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A theoretical foundation for multi-scale regular vegetation patterns


Self-organized regular vegetation patterns are widespread1 and thought to mediate ecosystem functions such as productivity and robustness2,3,4, but the mechanisms underlying their origin and maintenance remain disputed. Particularly controversial are landscapes of overdispersed (evenly spaced) elements, such as North American Mima mounds, Brazilian murundus, South African heuweltjies, and, famously, Namibian fairy circles5,6,7,8,9,10,11,12,13. Two competing hypotheses are currently debated. On the one hand, models of scale-dependent feedbacks, whereby plants facilitate neighbours while competing with distant individuals, can reproduce various regular patterns identified in satellite imagery1,14,15. Owing to deep theoretical roots and apparent generality, scale-dependent feedbacks are widely viewed as a unifying and near-universal principle of regular-pattern formation1,16,17 despite scant empirical evidence18. On the other hand, many overdispersed vegetation patterns worldwide have been attributed to subterranean ecosystem engineers such as termites, ants, and rodents3,4,7,19,20,21,22. Although potentially consistent with territorial competition19,20,21,23,24, this interpretation has been challenged theoretically and empirically11,17,24,25,26 and (unlike scale-dependent feedbacks) lacks a unifying dynamical theory, fuelling scepticism about its plausibility and generality5,9,10,11,16,17,18,24,25,26. Here we provide a general theoretical foundation for self-organization of social-insect colonies, validated using data from four continents, which demonstrates that intraspecific competition between territorial animals can generate the large-scale hexagonal regularity of these patterns. However, this mechanism is not mutually exclusive with scale-dependent feedbacks. Using Namib Desert fairy circles as a case study, we present field data showing that these landscapes exhibit multi-scale patterning—previously undocumented in this system—that cannot be explained by either mechanism in isolation. These multi-scale patterns and other emergent properties, such as enhanced resistance to and recovery from drought, instead arise from dynamic interactions in our theoretical framework, which couples both mechanisms. The potentially global extent of animal-induced regularity in vegetation—which can modulate other patterning processes in functionally important ways—emphasizes the need to integrate multiple mechanisms of ecological self-organization27.

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Figure 1: Social-insect nest distributions, in theory and in nature.
Figure 2: Simulation of FC emergence from termite engineering and vegetation feedbacks.
Figure 3: FC life cycle in the coupled termite-vegetation model.
Figure 4: Predicted and observed regular patterning of FC matrix vegetation.


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This research is a product of US National Science Foundation grant DEB-1355122 to C.E.T. and R.M.P., with seed funding provided by the Princeton Environmental Institute’s Grand Challenges Program. J.A.B. was supported by the Marine Alliance for Science and Technology for Scotland (MASTS) pooling initiative, funded by the Scottish Funding Council (HR09011) and contributing institutions. WorldView-2 satellite imagery was obtained through a grant from the DigitalGlobe Foundation to R.A.L. We thank the Government of Namibia, N. Oldendaal and NamibRand Nature Reserve ( for permission to conduct research and for providing rainfall data; A. Lamb, D. Doak, E. Lombardi, G. Barrenechea, P. Davies, S. Levin, R. Martinez-Garcia, I. Rodriguez-Iturbe, A. Sabatino, and J. Ware for discussions and assistance; I. Arndt for Australian termite-mound images used in analyses and shown in Extended Data Fig. 3; and F. Lanting for connecting us to NamibRand Nature Reserve and for use of the image in Fig. 3d.

Author information

Authors and Affiliations



C.E.T., J.A.B., and R.M.P. conceived the study and developed the models, with input from E.S. J.A.B. performed point-pattern analyses and all simulations. E.S. performed Voronoi and Fourier transform analyses. J.A.G., T.C.C., and R.A.L. contributed field data and remote-sensing analyses. C.E.T., J.A.B., and R.M.P. drafted the paper, and all authors provided comments.

Corresponding authors

Correspondence to Corina E. Tarnita or Juan A. Bonachela.

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Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks A. Hastings, N. Juergens, and M. Rietkerk for their contribution to the peer review of this work.

Extended data figures and tables

Extended Data Figure 1 Probability functions associated with conflicts in the termite model.

a, Probability for colony A and colony B to engage in a war as a function of each colony’s population biomass. b, If colonies A and B engage in a war, probability functions for colony A (red curve) or B (blue curve) to win the war as a function of the ratio of colony population biomass. When the two colonies have roughly the same biomass, they coexist (green area).

