The nonlinear physics of dryland landscapes
Ehud Meron
Institute for Dryland Environmental Research & Physics Department
Ben-Gurion University
Physics Colloquium, Toronto, March 5, 2009
Cistanche tubulosa
‫יחנוק‬
Seashore Paspalum
Squill
‫חצב‬
Motivation
Focusing on the direct response of any individual to the changing
climatic conditions is insufficient because of indirect processes at
the population and community levels that affect species diversity:
Climate change  vegetation patterns  resource distributions,
seed dispersal, consumer pressure  species diversity
Climate change  inter-specific plant interactions  transitions
from competition to facilitation  species diversity
More generally, environmental changes affect species assemblage
properties by inducing indirect processes involving various levels of
organization often across different spatial scales.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Innocent questions such as
how climate changes affect species diversity (bears on ecosystem
function and stability) ?
are quite complex:
Motivation
Added value of mathematical modeling:
Laboratory and field experiments are limited by duration, spatial
extent and by uncontrollable environmental factors.
Mathematical models circumvent these limitations and allow:
1. Identifying asymptotic behaviors (rather than transients).
2. Isolating factors and elucidating mechanisms of ecological
processes
3. Studying various scenarios of ecosystem dynamics
4. Proposing and testing management practices
Mathematical modeling has its own limitations:
Models simplify the complex reality, quite often oversimplify it
The challenge is to propose simple models that not only reproduce
observed behaviors but also have predictive power –
usually requires identifying and modeling basic feedbacks
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Study processes of this kind by mathematical modeling as a
complementary tool to field and laboratory experiments.
Outline
2.
3.
4.
5.
Background:
Vegetation patterns, feedbacks between biomass and water, and
between above-ground below-ground biomass.
Population level:
Introduction of a spatially explicit model for a plant population,
applying it to vegetation pattern formation along a rainfall
gradient and to desertification.
Two-species communities:
Extending the model to two populations representing species
belonging to different functional groups – the woody-herbaceous
system. Using it to study mechanisms affecting species diversity
(not yet community level properties).
Many-species communities:
Extending the model to include trait-space and use it to derive
community level properties such as species diversity along a
rainfall gradient.
Conclusion
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
1.
Background: Vegetation patterns
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Aerial photograph of
vegetation bands in Niger of
‘tiger bush’ patterns on hill
slopes (Clos-Arceduc, 1956)
Recent studies:
Catena Vol. 37, 1999
Valentin et al. Catena 1999,
Rietkerk et al. Science 2004
A worldwide phenomenon
observed in arid and semi-arid
regions, 50–750 mm rainfall
(Valentin et al. 1999)
Salt formation in the Atacama desert (Marcus Hauser)
Background: Biomass-water and below-aboveground feedbacks
(2) Root augmentation
Precipitation
Precipitation
Soil crusts reduce
infiltration
infiltration
Positive feedback
Biomass
Soil
water
Positive feedback
Biomass
Water
uptake
Water
infiltration
Root
extension
Quantified by an infiltration
contrast parameter c
Quantified by a root
augmentation parameter

Both feedbacks can induce vegetation patterns because they involve water
transport  help patch growth but inhibit growth in the patch surroundings
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
(1) Infiltration
Population level: a spatially explicit model
models: Lefever & Lejeune (1997); Klausmeier, (1999);
HilleRisLambers et al. (2000), Okayasu & Aizawa (2001); Von Hardenberg et al. (2001); Rietkerk et al. (2002);
Lejeune et al. (2002); Shnerb et al. (2003)]
b
 Gb b(1  b)  b   b  2b
Biomass
t
w
 Ih  Lw  wGw   w 2 w
Soil-water content
t
h
 p  Ih   h  2 h 2  2 h h    2 h h 2
t

 


Gb (r , t )    g (r , r ' , t )w(r ' , t )dr '

  2


r  r
 
1


g (r , r , t ) 
exp 
 2
2

2
1


b
(
r
, t ) 



Root augmentation as plant grows
~ root to shoot ratio  ~ dL db


b( r , t )  q / c
I (r , t )  

b( r , t )  q

Surface-water height

 


