Localised labyrinthine patterns in ecosystems

Abstract

Self-organisation is a ubiquitous phenomenon in ecosystems. These systems can experience transitions from a uniform cover towards the formation of vegetation patterns as a result of symmetry-breaking instability. They can be either periodic or localised in space. Localised vegetation patterns consist of more or less circular spots or patches that can be either isolated or randomly distributed in space. We report on a striking patterning phenomenon consisting of localised vegetation labyrinths. This intriguing pattern is visible in satellite photographs taken in many territories of Africa and Australia. They consist of labyrinths which is spatially irregular pattern surrounded by either a homogeneous cover or a bare soil. The phenomenon is not specific to particular plants or soils. They are observed on strictly homogenous environmental conditions on flat landscapes, but they are also visible on hills. The spatial size of localized labyrinth ranges typically from a few hundred meters to ten kilometres. A simple modelling approach based on the interplay between short-range and long-range interactions governing plant communities or on the water dynamics explains the observations reported here.

Full text

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Localised labyrinthine patterns
in ecosystems
M. G. Clerc1, S. Echeverría‑Alar 1* & M. Tlidi2
Self‑organisation is a ubiquitous phenomenon in ecosystems. These systems can experience
transitions from a uniform cover towards the formation of vegetation patterns as a result of
symmetry‑breaking instability. They can be either periodic or localised in space. Localised vegetation
patterns consist of more or less circular spots or patches that can be either isolated or randomly
distributed in space. We report on a striking patterning phenomenon consisting of localised vegetation
labyrinths. This intriguing pattern is visible in satellite photographs taken in many territories of Africa
and Australia. They consist of labyrinths which is spatially irregular pattern surrounded by either a
homogeneous cover or a bare soil. The phenomenon is not specic to particular plants or soils. They
are observed on strictly homogenous environmental conditions on at landscapes, but they are also
visible on hills. The spatial size of localized labyrinth ranges typically from a few hundred meters to
ten kilometres. A simple modelling approach based on the interplay between short‑range and long‑
range interactions governing plant communities or on the water dynamics explains the observations
reported here.
e appearance of order and structures that involve nonequilibrium exchanges of energy and/or matter have been
widely observed in many natural systems including uid mechanics, optics, biology, ecology, and medicine1–6.
Vegetation populations and vegetation patterns belong to this eld of research. Being oen undetectable at the
soil level, large-scale vegetation patterns have been rst observed thanks to the usability of aerial photographs
in the early ies7. ey appear as extended bands of vegetation alternating periodically with vegetated areas
and unvegetated bands. ese large-scale botanical organisations have been reported in many semi-arid and
arid ecosystems of Africa, Australia, America, and Asia. It is now widely admitted that the origin of these large
scale botanical organisations is attributed to a nonequilibrium symmetry-breaking instability leading to the
establishment of stable periodic spatial patterns. Extended and periodic vegetation pattern arising in semi-arid
and arid ecosystems has been the subject of numerous studies and is by now fairly well-understood issue (see
recent overviews8–10 and references therein).
Vegetation patterns are not always periodic and extended in space. ey can be spatially localised and ape-
riodic consisting of isolated or randomly distributed patches on bare soil11–13 or gaps embedded in a uniform
vegetation cover14,15. ey are generated in a regime where the homogeneous cover coexists with periodic vegeta-
tion patterns. e interaction between well-separated patches is always repulsive16,17 while for gaps the interaction
alternates between attractive and repulsive depending on the distance separating gaps14,17. e localised patches
has a more or less circular shape. However, for a moderate aridity condition, the circular shape can exhibit defor-
mation followed by splitting of a single into two new patches. Newer patches will in their turn exhibit deformation
and self-replication18–20 until the system reaches a periodic distribution of patches that occupies the whole space
available in landscapes19,20. is process leading to spotted periodic patterns can be seen as warnings of ecosystem
degradation and may lead to outcome of vegetation recovery. Besides patches self-replication, circular spots can
exhibit deformation leading to the formation of arcs and spirals like in isotropic and uniform environmental
conditions21. e vegetation spirals are not waves since they do not rotate21.
