What can pattern formation
theory tell us about
desertification and restoration
of degraded landscapes?
Ehud Meron
Department of Solar Energy &
Environmental Physics
and
Physics Department
Ben-Gurion University of the Negev
Spatio-Temporal Dynamics in
Ecology
8-12 December 2014, Leiden
Desertification - an irreversible
decrease in biological productivity
induced by climatic variations and
anthropogenic disturbances
A wide scope problem – most drylands,
which occupy about 2/5 of the Earth’s
terrestrial area and are home to about
1/3 of the human population are
susceptible to desertification.
Involves four research directions:
1. Understanding desertification
2. Devising warning signals
3. Preventive measures
4. Reversing desertification
Claim:
The concepts and tools of
pattern formation theory are crucial
for understanding desertification and
restoration
Desertification in the northern Negev
Collaborators
Colleagues:
Golan Bel (BGU)
Aric Hagberg (LANL)
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Students:
Yuval Zelnik
Yair Mau (now postdoc at Duke U)
Lev Haim (now at Soroka University Medical Center)
Shai Kinast (Now at NCRN)
Outline
Desertification in spatially extended ecosystems:
1. Pattern formation aspects: local disturbances
induce front dynamics gradual desertification
2. The Namibian fairy-circle ecosystem
Reversing desertification by water harvesting:
1. A spatial resonance problem
2. Restoring in stripe vs. in rhombic patterns
Conclusion
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Vegetation states of different productivity:
1. Feedbacks inducing pattern-forming instabilities
2. Mathematical modeling
3. Uniform and nonuniform states along the rainfall
gradient
Vegtetation states: pattern-forming feedbacks
Three different transport
mechanisms:
+
Local
vegetation
growth
Water transport
towards growing
vegetation
+
Kinast, Zelnik, Bel, Meron, PRL 2014
Water uptake and
conduction by laterally
extended roots
Water transport helps local vegetation growth but inhibits growth
in the patch surroundings mechanism for pattern formation
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Vegetation pattern formation
results from instabilities driven
by positive feedbacks
Vegtetation states: mathematical modeling
Gilad, Hardenberg, Provenzale, Shachak, Meron PRL 2004, JTB 2007
𝜁
𝜕𝑡 𝑏 = 𝐺𝑏 𝑏 1 − 𝑏/𝜅 − 𝑏 + 𝛻 2 𝑏
Biomass b  bx, y, t 
𝜕𝑡 𝑤 = 𝐼ℎ − 𝐿𝑤 − 𝐺𝑤 𝑤 + 𝛿𝑤 𝛻 2 𝑤
Soil-water content w  wx, y, t 
𝜕𝑡 ℎ = 𝑝 − 𝐼ℎ − 𝜵 ⋅ 𝑱
𝑱 = −2𝛿ℎ ℎ 𝜵(ℎ + 𝜁)
Surface-water height ℎ = ℎ(𝑥, 𝑦, 𝑡)
Three
water
𝐺𝑏 (𝒙, 𝑡)
= 𝜈∫transport
𝑔 𝒙, 𝒙′ , 𝑡 mechanisms:
𝑤 𝒙′ , 𝑡 𝑑𝒙′
′
Overland
flow
𝐺
𝑤 (𝒙, 𝑡) = 𝛾∫ 𝑔 𝒙′, 𝒙, 𝑡 𝑏 𝒙 , 𝑡 𝑑𝒙′
Conduction by roots
′ 2
𝒙−𝒙
1
′
Soil-water
diffusion
𝑔 𝒙, 𝒙 , 𝑡 =
2 𝑒𝑥𝑝 −
2
𝜋𝑠0
𝑠 𝑏(𝒙,𝑡)
Root augmentation as plant grows
𝜂=
𝑠0−1 𝑑𝑠/𝑑𝑏|𝑏=0
- root to shoot ratio
𝑠 𝑏 ≈ 𝑠0 (1 + 𝜂𝑏)
c= 1 no contrast

𝑏(𝒙, 𝑡) + 𝑞/𝑐
𝐼(𝒙, 𝑡) = 𝛼
𝑏(𝒙, 𝑡) + 𝑞
Walker 1980; Rietkerk et al. AN 2002
h
Infiltration contrast
between vegetation
patch and bare soil
c>>1
I
 /c
0
high contrast
b
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
A model that captures all three feedbacks (in dimensionless form):
Vegtetation statess: basic vegetation states
bare
soil
uniform
spot
stripe
gap
pattern pattern pattern vegetation
Localized structures
– building blocks for
extended patterns
Spots in Zambia
biomass
Five basic states along the
rainfall gradient:
Bare soil
1
𝑝𝑐
Precipitation p
Stripes in Niger
Gaps in Senegal
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Model results Gilad, Hardenberg, Provenzale, Shachak, Meron PRL 2004, JTB 2007
Bifurcation diagram
Desertification: what pattern formation theory can tell us?