Extended Data Figure 2 Results of the termite self-organization model with a fixed resource density level, Pcst.

a, b, Temporal behaviour of mature colonies in the termite model for Pcst = 50 g m−2. a, Average diameter of mature nests (blue shading, ±1 s.d.). b, Average distance between nearest neighbours, (where neighbours are nests that share territory borders). Both observables reach a clear stationary state after a transient period of ~200 years. cf, Emergent behaviour for the colonies at the stationary state as a function of the (annually) available level of resources. c, Average mound diameter, which reflects foraging-territory area, decreases as resource availability increases. d, Mean nearest-neighbour distance also decreases with increasing resource density. e, Termite population density (number of individuals per square metre) increases with available resources. f, Average colony biomass density (individuals/colony) increases with available resource density, and reaches a saturation value around Bmax(1−m/μ) (see Supplementary Information). g, Frequency distribution of neighbour numbers from Voronoi analysis for the model with different resource densities (inset: mean number of neighbours). Higher resource densities result in a higher number of colonies (numbers in parentheses in the legend) and therefore more powerful results. Results are obtained by averaging over 100 simulations for each resource level; error bars, ±1 s.d.

Extended Data Figure 3 Average number of neighbours in various field locations.

Upper left panel: average number of neighbours (±1 s.d.) from Voronoi analysis of model and field data; number of nests at each location is shown in parentheses. All other panels: satellite imagery and/or photographs used for data analysis. Mounds are highlighted for ease of observation. If a white rectangle is present then only the points within the rectangle were analysed; otherwise, the whole image was analysed. All scale bars, 100 m. Aerial photographs of Amitermes mounds in Australia courtesy of I. Arndt. Pleiades-1 satellite imagery of Kenya copyright 2013 CNES/Astrium (GeoTIFF file supplied by Apollo Mapping, Boulder, Colorado, USA). Multispectral WorldView-2 satellite imagery of Mozambique courtesy of the DigitalGlobe Foundation. Google Earth satellite imagery: images for Arizona copyright 2012 DigitalGlobe, for Brazil copyright 2016 CNES/Astrium, and for Namibia copyright 2016 DigitalGlobe.

Extended Data Figure 4 Spatial point-pattern analyses of various field locations.

Left: pair correlation function as a function of distances between nests. Right: Ripley’s L function for the same examples. Ninety-five per cent pointwise simulation envelopes (shaded areas) were calculated using the default function from the R package spatstat. These envelopes allow us to reject the null hypothesis (complete spatial randomness) at a confidence level of 95%; thus, if the focal function (red line) falls out of the envelope for a given distance r, the function differs from the expectation for a completely random point distribution. Both sets of panels show peaks (left panels) or valleys (right panels) of regularity that indicate the presence of overdispersion for each of these examples. Note the different number of nests present in the samples from each location (Extended Data Fig. 3), which leads to different levels of noise in the calculation of the two statistics.

Extended Data Figure 5 Rainfall data from NamibRand Nature Reserve.

Top: 10-year time-series of monthly rainfall totals 2004–2014, averaged across multiple sites within NamibRand Nature Reserve (data provided by V. Hartung). Bottom: mean monthly rainfall (that is, averaged for each month across all years) in NamibRand Nature Reserve from 2004 to 2014 (green line, ± 1 s.d. in red) and proposed rainfall function (blue). The noise term included in Rainfall(t) (equation (5) in Methods) ensures that the rainfall function variability is high during the rainy season and low in the dry season, consistent with the data.

Extended Data Figure 6 Vegetation dynamics with and without termite engineering.

a, b, Comparison of the stationary pattern obtained with the vegetation model alone using (a) the original symmetric implementation for the root kernel and (b) the modified root kernel that is allowed to grow asymmetrically. c, d, Stationary pattern obtained with the naive setup (that is, one single, static colony in the centre of the system; constant rainfall); c, the resulting pattern using the original, symmetric root kernel; d, the pattern obtained when the asymmetric root system growth is implemented. e, f, Simulation run measuring the recovery time after the death of a colony in the coupled model with variable rainfall and asymmetric roots; e, system a few months before reaching stationarity; a ring of taller and denser vegetation is formed around the gap, and matrix vegetation is reaching its stationary clumpy distribution; f, several decades after colony death, the gap closes fully, and the remaining large matrix clumps disappear shortly thereafter. Brown, soil; green, vegetation. Colour intensity indicates vegetation density. Parameters are as in Extended Data Tables 1 and 2.