In this work, we unveil a new type of vegetation pattern consisting of a localised labyrinth embedded either
in a homogeneous cover or surrounded by bare soil. is phenomenon is observed in Africa and Australia by
remote sensing imagery. An example of such a botanical self-organisation phenomenon is shown in Fig.1. ey
consist of either an irregular distribution of vegetation surrounded by a uniform cover (see Fig.1a, b), or by a
bare state (Fig.1c, d). We show that localised labyrinths embedded in a uniform cover can be stable even if the
environment is isotropic and their formation does not depend on the topography. However, when a localised
labyrinth is surrounded by a bare state, they can expand or shrink. We analyse this phenomenon by using three
OPEN
Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y
        Département de Physique, Faculté des
             *email:
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well-known self-organisation vegetation models, which support localised labyrinths. We show that localised
labyrinths are permanent structures, and they can be observed worldwide involving a range of species and spatial
scales. We interpret this phenomenon as a spatial compromise between the extended labyrinth that occupies the
whole space available and stable homogeneous states. More precisely, the mechanism leading to their formation
is attributed to the pinning-depinning transition that takes place in a parameter space where models exhibit
bistability between extended disordered pattern and homogeneous cover.
Field observations of localised labyrinths
Localised labyrinths observed in nature are large-scale self-organisation patterns. ey are satellite images from
Africa and Australia obtained by the use of Google Earth soware. e landscape of Central Cameroon (zone
of forest-savanna mosaic22), shown in Fig.1a, displays contrasted phases of bare and densely vegetated areas
with well-dened scale and symmetry surrounded by more or less uniform woodland. e climate in the zone
where we observe the localised labyrinth is humid, with annual averaged precipitation of 1800 mm23. e annual
averaged of potential evapotranspiration is between 1500 and 1600 mm24. e localised labyrinth we observe
in Western Australia (see Fig.1b) consists of localised woodland embedded in the shrubland of Mulga Bush
(Acacia Aneura)25. In this zone the climate is arid, where the mean annual precipitation is 250 mm26 and the
mean annual potential evapotranspiration is between 1200 and 1300 mm27. Besides, the localised labyrinth can
be surrounded by bare zones as shown in landscapes of Southwest Niger in a brush-grass Savanna zone28 (Fig.1c,
d). In this region the climate is semi-arid, the mean annual rainfall is 605 mm, in between June and September29,
and the annual mean potential evapotranspiration near this zone is 1900 mm30.All the climate data is summa-
rized in Table1 in Methods section. Sparsely populated or bare areas alternate with dense vegetation irregular
bands or patches made of microchloa Indica. e eld observations suggest that localised labyrinthine structures
are formed both in a at landscape and with topographic variation (see Fig.2). By their spatial regularity, by
their spatial scales ranging from a few hundred meters to ten kilometres, as well as by the composition of their
vegetation (tree, shrubs, herbs, and grasses), localised labyrinthine patterns are permanent structures, and they
Figure1. Localised labyrinth vegetation patterns. Top views of (a) Central Cameroon (3
◦
59′ 22.05″ N 12
◦
17′ 20.99″ E), (b) Western Australia (29
◦
33′ 36.16″ S 117
◦
15′ 32.60″ E), (c) and (d) Southwest Niger (12
◦
34′
45.10″ N 2
◦
41′ 28.71″ E and 12
◦
22′ 6.72″ N 3
◦
28′ 39.35″ E, respectively). e inset (d) show a zoom of the
characteristic labyrinth pattern. All the images were retrieved from Google Earth soware (https:// earth. google.
com/ web/) with a resolution of
1920 ×1080
pixels (total areas of (a) 196.5
km2
, (b) 7.4
km2
, (c) 12.3
km2
, and
(d) 24.6
km2
).e satellite images were taken on 17 of February, 2021; 22 of September, 2018; 15 of November,
2016; and 12 of February, 2020, respectively. e upper-right insets show the localised patterns to emphasize the
topography of the landscape.
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Figure2. Elevation proles of the localised labyrinths observed in nature (Fig.1). ey were obtained using Google Earth
soware (https:// earth. google. com/ web/). In each zone two elevation proles are shown for two arbirtary cross-sections (L1
and L2). (a) e localised labyrinth in central Cameroon has large uctuations in height ranging from 665 to 745 m. e
homogenous cover that surrounds the localised labyrinth also has uctuations in height of the same order. e size of the
major axis of the localised pattern is 16.7 km. (b) In Western Australia the localised labyrinth is in a gentle slope (
0.8%
), t he
size of its major axis is 0.6 km. (c) and (d) shows the elevation proles of the localised labyrinths in Southwest Niger. Both
