Ehud Meron Department of Solar Energy & Environmental Physics and Physics Department Ben-Gurion University of the Negev 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 Desertification - an irreversible decrease in biological productivity induced by climatic variations and anthropogenic disturbances What can pattern formation theory tell us about desertification and restoration of degraded landscapes? Spatio-Temporal Dynamics in Ecology 8-12 December 2014, Leiden Claim: The concepts and tools of pattern formation theory are crucial for understanding desertification and restoration Desertification in the northern Negev Desertification in the northern Negev
Transcript
Slide 1
Ehud Meron Department of Solar Energy & Environmental
Physics and Physics Department Ben-Gurion University of the Negev A
wide scope problem most drylands, which occupy about 2/5 of the
Earths 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
Desertification - an irreversible decrease in biological
productivity induced by climatic variations and anthropogenic
disturbances What can pattern formation theory tell us about
desertification and restoration of degraded landscapes?
Spatio-Temporal Dynamics in Ecology 8-12 December 2014, Leiden
Claim: The concepts and tools of pattern formation theory are
crucial for understanding desertification and restoration
Desertification in the northern Negev
Slide 2
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Collaborators Students: Yuval Zelnik Yair Mau (now postdoc at Duke
U) Lev Haim (now at Soroka University Medical Center) Shai Kinast
(Now at NCRN) Colleagues: Golan Bel (BGU) Aric Hagberg (LANL)
Slide 3
Outline 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 Reversing
desertification by water harvesting: 1.A spatial resonance problem
2.Restoring in stripe vs. in rhombic patterns Conclusion
Desertification in spatially extended ecosystems: 1.Pattern
formation aspects: local disturbances induce front dynamics gradual
desertification 2.The Namibian fairy-circle ecosystem
Slide 4
Vegtetation states: pattern-forming feedbacks Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud Water transport helps
local vegetation growth but inhibits growth in the patch
surroundings mechanism for pattern formation Three different
transport mechanisms: Kinast, Zelnik, Bel, Meron, PRL 2014 Water
uptake and conduction by laterally extended roots Local vegetation
growth Water transport towards growing vegetation + + Vegetation
pattern formation results from instabilities driven by positive
feedbacks
Slide 5
Vegtetation states: mathematical modeling Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud h Soil-water content
Biomass Infiltration contrast between vegetation patch and bare
soil I 0 b c= 1 no contrast c>>1 high contrast Walker 1980;
Rietkerk et al. AN 2002 Three water transport mechanisms: Overland
flow Conduction by roots Soil-water diffusion A model that captures
all three feedbacks (in dimensionless form): Gilad, Hardenberg,
Provenzale, Shachak, Meron PRL 2004, JTB 2007
Slide 6
Vegtetation statess: basic vegetation states Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud Bifurcation diagram
Model results Gilad, Hardenberg, Provenzale, Shachak, Meron PRL
2004, JTB 2007 Precipitation p 1 Uniform vegetation Bare soil
Periodic pattern biomass Spots in ZambiaStripes in NigerGaps in
Senegal Five basic states along the rainfall gradient: bare soil
uniform vegetation gap pattern spot pattern stripe pattern
Localized structures building blocks for extended patterns
Slide 7
Rather than a global shift to the alternative stable state,
local domains of the alternative state can form. Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud Desertification: what
pattern formation theory can tell us? The common view of
desertification: unproductive state productive state pfpf p pcpc
Subsequent dynamics transition-zone or front dynamics General
results for uniform states: Single fronts - propagate in general
productive unproductive space Does not capture an important aspect
- disturbances are likely to be local: Three aspects of front
dynamics: 1.Dynamics of a single front 2.Front interactions 3.Front
instabilities
Slide 8
Desertification: what pattern formation theory can tell us? Ben
Gurion University, Ehud Meron - www.bgu.ac.il/~ehud pfpf pmpm
productive unproductive p Desertification can be gradual and occur
before the tipping point! (Bel, Hagberg, Meron, Theor. Ecol. 2012)
Gradual process Front interactions Unproductive domains merged
Unproductive domains have merged because of attractive
interactions. Repulsive interactions can lead to asymptotic
patterns Time t Space Simulation of a n activator-inhibitor model
(FHN) with fast inhibitor diffusion Desertification can be
incipient asymptotic state still includes productive domains
Slide 9
Desertification: what pattern formation theory can tell us? Ben
Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Front
instabilities Hagberg & Meron PRL 1994; Chaos 1994;
Nonlinearity 1994; PRL 1997. Paja brava grass patterns in Bolivia
Back to single front dynamics but for bistability of uniform and
patterned states: Periodic pattern Uniform vegetation Bare