productive state
Does not capture an important aspect disturbances are likely to be local:
b
Rather than a global shift to the
alternative stable state, local domains
of the alternative state can form.
unproductive state
Three aspects of front dynamics:
1. Dynamics of a single front
2. Front interactions
3. Front instabilities
pf
p
pc
General results for uniform states:
Single fronts - propagate in general
c
productive
0
b
unproductive
space
pm
p
Subsequent dynamics –
transition-zone or front
dynamics
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
The common view of desertification:
Desertification: what pattern formation theory can tell us?
p  pm
p  pm
𝑏
𝑏
𝑝 < 𝑝𝑚
Unproductive
domains merged
productive
𝑏
Gradual
process
Space
unproductive
pf
Time t
𝑏
p pm
Simulation of a n activator-inhibitor model
(FHN) with fast inhibitor diffusion
Desertification can be incipient
Desertification can be gradual
and occur before the tipping point! – asymptotic state still includes
(Bel, Hagberg, Meron, Theor. Ecol. 2012)
productive domains
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
𝑏
Front interactions
Unproductive domains have
merged because of attractive
interactions.
Repulsive interactions can lead to
asymptotic patterns
Desertification: what pattern formation theory can tell us?
Back to single front dynamics
but for bistability of uniform
and patterned states:
Hagberg & Meron
PRL 1994; Chaos 1994;
Nonlinearity 1994; PRL 1997.
Uniform vegetation
Periodic pattern
Bare soil
Paja brava grass patterns in Bolivia
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Front instabilities
Desertification: the Namibian fairy-circle ecosystem
0
Pomeau, Physica D 1986;
Knobloch, Nonlinearity 2008
p
Concrete system:
Namibian Fairy Circle (NFC) ecosystem
Sandy soil, confined root zones
model equations simplify to:
𝜕𝑡 𝑏 = 𝐺𝑏 𝑏 1 − 𝑏/𝜅 − 𝑏 + 𝛻 2 𝑏
𝜕𝑡 𝑤 = 𝑝 − 𝐿𝑤 − 𝐺𝑤 𝑤 + 𝛿𝑤 𝛻 2 𝑤
𝐺𝑏 ≈ 𝜈𝑤 1 + 𝜂𝑏 2 , 𝐺𝑤 ≈ 𝜈𝑏 1 + 𝜂𝑏
2
Fairy circles = gap patterns Tlidi,
Lefever, Vladimirov LNP 2008; Getzin K.
Wiegand, T. Wiegand, Yizhaq, von
Hardenberg & Meron, Ecography 2015,
Zelnik, Meron & Bel submitted
Soil-water content in FC higher
than in vegetation matrix
Cramer and Barger PLoS ONE 2013;
Juergens Science 2013
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
c
Single fronts can be stationary
in a parameter range
Desertification: the Namibian fairy-circle ecosystem
Within the bistability range of
uniform vegetation and periodic
pattern – many more solution
branches of hybrid states:
space
Homoclinic snaking
space
(Edgar Knobloch)
Note that the bare soil state remains stable at high rainfall rates
bistability of uniform vegetation and bare soil pattern formation
results for bistability of uniform states may apply FC induced by
front repulsion. Fernandez-Oto, Tlidi, Escaff and Clerc, Phil. Trans A 2014.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Choosing parameters that fit the NFC ecosystem we find the
bifurcation diagram (Zelnik, Meron, Bel, submitted):
Desertification: the Namibian fairy-circle ecosystem
Periodic
pattern
Any indications of such processes in the NFC ecosystem?