patterns are in small hills of about 10 m, surrounded by a bare state. ese localised labyrinths emerge in plain terrains.
e sizes of the major axes are 1.0 km and 2.0 km, respectively. See the "Methods" section for details on the accuracy of the
elevation data.
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can be observed even in non-arid climates. ey have neither been observed nor reported. Understanding their
formation and maintenance is an important ecological issue.
e mechanism underlying the emergence of the localised labyrinth can be captured by using self-organisa-
tion mathematical models that can explain vegetation pattern formation within a unied conceptual framework.
In this respect, two approaches will be used. e rst is based on the relationship between the plants’ aerial-
subterranean structures, the facilitative and competitive feedbacks which act at the community level, and the
plants’ spatial propagation by seed dispersion31,32. e second approach incorporates explicitly water transport
by below ground diusion and/or above ground run-o33–35. ese models are in reasonable agreement with
the eld observations36–38.
Mathematical modelling of ecosystems
e absence of the rst principles for biological systems in general, and in particular for vegetation populations
where phenomena are interconnected makes their mathematical modelling complex. e theory of vegetation
pattern formation rests on the self-organisation hypothesis and symmetry-breaking instability that provoke the
fragmentation of the uniform cover. e symmetry-breaking instability takes place even if the environment is
isotropic31,33,35. is instability may be an advection-induced transition that requires the pre-existence of the
environment anisotropy due to the topography of the landscape34,39,40. Generally speaking, this transition requires
at least two feedback mechanisms having a short-range activation and a long-range inhibition. In this respect,
we consider three dierent vegetation models that are experimentally relevant systems: (i) the generic interac-
tion redistribution model describing vegetation pattern formation which incorporates explicitly the facilitation,
competition and seed dispersion nonlocal interactions (ii) the local nonvariational partial dierential model
described by a nonvariational Swi–Hohenberg type of model equation, and (iii) the reaction–diusion system
that incorporate explicetely water transport.
The interaction‑redistribution approach. e integrodierential model. is approach consists of
considering a well-known logistic equation with nonlocal plant-to-plant interactions. ree types of interactions
are considered: the facilitative
M
f
(r,t)
, the competitive
Mc(r,t)
, and the seed dispersion
Md(r,t)
nonlocal inter-
actions. To simplify further the mathematical modelling, we consider that the seed dispersion obeys a diusive
process
Md(
r
,t)≈∇
2b(
r
,t)
, with D the diusion coecient, b the biomass density, and
∇2
=∂
2
/∂x
2
+∂
2
/∂y
2
is the Laplace operator acting in the (x,y) plane. e interaction-redistribution reads
where
i
=
f,c
.
ξi
represents the strength of the interaction,
Ni
is a normalisation constant. We assume that their
Kernels
φi(r,t)
are exponential functions with
Li
the range of their interactions. e facilitative interaction
Mf(r,t)
favouring vegetation development. ey involve the accumulation of nutrients in the neighbourhood
of plants, the reciprocal sheltering of neighbouring plants against climatic harshness which improves the water
budget in the soil. e range of the facilitative interaction
Lf
operates on the crown size. e competitive interac-
tion operates over a length
Lc
and involves the below-ground structures, i.e., the rhizosphere. In nutrient-poor
or/and in water-limited territories, lateral spreading may extend beyond the radius of the crown. is extension
of roots relative to their crown size is necessary for the survival and the development of the plant in order to
extract enough nutrients and/or water from the soil. When incorporating these nonlocal interactions in the
paradigmatic logistic equation, the spatiotemporal evolution of the normalised biomass density
b(r,t)
in isotropic
environmental conditions reads14
e normalisation is performed with respect to the total amount of biomass supported by the system. e
rst two terms in the logistic equation with nonlocal interaction Eq.(2) describe the biomass gains and losses,
respectively. e third term models seed dispersion. e aridity parameter
µ
accounts for the biomass loss and
gain ratio, which depends on water availability and nutrients soil distribution, topography, etc. e homogene-
ous cover solutions of Eq.(2) are:
bo=0
which corresponds to the state totally devoid of vegetation, and the
homogeneous cover solutions satisfy the equation
with
�=ξf−ξc
measures the community cooperativity if
�>0
or anti-cooperativity when
�<0
. e bare
state
bo=0
is unstable (stable)
µ<1
(µ > 1
). e homogeneous cover state with higher biomass density is stable
and the other is unstable. ese solutions are connected by a saddle-node or a tipping point whose coordinates
are given by