Gw (r , t )    g (r ' , r , t )b(r ' , t )dr '

c= 1 no contrast
L ~ 1  b
Infiltration contrast between
vegetation patch and bare soil
h

c>>1
I
 /c
0
high contrast
b
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Gilad et al. (PRL 2004, JTB 2007) [Earlier
Population level: Vegetation states along a rainfall gradient
Uniform states:
Bare-soil state (b = 0)
Fully vegetated state (b  0)
Plain topography
Pattern states: Spots, stripes, gaps
Constant slope:
Same uniform states
Pattern states: Spots, bands  slope
Mechanism of migration:
Precipitation
Downhill
Slope
~ 1 cm/yr
infiltration
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Plane topography:
Population level: Vegetation states along a rainfall gradient
Precipitation range where both
bare-soil and spots are stable
Bistability range for any other
consecutive pair of states:
spots & stripes, stripes & gaps,
gaps & uniform vegetaqtion
Implications:
1. Stable localized
structures (120)
S
spot pattern Max(b)
B
Squill
‫חצב‬
Mathematically – homoclinic snaking
(Knobloch, Nonlinearity 2008).
Equivalent to localized structures in
nonlinear optics, fluid dynamics, etc.
2. Spatially mixed patterns
(240, 360)
http://desert.bgu.ac.il
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Multistability of states:
Population level: Vegetation states along a rainfall gradient
A dynamical-system view of
desertification
Desertification - an irreversible
decrease in biological productivity
induced by a climate change.
S
Spot pattern
B
Desertification in the
northern Negev
The positive feedbacks that induce vegetation patterns are also
responsible for the bistability range of bare soil and spots:
The stronger the feedbacks the wider the bistability range and the
less vulnerable to desertification the system is.
Many more causes and forms of desertification: gully formation by
erosion, active sand dunes, the human factor, …
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
3. State transitions
Population level: Observations of vegetation patterns
Spots
Stripes
Gaps
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Stripes of Paspalum vaginatum
Population level: Observations of vegetation patterns
Mixed spots and stripes
Rietkerk
Barbier
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Mixed gaps and stripes
Spots
Population level: Soil-water patterns
Effects of the biomass-water feedbacks:
C=10
C=1.1
Infiltration contrast
Aridity
stress
14m
Strong augmentation
Competition
Weak infiltration
3.5m
3.5m
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Root augmentation (water uptake)
Strong infiltration
Facilitation
Weak augmentation
Community level: a model for several functional groups
Two functional groups:
b1 - woody, b2 - herbaceous
b1
Spots
b1
b2
Uniform
woody
b2
b1
b1
b2
Uniform
herbaceous
b2
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
bi
 Gbi bi (1  bi )  i bi   bi  2b
i  1,..., n
t
w
n
 Ih  Lw  wi 1 Gwi   w 2 w
# of functional groups (fg)
t
h
 p  Ih   h  2 h 2  2 h h    2 h h 2
t
Community level: Competition vs. facilitation
Woody species alone:
Ameliorates its microenvironment as aridity
increases.
Mechanism:
Infiltration remains high,
but uptake drops down
Consistent
field
observations
because of with
smaller
woody
of
annual plant–shrub interactions
patch.
along anaridity gradient:
Holzapfel,
I Tielbörger, Parag, Kigel,
Sternberg, 2006
 /c
Facilitation in
0 stressed environments:
b
Pugnaire & Luque, Oikos 2001,
Callaway and Walker 1997
Woody-herbaceous
system:
Bruno et al. TREE 2003
Competition
 facilitation
Woody
patches
can buffer species
diversity loss as aridity increases
Facilitation
Competition
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Inter-specific interactions along a rainfall gradient:
Community level: Competition vs. facilitation
Woody alone
Clear cutting on a slope in a bistability
range of spots and bands:
b1
b2
Mechanism: spots “see” bare areas uphill twice
as long as bands and infiltrate more runoff.
Woodyherbaceous
Species coexistence and diversity are affected by global pattern
transitions. Coexistence appears as a result of bands  spots
transition.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Inter-specific interactions and pattern
transitions:
Downhill
Large communities
Extend the space over which biomass
variables are defined to include a
trait subspace



B  B x,  , t
physical subspace

1
Species A
trait subspace
Species B
2
and use this trait space to distinguish among
different species within a functional group.
A simple system:
1.
A single functional group with one-dimensional trait space 
Big plants, short roots
Small plants, long roots
2. Homogeneous system (no spatial patterns)

Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Current form of model cannot provide information about species
assemblage properties such as species diversity.
Deriving community-level properties
Width  Richness
Height  Abundance
Position  Composition
B
Big plants, short roots
Small plants, long roots
ξ
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Pulse solutions: provide information on species assemblage properties:
Species diversity along a rainfall gradient
Precipitation rate
As precipitation rate increases:
1. Species diversity (width) increases
2. Abundance (height) increases
3. Average composition moves to lower ξ values, i.e. to species
investing more in above-ground biomass and less in roots.
Derive diversity-resource relations
Can woody patches buffer species
diversity loss?
Richness
(herbs)
In the presence of
Woody patches
Herbs only
Precipitation
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Stationary pulse solutions at increasing precipitation rates:
Conclusion
Bottom-up processes:
plant interactions  vegetation pattern formation
Top-down processes
pattern transitions  plant interactions
Various aspects of this complexity can be addressed using a single
platform of nonlinear mathematical models that capture basic
feedbacks between biomass and water and between above-ground
and below-ground biomass.
Theoretical results are consistent with many field observations,
but controlled experiments are needed!
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Eco-physical phenomena involve various levels of organization,
different time scales and different spatial scales. This results in
many indirect processes that bear on the questions that we ask,
including
Acknowledgement
Segoli
Antonello
Provenzale
Moshe
Jonathan
shachak
Nathan
Sheffer
Hezi
Assaf
Yizhaq
Kletter
Jost von
Hardenberg
Erez
Gilad
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Moran
Efrat
Funded by:
The Center for Complexity Science
Israel Ministry of Science (Eshcol program)
References
1.
J. Von Hardenberg, E. Meron, M. Shachak, Y. Zarmi, “Diversity of Vegetation
Patterns and Desertification” Phys. Rev. Lett. 89, 198101 (2001).
2.
E. Meron, E. Gilad, J. Von Hardenberg, M. Shachak, Y. Zarmi, “Vegetation Patterns
Along a Rainfall Gradient”, Chaos Solitons and Fractals 19, 367 (2004).
3.
E. Gilad, J. Von Hardenberg, A. Provenzale, M. Shachak, E. Meron, “Ecosystem
Engineers: From Pattern Formation to Habitat Creation”, Phys. Rev. Lett. 93,
098105 (2004).
4.
H. Yizhaq, E. Gilad, E. Meron, “Banded vegetation: Biological Productivity and
Resilience”, Physica A 356, 139 (2005).
5.
E. Meron & E. Gilad, “Dynamics of plant communities in drylands: A pattern
formation approach”, in Complex Population Dynamics: Nonlinear Modeling in
Ecology, Epidemiology and Genetics, B. Blasius, J. Kurths, and L. Stone, Eds. ,
World-Scientific, 2007.
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Israel Science Foundation
James S. McDonnel Foundation (Complex Systems program)
References
E. Gilad, J. Von Hardenberg, A. Provenzale, M. Shachak, E. Meron, “A
mathematical Model for Plants as Ecosystem Engineers”, J. Theor. Biol. 244, 680
(2007).
7.
E. Gilad, M. Shachak, E. Meron, “Dynamics and spatial organization of plant
communities in water limited systems” , Theo. Pop. Biol. 72, 214-230 (2007).
8.
E. Meron, E. Gilad, J. Von Hardenberg, A. Provenzale, M. Shachak, “Model studies
of Ecosystem Engineering in Plant Communities”, in Ecosystem Engineers: Plants
to Protists , Eds: K. Cuddington et al., Academic Press 2007.
9.
E. Sheffer E., Yizhaq H., Gilad E., Shachak M. and & Meron E., “Why do plants in
resource deprived environments form rings?” Ecological Complexity 4, 192-200
(2007).
10. E. Meron, H. Yizhaq and E. Gilad E., “Localized structures in dryland vegetation:
forms and functions”, Chaos 17, 037109 (2007)
11. Kletter A., von Hardenberg J., Meron E., Provenzale A., "Patterned vegetation and
rainfall intermittency", J. Theoretical Biology 2008.
12. Shachak M., Boeken B., Groner E., Kadmon R., Lubin Y., Meron E., Neeman G.,
Perevolotsky A., Shkedy Y. and Ungar E., " Woody Species as Landscape
Modulators and their Effect on Biodiversity Patterns", BioScience 58, 209-221
(2008).
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
6.
Biological soil crusts
Soil crust
Karnieli
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Areal photographs
Egypt-Israel border
Desertification induced by drought
Moshe Shachak (2009)
Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud
Remains of a spot pattern of Noaea mucronata in the northern Negev