Extended Data Figure 7 The effect of decreasing termite-induced plant mortality or increasing rainfall in the coupled system.

When on-nest enhanced plant mortality is low and/or rainfall is high, vegetation growth outpaces termite engineering and, consequently, vegetation is found also on nests, disrupting (and for high enough rainfall values completely removing) the bare discs. a, Low mortality enhancement (ν = 1.1); b, intermediate mortality enhancement (ν = 1.25); values in a and b are both lower than in Extended Data Table 1 but have the same average rainfall as Extended Data Fig. 5; c, intermediate mortality enhancement (ν = 1.25) and average rainfall increased by 10%. Brown, soil; green, vegetation. Colour intensity indicates vegetation density. df, Corresponding underlying termite territories and nests. Blue dots, established nests; red dots, incipient nests (including the initial diggings of an alate pair, leading to occasionally high local densities as shown in f). Snapshots taken for a peak in vegetation after the system has reached stationarity. Rest of the parameters as in Extended Data Tables 1 and 2.

Extended Data Figure 8 Distribution of FC lifetimes measured in the coupled model.

For n = 9 replicates of the merged model, we kept track for ~300 years (until the end of the simulation) of 100 randomly selected FCs that were born after the stationary state (reached after ~100 years). Fifty-three of these FCs disappeared before the end of the simulation, allowing lifespan estimates for that subset. The resulting lifespans range from <5 years to >165 years, within reported estimates for Namibian FCs. Note that the distribution is truncated on the right tail owing to the limit of available simulation times; however, the overall shape of the distribution should not be strongly affected since such long-lasting FCs are very infrequent.

Extended Data Table 1 List of parameters for the termite model and associated literature sources7,29,31,35,37,44,45,46,47,48,49,50,51,52,53,54 and estimation procedures
Extended Data Table 2 List of parameters for the vegetation parts of the merged model and associated literature sources9,10,15,55,56,57,58,59,60

Related audio

Supplementary information

Supplementary Information

This file contains Supplementary Text and Data and additional references. (PDF 687 kb)

Dynamics of the termite model.

Pink dots represent the center of young (incipient) colonies; blue dots represent the center of mature (established) colonies. Pale blue = foraging territory; darker blue = territory boundary. (MOV 3111 kb)

Dynamics of the termite-vegetation model from initial state to stationarity.

Starting with a single colony and a homogeneous distribution of vegetation, the coupled system self-organizes to patterned vegetation and regularly-organized FCs. The birth and death of colonies marks the emergence and disappearance of FCs. The bottom panel shows the rainfall availability (orange = data-inferred function; red point = instantaneous value of the stochastic rainfall function; grey = historical rainfall within the year). The middle panel, shows the dynamics of termite territories, which influence (and are influenced by) the dynamics of vegetation (top panel). Note the gaps matching the location of the centers; they are surrounded by taller vegetation, which defines the FCs. (MOV 29840 kb)

Close-up view of the coupled dynamics of colonies and vegetation from initial state to stationarity, focused on the matrix vegetation.

Territories expand and coexist, giving rise to neighboring FCs. Shrinkage of territories due to competition gives the opportunity for young colonies to establish in the available space. Matrix vegetation self-organizes in response to scale-dependent feedbacks along the differences in soil moisture induced by the FCs. Panels are same as in Supplementary Video 2. (MOV 26784 kb)

Comparison of the response of coupled (left) and pure-SDF (right) 880 systems to 10 consecutive dry years (20% reduction of average rainfall) followed by return to baseline rainfall.

The same reduction in rainfall leads the SDF system to almost complete extinction of vegetation, which remains in densities that are much lower than in the FC case. After rainfall returns to baseline, vegetation in both systems recovers, but the FC ecosystem is fully restored to stationarity much faster than the SDF system. After the FC system reaches stationarity, the video focuses on only the pure-SDF system. For both systems, dry years are represented by blue lines in the rainfall function plot; baseline years are represented by grey lines. Snapshots taken the same month each year, represented by the red point on the rainfall (bottom) panel. (MOV 10940 kb)

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Tarnita, C., Bonachela, J., Sheffer, E. et al. A theoretical foundation for multi-scale regular vegetation patterns. Nature 541, 398–401 (2017).

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