b
sn
=(� −1)/�,µ
sn
=e
�
−
1
/�

. e linear stability analysis of vegetated cover (
bs
) with respect
to small uctuations of the from
b(r,t)
=
bs
+
δb exp
{
σt
+
ik
·
r}
with
δb
small, yields the dispersion relation
Given the spatial isotropy, the growth rate
σ(k)
is a real quantity. is eigenvalue may become positive for a nite
band of unstable modes which triggered the spontaneous amplication of spatial uctuations towards the forma-
tion of periodic structures with a well-dened wavelength. At the symmetry-breaking instability the value of the
critical wavenumber
kc
marking the appearance of a band of unstable modes, and hence the symmetry-breaking
(1)
M
i=exp

ξi
Ni
b(r+r′,t)φi(r,t)dr′

, with φi(r,t)=exp(−r/Li
)
(2)
∂t
b(r,t)=b(r,t)[1−b(r,t)]M
f
(r,t)−µb(r,t)M
c
(r,t)+D∇
2
b(r,t)
.
(3)
µ
=
(1
−
b)exp(�b),
(4)
σ(
k)=

bs(1−bs)ξf−bs−
b
s
(1−b
s
)ξ
c
(1+
L
2
ck
2)3/2

eξfbs−Dk2
.
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instability, can be evaluated by two conditions:
σ(kc)=0
and
∂σ /∂ k|kc=0
. ese conditions yield the most
unstable mode
is critical wavenumber determines the wavelength of the periodic vegetation pattern
2π/kc
that emerges from
the symmetry-breaking instability. Replacing
kc
in the condition
σ(kc)=0
, we can then calculate the critical
biomass density
bc
. e corresponding critical aridity parameter
µc
is provided explicitly by the homogeneous
steady states Eq.(3).
Local model: a nonvariational Swi–Hohenberg model. e integrodierential equation(2) can be reduced
by means of a multiple-scale analysis to a simple partial dierential equation, in the form of nonvariational
Swi–Hohenberg equation. is reduction has been performed in the neighbourhood of the critical point asso-
ciated with the nascent bistability14,32. e coordinates of the critical point are: the biomass density
bc=0
, the
cooperativity parameter
c=1
, and the aridity parameter
µc=1
. ese coordinates are obtained from Eq.(3)
by satisfying the double condition
∂µ/∂ bs=0
and
∂2
µ/∂
b2
s
=
0
. To apply a multiple-scale analysis it is neces-
sary to dene a small parameter that measures the distance from criticality and expand b,
µ
, and

in the Taylor
series around their critical values. e symmetry-breaking instability should be close to that critical point. To
full this condition, we must consider a small diusion coecient in order to include the symmetry-breaking
instability in the description of the dynamics of the biomass density. is reduction is valid in the double limit of
nascent bistability and close to the symmetry-breaking instability. In this double limit, the time-space evolution
of biomass density obeys a non-variational Swi–Hohenberg model14
where
η
and
κ
are, respectively, the deviations of the aridity and cooperativity parameters from their values at
the critical point. e linear and nonlinear diusion coecients
ν
,
γ
, and
α
depend on the shape of kernels17. In
addition to the bare state
u=0
, the homogeneous covers obey
where the two homogeneous solutions
u±
are connected through the saddle-node bifurcation

u
sn
=κ/2, η
sn
=κ
2
/4

, with
κ>0
. e solution
u−
is always unstable even in the presence of small spatial
uctuations. e linear stability analysis of vegetated cover (
u+
) with respect to small spatial uctuations, yields
the dispersion relation
Imposing
∂σ /∂ k
|
kc
=
0
and
σ(kc)=0
, the critical mode can be determined
where
uc
satises
4
α
u2
c
(
2uc
−κ) =(
2
γ
uc
−ν)
2
.e corresponding aridity parameter
ηc
can be calculated from
Eq.(7).
The reaction–diusion approach. e second approach explicitly adds the water transport by below
ground diusion. e coupling between the water dynamics and the plant biomass involves positive feedbacks
that tend to enhance water availability. Negative feedbacks allow for an increase in water consumption caused by
vegetation growth, which inhibits further biomass growth.
e modelling considers the coupled evolution of biomass density
b(r,t)
and groundwater density
w(r,t)
. In
its dimensionless form, this model reads33
e rst term in the rst equation describes plant growth at a constant rate (
γ /ω
) that grows linearly with w
for dry soil. e quadratic nonlinearity
−b2
accounts for saturation imposed by poor nutrients soil. e term
proportional to
θ
accounts for mortality, grazing or herbivores. e mechanisms of dispersion are modelled by
a simple diusion process. e groundwater evolves due to a precipitation input p. e term
(1−ρb)w
in the
second equation accounts for the evaporation and drainage, that decreases with the presence of vegetation. e
term
w2b
models the water uptake by the plants due to the transpiration process. e groundwater movement
follows the Darcy’s law in unsaturated conditions; that is, the water ux is proportional to the gradient of the
water matric potential41. e matric potential is equal to w, under the assumption that the hydraulic diusivity
(5)
k
2
c=
1
L2
c


3bseξfbs(1−bs)ξcL2
c
2D2/5
−1

.
(6)
∂tu(
r
,t)
=−
u(
r
,t)(η
−
κu(
r
,t)
+
u(
r
,t)2)
+
[ν
−
γu(
r
,t)]
∇
2u(
r
,t)
−
αu(
r
,t)
∇
4u(
r
,t),
(7)
u
±=κ±

