The nonlinear physics of dryland landscapes

Ehud Meron Institute for Dryland Environmental Research & Physics Department

Ben-Gurion University

Squill חצב

Cistanche tubulosaיחנוק Seashore Paspalum

Physics Colloquium, Toronto, March 5, 2009

Motivation Ben G

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Innocent questions such as

how climate changes affect species diversity (bears on ecosystem function and stability) ?are quite complex:

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.

Mathematical models circumvent these limitations and allow:

1. Identifying asymptotic behaviors (rather than transients).2. Isolating factors and elucidating mechanisms of ecological

processes3. Studying various scenarios of ecosystem dynamics4. Proposing and testing management practices

Laboratory and field experiments are limited by duration, spatial extent and by uncontrollable environmental factors.

Added value of mathematical modeling:

Motivation Ben G

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Study processes of this kind by mathematical modeling as a complementary tool to field and laboratory experiments.

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

Outline Ben G

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1. Background:Vegetation patterns, feedbacks between biomass and water, and between above-ground below-ground biomass.

2. 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.

3. Two-species communities: Extending the model to two populations representing speciesbelonging to different functional groups – the woody-herbaceoussystem. Using it to study mechanisms affecting species diversity (not yet community level properties).

4. 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.

5. Conclusion

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)

Background: Vegetation patterns Ben G

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Salt formation in the Atacama desert (Marcus Hauser)

Precipitation

infiltration

(1) Infiltration

Soil crusts reduce infiltration

BiomassBiomass

Water infiltration Water infiltration

Soil waterSoil water

Positive feedback

(2) Root augmentation

BiomassBiomass

Rootextension Rootextension

Water uptakeWater uptake

Positive feedback

Precipitation

Both feedbacks can induce vegetation patterns because they involve water

transport help patch growth but inhibit growth in the patch surroundings

Quantified by an infiltration

contrast parameter cQuantified by a root

augmentation parameter

Background: Biomass-water and below-aboveground feedbacks B

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222

2

2

22

)1(

hhhIhpt

h

wwGLwIht

w

bbbbGt

b

hhh

ww

bb

Gilad et al. (PRL 2004, JTB 2007) [Earlier 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)]

Root augmentation as plant grows ~ root to shoot ratio dbdL~

2

2

),(12exp

2

1),,(

trb

rrtrrg

'),'(),,'(),('),'(),',(),( rdtrbtrrgtrGrdtrwtrrgtrG wb

bL 1~

qtrb

cqtrbtrI

),(

/),(),(

Infiltration contrast between

vegetation patch and bare soil

I

c/0 b

c= 1 no contrast

c>>1 high contrast

Population level: a spatially explicit model Ben G

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h

Surface-water height

Soil-water content

Biomass

Plain topography

infiltration

Precipitation

Mechanism of migration:

~ 1 cm/yr

Population level: Vegetation states along a rainfall gradient B

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Uniform states:Bare-soil state (b = 0) Fully vegetated state (b 0)

Pattern states: Spots, stripes, gaps

Plane topography:

Slope

Constant slope:

Same uniform statesPattern states: Spots, bands

slope

Dow

nhill

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Population level: Vegetation states along a rainfall gradient

spot pattern Max(b)

B

SPrecipitation range where

bothbare-soil and spots are stableBistability range for any otherconsecutive pair of states: spots & stripes, stripes & gaps, gaps & uniform vegetaqtion

Multistability of states:

2. Spatially mixed patterns (240, 360)

Mathematically – homoclinic snaking(Knobloch, Nonlinearity 2008).

Equivalent to localized structures in nonlinear optics, fluid dynamics, etc.

Implications:1. Stable localized

structures (120)Squill חצב

http://desert.bgu.ac.il

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A dynamical-system view of desertification

S

B

Spot pattern

Population level: Vegetation states along a rainfall gradient3. State transitions

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

byerosion, active sand dunes, the human factor, …

Desertification - an irreversible decrease in biological productivity induced by a climate change.