Birth of FC
Death of FC
Instances of
hybrid-state
transitions
Tschinkel, PLoS ONE 2012
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
This suggests another form of gradual desertification in a fluctuating
environment - temporal escapes outside the snaking range where
fronts are not pinned (Gandhi, Knobloch & Beaume 2015).
Front propagation then leads to the creation of
Uniform
additional gaps and to a cascade of hybrid state
vegetation
transitions to lower-productivity states.
Desertification: the Namibian fairy-circle ecosystem
Zelnik, Meron, Bel, submitted
Observations
(Namibia)
Birth of FC = front
propagation outside
snaking range ?
Model
simulations
2004 image as
init. cond. within
snaking range
Drought outside Drought is over,
the snaking
back to snaking
range
range
Longer time
within the
snaking range
Escape from and return to snaking range explain observations . Repeated
droughts gradual desertification involving a cascade of such events.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Drought in 2007: 41mm/y vs.
100-300mm/y in other years
Reversing desertification - a spatial resonance problem
Since the unmodulated system tends anyway to form patterns, this
is a spatial resonance problem analogous to temporally forced
oscillatory systems.
Implicit in this approach is the intuitive assumption that vegetation
growth is likely to follow the template of favorable growth
conditions dictated by the periodic ground modulations, and form a
1:1 resonant pattern - vegetation band at each embankment.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
The common approach: water harvesting by ground modulations, e.g.
periodic stripe-like embankments that capture runoff and along
which the vegetation is planted.
Restoration by water harvesting can fail
Question: is this the best restoration practice? Are there other
practices more resilient to environmental fluctuations?
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Restoration in the northern Negev by the JNF-KKL
Simplified model
Approximate the root kernel by a delta function and set 𝜂 = 0:
𝜕𝑡 𝑏 = 𝜈𝑤𝑏 1 − 𝑏/𝜅 − 𝑏 + 𝛻 2 𝑏
𝜕𝑡 𝑤 = 𝐼ℎ − 𝐿𝑤 − 𝜈𝑏𝑤 + 𝛿𝑤 𝛻 2 𝑤
𝜕𝑡 ℎ = 𝑝 − 𝐼ℎ − 𝜵 ⋅ 𝑱
𝑱 = −2𝛿ℎ ℎ 𝜵(ℎ + 𝜁)
Instead of modulating the topography function 𝜁 = 𝜁(𝑥, 𝑦) to simulate
embankments, modulate the infiltration rate.
Represents periodic crust removal to form bands of higher
infiltration rates – a mimetic forcing, mimics what the natural
vegetation does anyway when exists.
For simplicity we choose
harmonic modulations and
assume 𝜁 = constant
𝐼=𝛼
𝑏+ 1+
𝛾𝑓
𝑞
[𝑐𝑜𝑠(𝑘
𝑥)+1]
𝑓
2
𝑐
𝑏+𝑞
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Assume plant species with laterally confined roots and small
root-to-shoot ratios pattern formation induced by the
infiltration feedback
Restoring in 2d resonant patterns
Instead
Yes!
restore
embankment
vegetation
1:1 resonant 1d
pattern
2:1 resonant 2d
pattern
𝑥
The 2d patterns are 2:1 resonant in the sense that the pattern’s
periodicity in the forcing direction 𝑥 is exactly twice the forcing
periodicity, in a range of forcing periods.
Claim:
The 2d resonant patterns are more resilient to rainfall fluctuations
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Is there a better alternative approach to the 1:1 practice?
2d resonant solutions of the model equations
Pattern’s periodicity in the forcing
direction = 2 × forcing periodicity
Exist in a wide forcing wavenumber range
that overlaps with 1:1 resonant stripes.