κ2−4η
2,
(8)
σ(k)
=
u+(κ
−
2u+)
−
(ν
−
γu+)k2
−
αu+k4.
(9)
k
c=

γ−ν/uc
2α,
(10)
∂b
∂t
=γ
w
1+ωw
b−b2−θb+∇
2b
,
(11)
∂w
∂t
=p−(1−ρb)w−w2b+δ∇2(w−βb)
.
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is constant41. To model the suction of water by the roots, a correction to the matric potential is included;
−βb
,
where
β
is the strength of the suction.
Results
Localised labyrinthine vegetation pattern. In our analysis, we focus on the simplest vegetation model
that has been derived from theinteraction-redistribution approach, namely the non-variational Swi–Hohen-
berg Eq.(6) described above. is model is appropriate to describe the space-time dynamics of the biomass
under resource-limited landscapes such as nutrient limitation or water deprivation. In this case, the average bio-
mass density is low comparing the carrying capacity closed-packing density of unstressed vegetation. e simu-
lated stationary localised vegetation labyrinth is shown in Fig.3a. Moreover, to conrm the eld observation
and to show that this phenomenon is model-independent, we conducted numerical simulations of the other two
models, the integrodierential (Eq.(2)) based on the facilitative, competitive, and seed dispersion interactions;
and the reaction–diusion type that explicitly incorporates water transport (Eq.(11)). e results are shown in
Fig.3b and 3c. e parameters used to simulate the dierent localised labyrinths are listed in Tables2,3, and4 in
the "Methods" section. e localised labyrinth consists of one spatially disordered state surrounded by a quali-
tatively dierent state. Note that the localised labyrinthine patterns shown in Fig.3 do not have a round shape.
e fact that this shape is not round is attributed to the presence of defects in the disordered pattern since they
modify the interface energy. Investigations of fronts propagation between labyrinths and homogeneous states
mediated by defects are missing in the literature. e interface separating these two states is stationary leading to
a xed size of a localised labyrinth. It neither grows and invades the uniform cover nor shrinks. e stabilization
of localised labyrinth is attributed to the interface pinning phenomenon42,43. is phenomenon is characterized
by an interface that connects a homogeneous state and a periodic one, which is motionless on a nite region of
parameters, pinning range. is pinning eect occurs due to the competition between a global energy symmetry
breaking between states that favors the interface propagate in one direction and the spatial modulations that
block the interface by introducing potential barriers42.
To determine the stability domain of the localised labyrinth, we establish the bifurcation diagram shown in
Fig.4a, where we plot the biomass density as a function of the aridity parameter
η
. e aridity refers not only
to water scarcity but can be also attributed to the nutrient-poor soil. When the aridity is low obviously the uni-
form vegetated state is stable (blue line) and the bare state (broken line) is unstable. When the aridity parameter
is further increased, the homogeneous cover becomes unstable with respect to small uctuations. Above this
symmetry-breaking instability, several branches of solutions emerge sub-critically for
η<η
c
. Example of veg-
etation patterns that appears follows the well-known sequence made sparse vegetation spots that can be either
periodic or localised in space (see i, Fig.4a), banded vegetation (see ii, Fig.4a) or a periodic distribution of
localised patches setting on the bare state (see iii, Fig.4a).
An extended labyrinthine pattern can be generated subcritically as indicated by the red line in the bifurcation
diagram (see Fig.4a). e situation which interests us requires that this extended labyrinth exhibits a coexistence
with the uniform vegetated state. e coexistence between these two qualitatively dierent states is the prereq-
uisite condition for the formation of a stable localised labyrinth. However, this condition is necessary but not
sucient, the interface separating these two states exhibits a pinning phenomenon42. Indeed, as shown in the
inset of Fig.4a, there exists a nite range of the aridity parameter oen called the pinning zone
η−
p
<η<η
+
p
,
where localised labyrinthine patterns are stable. Examples of localised labyrinth obtained by numerical simula-
tions for xed values of the control parameters are shown in Fig.4a (iv, v, vi). e motionless interface is not
necessarily circular, and contains bands perpendicular to it and circular patches. Similar bifurcation diagram is
obtained from the integrodierential model (see Fig.4b).
Finally, we discuss the situation where the aridity is not homogenous due to the topography. For this purpose,
we choose a top hat-like shape for the aridity parameter as shown in Fig.5a. In this case, numerical simulations
of the integrodierential model Eq.(2) show a stable localised labyrinthine pattern (see Fig.5a). Note that the
localised labyrinthine structures surrounded by bare soil shown in Fig.1c, d are unstable since the interface
propagates. e interface can not be pinned in the absence of spatial oscillations around the bare state. Oscilla-