soil
Slide 10
Desertification: the Namibian fairy-circle ecosystem Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud Single fronts can be
stationary in a parameter range Pomeau, Physica D 1986; Knobloch,
Nonlinearity 2008 Concrete system: Namibian Fairy Circle (NFC)
ecosystem Fairy circles = gap patterns Tlidi, Lefever, Vladimirov
LNP 2008; Getzin K. Wiegand, T. Wiegand, Yizhaq, von Hardenberg
& Meron, Ecography 2015, Zelnik, Meron & Bel submitted
Sandy soil, confined root zones model equations simplify to: Cramer
and Barger PLoS ONE 2013; Juergens Science 2013 Soil-water content
in FC higher than in vegetation matrix
Slide 11
Desertification: the Namibian fairy-circle ecosystem 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) : Within the bistability range of
uniform vegetation and periodic pattern many more solution branches
of hybrid states: Homoclinic snaking (Edgar Knobloch) space 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. space
Slide 12
Desertification: the Namibian fairy-circle ecosystem Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud Any indications of
such processes in the NFC ecosystem? Birth of FC Tschinkel, PLoS
ONE 2012 Death of FC Instances of hybrid-state transitions This
suggests another form of gradual desertification in a fluctuating
environment - temporal escapes outside the snaking range where
fronts are not pinned Front propagation then leads to the creation
of additional gaps and to a cascade of hybrid state transitions to
lower-productivity states. Uniform vegetation Periodic pattern
(Gandhi, Knobloch & Beaume 2015).
Slide 13
Desertification: the Namibian fairy-circle ecosystem Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud Observations (Namibia)
Drought in 2007: 41mm/y vs. 100-300mm/y in other years Birth of FC
= front propagation outside snaking range ? Model simulations 2004
image as init. cond. within snaking range Drought outside the
snaking range Drought is over, back to snaking 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. Zelnik, Meron, Bel,
submitted
Slide 14
Reversing desertification - a spatial resonance problem 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. 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.
Slide 15
Restoration by water harvesting can fail Ben Gurion University,
Ehud Meron - www.bgu.ac.il/~ehud Restoration in the northern Negev
by the JNF-KKL Question: is this the best restoration practice? Are
there other practices more resilient to environmental
fluctuations?
Slide 16
Simplified model 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 Represents periodic crust removal to form
bands of higher infiltration rates a mimetic forcing, mimics what
the natural vegetation does anyway when exists.
Slide 17
Restoring in 2d resonant patterns Ben Gurion University, Ehud
Meron - www.bgu.ac.il/~ehud Is there a better alternative approach
to the 1:1 practice? 2:1 resonant 2d pattern restore 1:1 resonant
1d pattern Instead embankment vegetation Claim: The 2d resonant
patterns are more resilient to rainfall fluctuations Yes!
Slide 18
2d resonant solutions of the model equations Ben Gurion
University, Ehud Meron - www.bgu.ac.il/~ehud Manor et al. EPL 2008,
Mau et al. submitted 2:1 resonant solutions (biomass): (Mau,
Hagberg & Meron PRL 2012) Describe rhombic patterns that
consist of three resonating modes: which makes rhombic patterns
robust.
Slide 19
Stripe pattern Rhombic pattern p Bare soil Collapse of restored
1:1 stripes Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
In what sense rhombic patterns are more resilient? (Mau, Haim,
Meron, submitted) Responses of restored stripes to precipitation
downshifts can involve ecosystem collapse. Collapse to bare soil
Responses of restored rhombic patterns to precipitation uphifts
involve a smooth transition to stripe patterns.
Slide 20
Amplitude equations Ben Gurion University, Ehud Meron -
www.bgu.ac.il/~ehud 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. Use the amplitude equations to study
the mechanism of the collapse process. Indeed they give a similar
bifurcation diagram p Rhombic pattern Stripe pattern Bare soil
Slide 21
p Dynamics in phase space Ben Gurion University, Ehud Meron -
www.bgu.ac.il/~ehud Rhombic pattern - R Stripe pattern - S Bare
soil - B SS R R B The mechanism of collapse the disappearance of
unstable stripe solutions
Slide 22
Conclusion Ben Gurion University, Ehud Meron -
www.bgu.ac.il/~ehud 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. 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. Desertification is not necessarily
abrupt can occur gradually by front propagation. pfpf pmpm
productive unproductive p Bistability of uniform states productive
less productive p Bistability of uniform and patterned states
Repulsive front interactions incipient shifts Less is more: less
intervention, less areal coverage, more resilience.
Slide 23
Conclusion 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. 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. Introduction I Overview II Pattern
formation theory III Applications to Ecology Nonlinear Physics of
Ecosystems Ehud Meron