𝑦
𝑥
(Mau, Hagberg & Meron PRL 2012)
Describe rhombic patterns that consist of three resonating modes:
two oblique modes,
exp 𝑖𝒌± ⋅ 𝒙
𝒌± = 𝑘𝑥 𝒙 ± 𝑘𝑦 𝒚, 𝑘𝑥 =
and a stripe mode,
exp −𝑖𝒌𝑓 ⋅ 𝒙
𝑘0
𝑘𝑓
2
, |𝒌± | = 𝑘0 ,
𝑘𝑦
𝑘𝑥
𝑘0 − selected pattern wavenumber
in unmodulated system
The modes satisfy the resonance condition: 𝒌+ +𝒌− − 𝒌𝑓 = 0,
which makes rhombic patterns
𝐵𝑖𝑜𝑚𝑎𝑠𝑠 ≈ 𝐴𝑒 −𝑖𝑘0𝑥 + 𝑎+ 𝑒 𝑖𝒌+ ⋅𝒓 + 𝑎− 𝑒 𝑖𝒌− ⋅𝒓 + 𝑐. 𝑐. robust.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
2:1 resonant solutions (biomass):
Collapse of restored 1:1 stripes
Stripe pattern
𝑏
Rhombic
pattern
Collapse to
bare soil
Bare soil
(Mau, Haim, Meron, submitted)
p
Responses of restored stripes to precipitation downshifts can
involve ecosystem collapse.
Responses of restored rhombic patterns to precipitation uphifts
involve a smooth transition to stripe patterns.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
In what sense rhombic patterns are more resilient?
Amplitude equations
kx
kx
𝑘𝑓 2
kx
𝑖(𝑘 𝑥+𝑘 𝑦)
𝑖(𝑘 𝑥−𝑘 𝑦)
−𝑖𝑘 𝑥
𝐵
: 𝑖𝑜𝑚𝑎𝑠𝑠 ≈ 𝐴𝑒 0 + 𝑎+ 𝑒 𝑥 𝑦 + 𝑎− 𝑒 𝑥 𝑦 + 𝑐. 𝑐.
and deriving amplitude equations for 𝐴, 𝑎+ , 𝑎− .
Relatively easy for simple models (e.g. the forced SH), much harder
for the vegetation model, but the amplitude equations are universal
and their structural form should apply to the vegetation context too.
2
+ 𝑎−
Rhombic
pattern
2
𝐴
Use the amplitude equations to
study the mechanism of the
collapse process.
Stripe pattern
+ 𝑎+
2
Indeed they give a similar
bifurcation diagram
Bare soil
p
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Get a better understanding by studying the
interaction between the stripe and oblique modes in
the vicinity of the 1:1 resonance where 𝑘𝑓 ≈ 𝑘0
(Lev Haim, PHD thesis):
Dynamics in phase space
2
+ 𝑎−
2
+ 𝑎+
2
The mechanism of
collapse –
Rhombic
pattern - R
𝐴
𝜌𝑅 = |𝑎+ | = 𝑎−
𝜌𝑆 = 𝐴
Stripe pattern - S
the disappearance of
unstable stripe solutions
Bare soil - B
p
R
𝜌𝑅
𝜌𝑅
R
S
𝜌𝑆
𝜌𝑆
B
𝜌𝑆
S
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Project the amplitude
dynamics onto the phase
plane spanned by
Conclusion
Bistability of
uniform states
Repulsive front
interactions
incipient shifts
productive
𝑏
productive
𝑏
unproductive
pf
p pm
Bistability of uniform and patterned states
less
productive
p
The NFC ecosystem as a case study: being uniform ,undisturbed and
describable by a relatively simple model, the NFC is an excellent case
study for studying vegetation pattern formation and desertification.
Restoration by water harvesting as a
spatial resonance problem: restoring in
a resonant 2d rhombic pattern is more
resilient to droughts in comparison with
the classical 1:1 stripe restoration.
Less is more: less intervention, less areal coverage, more resilience.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Desertification is not necessarily abrupt – can occur gradually by
front propagation.
Conclusion
Inasmuch as concepts of nonlinear dynamics, such as multi-stability,
tipping points, oscillations and chaos, have already been integrated
into ecological research, pattern formation theory should be
integrated too.
Nonlinear Physics of Ecosystems
Ehud Meron
Introduction
I Overview
II Pattern formation theory
III Applications to Ecology
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
The concepts and tools of pattern formation theory can be crucial
for the understanding of spatially extended ecosystems.