tions around this state are unphysical since the biomass density is a positive dened quantity. However, when
the aridity parameter possesses an inverted top hat-like shape, it is possible to pin the interface (see Fig.5b). In
this case, the localised labyrinthine pattern is surrounded by a mosaic extended state, and the mechanismof
stabilization is rather due to the inhomogeneity of the aridity parameter.
Deppining mechanism. e spatial location of the localised labyrinth immersed in the bulk of the stable
uniform vegetated state depends on the initial condition considered. When ecosystems operate out of the pin-
Table 1. Mean annual precipitation, potential evapotranspiration, and aridity index of the regions where
localised labyrinthine patterns are observed. For more details on the meteorological data see the references
given in the text.
Precipitation (mm) Potential evapotranspiration (mm) Aridity index Classication
Central Cameroon 1800 1500–1600 1.1–1.2 Humid
Western Australia 250 1200–1300 0.1–0.2 Arid
Southwest Niger 605 1900 0.3 Semi-arid
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ning zone, the interface separating the labyrinth and the homogeneous cover propagates due to the depinning
transition (see Fig.6a, b). In this case, depending on the aridity level, the interface propagates from one stable
state to another e transition is dierent when moving the aridity parameter slowly or abruptly. In the second
type of variation, when
η<η
−
p
, the homogeneous cover invades the system, while when
η>η
+
p
, the localised
labyrinth survives but it is now embedded by a periodic distribution of gaps (see Fig.6b).
Conclusions
In this paper we have reported for the rst time evidence of localised labyrinthine vegetation patterns observed
on satellite images from Africa and Australia. We have shown that these localised structures are robustly con-
sisting of either an irregular distribution of vegetation surrounded by a uniform cover or on the contrary sur-
rounded by a bare state. We have shown that the formation of localized labyrinthine patterns is not specic to
particular plants or soils. We have found localised labyrinths in ecosystems on at landscapes and hills. ree
relevant models which undergo localised vegetation labyrinthine patterns have been considered; (i) vegeta-
tion interaction-redistribution model of vegetation dynamics, which can generate patterns even under strictly
homogeneous and isotropic environmental conditions. It is grounded on a spatially explicit formulation of the
balance between facilitation and competition. Ecosystems experience transitions towards landscape fragmenta-
tion of landscapes (ii) the nonvariational Swi–Hohenberg model that can be derived from the model (i) in the
long-wavelength pattern forming regime, and (iii) reaction–diusion model that incorporates explicitly water
transport. We have shown that all these models despite their mathematical structure support the phenomenon
Figure3. Numerical observations of localised labyrinths. e model-independent structure is observed in (a)
a non-variational Swi–Hohenberg model, (b) integrodierential non-local model, and (c) reaction–diusion
model. In the three cases the labyrinth is supported by a uniform vegetated state. e parameters used in each
model are listed in the "Methods" section. From numerical simulations, the gure was created using Inkscape
1.0 (https:// inksc ape. org/ relea se/ inksc ape-1. 0/).
Table 2. List of parameter values of the simulations of the non-variational Swi–Hohenberg equation, shown
in Fig.3a (
200 ×200
grid,
η
=
1.01
), Fig.4a (
120 ×120
grid, [i, ii, iii] with space step
x=0.5
), and Fig.6
(
120 ×120
grid).
Cooperativity (
κ
)
ν
γ
α
Time step (
t
) Space step (
x
)
0.6 0.011 0.5 0.125 0.05 0.8
Table 3. List of parameter values of the simulations of the integrodierential model, shown in Fig.3b
(
512 ×512
grid,
µ
=
1.301
), Fig.4b (
256 ×256
grid), and Fig.5 (
256 ×256
grid).
Competition length (
Lc
) Diusion (D) Facilitation strength (
ξf
)Competition strength (
ξc
) Time step (
t
) Space step (
x
)
2.5 1 3 1 0.1 0.8
Table 4. List of parameter values of the simulation of the reaction–diusion model, shown in Fig.3c
(
512
×
512
grid).
γ
ω
θ
p
ρ
δ
β
Time step (
t
) Space step (
x
)
1.45 1.5 0.2 0.7 1.5 100 2.7 0.001 0.6
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of the localised labyrinth. We have established their bifurcation diagram and identied a parameter region,
where we have observed a coexistence between a homogeneous cover and an extended labyrinthine structure
which are both linearly stable. Within it, there exist a pinning zone of parameters where localised labyrinthine
vegetation patterns have been generated as a stable pattern. Note however that localised labyrinth is determined
by the initial condition, while their maximum peak biomass remains constant for a xed value of the system
parameters. is phenomenon results from front pinning between qualitatively dierent coexisting vegetation
states. Outside of the pinning region, we have shown that the localised labyrinth either shrink and leads to the
formation of regular distribution of circular spots or expand leading to the formation of an extended labyrinth.