Desertification in the northern Negev

Spots Stripes

Gaps

Stripes of Paspalum vaginatum

Population level: Observations of vegetation patterns Ben G

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Mixed gaps and stripes

Spots

Mixed spots and stripes

Rietkerk

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Population level: Observations of vegetation patterns

Barbier

Population level: Soil-water patterns

Strong infiltration Weak augmentation

Strong augmentation Weak infiltration

14m 3.5m 3.5m

Effects of the biomass-water feedbacks:

Infilt

rati

on c

ontr

ast

Root augmentation (water uptake)

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C=

10C

=1.

1

Facilitation

Competition

Aridity

stress

222

2

1

2

22

,...,1)1(

hhhIhpt

h

wGwLwIht

w

nibbbbGt

b

hhh

w

n

i

iw

biiiiib

ii

# of functional groups (fg)

Community level: a model for several functional groups B

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Two functional groups: b1 - woody, b2 - herbaceous

Uniform woody

Uniform herbaceous

Spots

b1 b2

b1 b2

b1 b2

b1 b2

Inter-specific interactions along a rainfall gradient:

Community level: Competition vs. facilitation Ben G

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Mechanism:

Infiltration remains high, but uptake drops down because of smaller woody patch.

Ic/0 b

Competition facilitation

Woody-herbaceous system:

Consistent with field observations of annual plant–shrub interactions along an aridity gradient: Holzapfel, Tielbörger, Parag, Kigel, Sternberg, 2006

Facilitation in stressed environments: Pugnaire & Luque, Oikos 2001, Callaway and Walker 1997 Bruno et al. TREE 2003

CompetitionFacilitation

Woody species alone:Ameliorates its micro-environment as aridity increases.

Woody patches can buffer species diversity loss as aridity increases

Woody alone

Downhill

Clear cutting on a slope in a bistability range of spots and bands:

Species coexistence and diversity are affected by global pattern transitions. Coexistence appears as a result of bands spots transition.

Mechanism: spots “see” bare areas uphill twice as long as bands and infiltrate more runoff.

b1

b2

Woody-herbaceous

Community level: Competition vs. facilitation Ben G

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Inter-specific interactions and pattern transitions:

Current form of model cannot provide information about species

assemblage properties such as species diversity.

Large communities Ben G

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Extend the space over which biomass

variables are defined to include a trait subspace

and use this trait space to distinguish amongdifferent species within a functional group.

txBB ,,

physical subspace

trait subspace

1

2

Species A

Species B

1. A single functional group with one-dimensional trait space A simple system:

2. Homogeneous system (no spatial patterns)

Small plants, long rootsBig plants, short roots

Pulse solutions: provide information on species assemblage properties:Width Richness Height Abundance Position

Composition

ξ

B

Small plants, long rootsBig plants, short roots

Deriving community-level properties Ben G

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Stationary pulse solutions at increasing precipitation rates:

Species diversity along a rainfall gradient Ben G

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Precipitation rate

As precipitation rate increases:

1. Species diversity (width) increases2. Abundance (height) increases3. 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

Richness

(herbs)

Precipitation

Herbs onlyIn the presence of Woody patches

Can woody patches buffer species

diversity loss?

Conclusion Ben G

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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-

groundand below-ground biomass.

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

Bottom-up processes:

plant interactions vegetation pattern formation

Top-down processes

pattern transitions plant interactions

Theoretical results are consistent with many field observations,

but controlled experiments are needed!

Jost von

Hardenberg

Hezi

Yizhaq

Jonathan

Nathan

Assaf

Kletter

Moshe

shachak

Erez

Gilad

Moran

Segoli Antonello

Provenzale

Efrat

Sheffer

Acknowledgement Ben G

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References

Israel Science Foundation

James S. McDonnel Foundation (Complex Systems program)

The Center for Complexity Science

Israel Ministry of Science (Eshcol program)

Funded by: Ben G

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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.

6. 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).

References Ben G

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Biological soil crusts

Karnieli

Soil crust

Areal photographs Egypt-Israel border

Ben G

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Ben G

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Desertification induced by drought

Remains of a spot pattern of Noaea mucronata in the northern Negev

Moshe Shachak (2009)