 3 already provides some insights into functional aspects of dryland ecosystems, such as response to severe droughts. It indicates that periodic solutions extend to precipitation values significantly lower than those of the uniform solution. Vegetation patterning is a population-level means (i.e., involving many individual plants) to cope with water stress, which keeps the ecosystem functional under drier conditions. By forming patterns, each vegetation patch benefits from both direct rainfall and water drawn from its bare-soil surroundings. This simple observation also accounts for the morphological transitions along the rainfall gradient that are present in model studies — first from hexagonal gap patterns to stripe patterns, and then from stripe patterns to hexagonal spot patterns [9-10]. Both transitions extend the areas of water-contributing bare-soil patches and therefore compensate for reduced rainfall.
The tendency of dryland vegetation to self-organize in spatial patterns raises many more questions related to functional aspects of dryland ecosystems. Here I address two questions of this kind, related to the reversal of degradation processes and restoration of already-degraded landscapes. Degradation processes involving plant mortality (or desertification) may occur abruptly as tipping-point phenomena that encompass the entire ecosystem, or gradually as a domino-like process that proceeds via front propagation. The first two snapshots of Figure 4, which show bare-soil area (yellow) expanding into vegetation area (green), demonstrate the latter scenario. The subsequent snapshots depict the development of a transverse front instability that reverses the desertification process through vegetation fingers’ regrowth back into bare soil, ultimately forming a functional vegetation state.
Figure 4. Snapshots of numerical model solutions showing a transverse front instability. Following a short phase in which the bare-soil domain expands into the uniform-vegetation domain, vegetation fingers develop and grow back into the bare-soil domain, thus forming a stationary labyrinthine pattern. The time indicated in every snapshot is in units of years. Further information is available in [8].
This finding raises the following practical question: Given an observation of a stable desertification front, how can one induce a transverse front instability to reverse the desertification process? One possibility is to introduce a new plant species to the narrow front zone that speeds up water transport—e.g., species with higher rates of water or uptake when the transport mechanism is soil-water diffusion—to create steeper soil-water gradients and thus faster water transport. Another possibility is related to the existence of a tristability precipitation range of bare soil, uniform vegetation, and periodic patterns (see Figure 3). In that range, linearly stable desertification fronts may be nonlinearly unstable, i.e., may become unstable to large enough undulations along the front line [11]. Periodic irrigation, clearcutting, or grazing along the front line can induce instability in this case. These results are very appealing from the point of view of ecosystem management, as they imply local manipulations in the front zone only—rather than extensive intervention across the whole ecosystem—and manipulations that are limited in time, as they are only needed to trigger the instability. Once the instability sets in, a process of self-recovery begins.
Figure 5. Vegetation restoration by water harvesting in the northern Negev (a desert in Israel). 5a. Local and aerial views. 5b. Schematic illustrations of continuous restoration in a stripe pattern. 5c. Fragmented restoration in a rhombic pattern. Black lines symbolize parallel embankments (periodic forcing), whereas green bands or segments denote planted vegetation. Restoration in a stripe pattern is characterized by a wave-vector \(\boldsymbol{k}_f=(k_f,0)\), while restoration in a rhombic pattern is characterized by two additional wave-vectors, \(\boldsymbol{k}_{\bar{+}}=(-k_x, \bar{+}k_y)\), where \(k_x=k_f/2\) and \(k_y=\sqrt{k^2_0-k^2_x}\), representing oblique modes [12].
What land management options are available once complete desertification takes place? A common approach to restore degraded vegetation is water harvesting by spatially-periodic ground modulations, such as parallel embankments. Vegetation is planted along these embankments, which also intercept overland water flow (see Figure 5a). This is a spatial resonance problem whereby a system that tends to self-organize in a periodic stripe pattern with a wavenumber \(k_0\) is subjected to an external periodic force of a different wavenumber \(k_f\). Unlike the temporal counterpart problem of forced oscillations, here the system can form two-dimensional patterns in response to the one-dimensional forcing.
This possible response raises another sensible question: What plantation patterns should result in restored systems of higher resilience to droughts? Should the plantation patterns follow the periodic configuration of ground modulations, i.e., vegetation bands along each embankment, as is normally done (see Figure 5b)? Or should other plantation patterns that do not fully overlap the periodic ground modulations be used instead? Model studies indicate that the second option is preferred. Planting vegetation in a rhombic pattern—as Figure 5c illustrates—results in higher resilience to droughts. One can intuitively understand this higher resilience of rhombic patterns in terms of reduced competition for water along the embankments, but the model analysis provides additional information; it uncovers the existence of a precipitation threshold, below which collapse to bare soil is likely when the restoration is made in stripe patterns. Interestingly, this threshold is related to the disappearance of a pair of unstable stripe solutions, which points towards the significance of tracking unstable states to address questions of ecosystem management [12].
The relationships between vegetation, pattern formation, and functional aspects of dryland ecosystems—such as resilience to droughts and sustainable human intervention—call for many more studies that consider additional aspects of ecosystem complexity [13]. These aspects include seasonal and interannual rainfall variability, soil heterogeneities, interspecific interactions and community dynamics, and soil erosion and deposition processes. Mathematical modeling and model analysis should continue to play crucial roles in understanding these relationships and gaining deep insights into sustainable management of dryland ecosystems.
Ehud Meron presented this work during a minisymposium at the 2019 SIAM Conference on Applications of Dynamical Systems, which took place in May in Snowbird, Utah.
References
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Ehud Meron is a professor of physics at Ben-Gurion University of the Negev, and a Phyllis and Kurt Kilstock Chair Professor in Environmental Physics of Arid Zones. His research interests include nonlinear dynamics, pattern formation, and theoretical ecology with a focus on dryland ecosystems. He is the author of Nonlinear Physics of Ecosystems (2015, CRC Press).