Finally, we have investigated the formation of localised labyrinth on a hill by considering an inhomogenous
aridity parameter. is forcing acts as a trapping potential for the labyrinthine pattern. Owing to its general
character, robust localised labyrinthine structures observed and predicted in our analysis should be observed in
other systems of various elds of natural science such as uid mechanics, optics, and medicine.
We have documented for the rst time the phenomenon of localised vegetation labyrinth by remote observa-
tions, using the Google Earth computer program, and numerical simulations of three dierent theoretical models
Figure4. Bifurcation diagram of vegetation models. (a) the non-variational Swi–Hohenberg model, and (b)
the integrodierential model. Gaps (i), labyrinths (ii), and spots (iii) is the standard sequence of patterns in
vegetation models.
u
and
b
stands for the average biomass in each model. e solid curves indicate where the
bare soil or uniform vegetation cover are stable, whereas the segmented curves indicate where these states are
unstable. In (a), the critical point
(ηc=0.038, uc=0.53)
stands for the instability threshold where the uniform
vegetated cover loses stability to a modulated state. In a narrow region, between
η
−
p=0.010
and
η+
p
=
0.013
,
where there is a multistability of states (labyrinth, uniform vegetation, bare soil) the emergence of localized
labyrinths is possible. In (b),
(µc=1.309, bc=0.62)
and
µ
−
p=1.2950
,
µ+
p
=
1.3044
. e insets with the
pinning zones enlarged show the existence of a family of localized labyrinths (triangles) with dierent average
biomasses. e insets (iv), (v), and (vi) show dierent localized labyrinthine patterns [(a) and (b)]. e other
parameters are provided in the "Methods" section. From numerical simulations, the gure was created using
Inkscape 1.0 (https:// inksc ape. org/ relea se/ inksc ape-1. 0/).
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which are based on ecologically realistic assumptions. ese models provide a clear explanation of how nonlinear
plant-plant interactions and the eects of plants on soil water can be crucial in determining the spatial distribu-
tion of plant communities. It is far from the scope of this contribution to provide parameters assessment and
comparison between the theoretical predictions and the eld observations. Work in this direction is in progress.
Extended and localised vegetation labyrinthine patterns opens a whole new area of research in self-organi-
sation in vegetation pattern formation, where eld observations will be fundamental to establish a connection
with the concepts developed in this work.
Methods
Google Earth data. e satellite images (cf. Fig.1) are retrieved from the open-access program Google
Earth (see the link https:// earth. google. com/ web/ and information there), courtesy of CNES/Airbus, Landsat/
Copernicus, and Maxar Technologies (Fig.1a), and CNES/Airbus (see Fig.1b–d).
e elevation proles in Fig.2 are obtained from Google Earth. is soware uses digital elevation data from
the Shuttle Radar Topography Mission at a resolution of 30 m44,45. e error, at a
90%
condence level, associated
to the absolute height data is less than 6 m for the territories considered here (Africa and Australia)44.
Climate data. Localised labyrinthine patterns are observed in Central Cameroon (Fig.1a), Western Aus-
tralia (Fig.1b), and Southwest Niger (Fig.1c, d). e climate types of these regions are humid, arid, and semi-
arid, respectively. e climatic classication is based on the aridity index (see Table1), which is the ratio of mean
annual precipitation and potential evapotranspiration46. Note that the aridity index is small (big) when the arid-
ity parameter (
η
or
µ
), dened in the interaction-redistribution approach subsection, is big (small).
Numerical simulations data. Numerical simulations of models under consideration were solved in
square grids with Runge-Kuttta 4 time integrator. e spatial derivatives were approximated using nite dier-
ence scheme with a three point stencil using periodic boundary conditions. In the integrodierential simulation,
the convolution integrals were solved in Fourier space through DFT algorithms. e detail of the parameters
used in the numerical simulations are listed in the Tables below.
Generation of numerical localised labyrinthine patterns. e localised labyrinthine patterns are
initialised in a regionof parameters where the uniform vegetation cover and the labyrinthine pattern coexist,
in particular, in a pinning zone (see Fig.4). e initial condition consists of a circular patch of labyrinthine pat-
tern in the centre of the simulation box, embedded in a homogenous background (see Fig.7). Aer a transient
accommodation of the biomass eld, the stable localised labyrinth emerges. e dynamics towards equilibrium
in the integrodierential model Eq.(2) is resumed in Fig.7 and the Supplementary Video S1.
Computation of the bifurcation diagrams. e bifurcation diagrams in Fig.4 were determined with
analytical and direct numerical integration techniques of the governing equations. e blue and black curves
account for the vegetated state and the bare one, respectively. e curves are solid when the corresponding state
is stable, and broken if unstable. e critical points in which the dierent states change their stability are deter-
mined by linear analysis, detailed in the interaction-redistribution approach subsection.
e red curve is the stable branch of labyrinthine patterns, and it is determined by direct numerical integra-
tion of the governing equations (using the algorithm explained above). Starting from a vegetated state with a
small amplitude noise perturbation, in the region where the uniform vegetation state is unstable, a stable laby-
rinthine pattern can emerge (see (ii) in Fig.4). e stability range of the labyrinth state, that is (i) and (iii) in
Figure5. Localised labyrinthine patterns generated by inhomogenous aridity in the integrodierential model.
e spatially forced pattern can be supported by (a) the vegetated state (top hat-like shape
µ
parameter), and
(b) the bare state (inverted top hat-like shape
µ
parameter). From numerical simulations, the gure was created
using Inkscape 1.0 (https:// inksc ape. org/ relea se/ inksc ape-1. 0/).
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Fig.4, are found by decreasing/increasing the aridity parameter starting from the labyrinthine pattern (see the
black arrows in Fig.4).
e blue triangles account for the stable branch of the localised labyrinthine pattern. e initial condition is
a stable localised labyrinth state (cf. state (iv) in Fig.4). e aridity is decreased until the localised labyrinthine
pattern becomes a localised hexagonal pattern, which determines the le boundary of the pinning region (
η
−
p
or
µ
−
p
). On the other hand, the right boundary of the pinning region (
η+
p
or
µ+
p
) is determined by increasing the
aridity until the localised labyrinthine pattern invades all the system (see Fig.6).
Figure6. Deppining transitions of a localised labyrinth state (
η=0.0102
) in a non-variational Swi–
Hohenberg model. is state is shown in the middle panel of (a) and (b). e localised pattern destabilize when
crossing the pinning region boundaries when varying slowly (a) or abruptly (b) the aridity parameter. In the
rst case (a), when decreasing
η
the localized labyrinth loses its internal structure due to shrinking of stripes (le
panel,
η=0.007
), and when increasing
η
some stripes begin to grow at the interface of the localized labyrinth
and a hexagon pattern starts to invade the uniform cover (right panel,
η=0.016
) . In the second case (b), when
decreasing
η
all the stripes and patches of sparse vegetation disappear in favor of a uniform vegetated cover
(le panel,
η=−0.03
), and when increasing
η
the vegetated cover becomes unstable and stripes emerge. is
process transform the localised labyrinth into an extended one (right panel,
η=0.05
). e other parameters
are provided in the "Methods" section. From numerical simulations, the gure was created using Inkscape 1.0
(https:// inksc ape. org/ relea se/ inksc ape-1. 0/).
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Received: 28 April 2021; Accepted: 10 August 2021
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Figure7. Initialization and stabilization of localised labyrinthine pattern in the integrodierential model
Eq.(2). e aridity parameter is
µ=1.3
. e other parameters are summarized in Table3 on "Methods"
section. e sequence
t1=1
to
t6=4·105
accounts for the evolution towards equilibrium of the localised
labyrinthine pattern, starting from a circular patch of a labyrinth state embedded in a vegetated background
(
t1
). e curve in the right shows the evolution of the average biomass density
b
, that is the double integral of
the two dimensional biomass eld b divided by the area of the simulation box (see the Integrodierential model
subsection). e stable labyrinthine pattern is reached in
t5≈105
iterations of the RK4 time integrator, when
there is no change in
b
. From numerical simulations, the gure was created using Inkscape 1.0 (https:// inksc
ape. org/ relea se/ inksc ape-1. 0/).
Content courtesy of Springer Nature, terms of use apply. Rights reserved

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www.nature.com/scientificreports/
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Acknowledgements
MGC thanks for the nancial support of FONDECYT Project 1210353 and ANID-Millennium Science Initiative
Program-ICN17_012. S.E.-A. thanks the nancial support of ANID by Beca Doctorado Nacional 2020-21201376.
MT received support from the Fonds National de la Recherche Scientique (Belgium). e authors gratefully
acknowledge the nancial support of Wallonie-Bruxelles International (WBI).
Author contributions
M.G.C. and M.T. designed the research. S.E.-A. conducted numerical simulations and the data analysis. All
authors worked on the theoretical description and draed the paper, and contributed to the overall scientic
interpretation and edited the manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Supplementary Information e online version contains supplementary material available at https:// doi. org/
10. 1038/ s41598- 021- 97472-4.
Correspondence and requests for materials should be addressed to S.E.-A.
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