The nonlinear physics of dryland landscapes Ehud Meron Institute for Dryland Environmental Research & Physics Department Ben-Gurion University Squill חצב Cistanche tubulosa יחנוק Seashore Paspalum Physics Colloquium, Toronto, March 5, 2009

Motivation Ben Gurion University, Ehud Meron - 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 processes 3.Studying various scenarios of ecosystem dynamics 4.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 Gurion University, Ehud Meron - 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 Gurion University, Ehud Meron 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 species belonging to different functional groups – the woody-herbaceous system. 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 Gurion University, Ehud Meron - Salt formation in the Atacama desert (Marcus Hauser)

Precipitation infiltration (1) Infiltration Soil crusts reduce infiltration Biomass Water infiltration Soil water Soil water Positive feedback (2) Root augmentation Biomass Root extension Root extension Water uptake Water 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 c Quantified by a root augmentation parameter  Background : Biomass-water and below-aboveground feedbacks Ben Gurion University, Ehud Meron -

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 Infiltration contrast between vegetation patch and bare soil I 0 b c= 1 no contrast c>>1 high contrast Population level: a spatially explicit model Ben Gurion University, Ehud Meron -  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 Ben Gurion University, Ehud Meron - Uniform states: Bare-soil state (b = 0) Fully vegetated state (b  0) Pattern states: Spots, stripes, gaps Plane topography: Slope Constant slope: Same uniform states Pattern states: Spots, bands  slope Downhill

Ben Gurion University, Ehud Meron - Population level: Vegetation states along a rainfall gradient spot pattern Max(b) B S 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 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

Ben Gurion University, Ehud Meron - A dynamical-system view of desertification S B Spot pattern Population level: Vegetation states along a rainfall gradient 3.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 by erosion, active sand dunes, the human factor, … Desertification - an irreversible decrease in biological productivity induced by a climate change. Desertification in the northern Negev

SpotsStripesGaps Stripes of Paspalum vaginatum Population level: Observations of vegetation patterns Ben Gurion University, Ehud Meron -

Mixed gaps and stripes Spots Mixed spots and stripes Rietkerk Ben Gurion University, Ehud Meron - Population level: Observations of vegetation patterns Barbier

Population level: Soil-water patterns Strong infiltration Weak augmentation Strong augmentation Weak infiltration 14m3.5m Effects of the biomass-water feedbacks: Infiltration contrast Root augmentation (water uptake) Ben Gurion University, Ehud Meron - C=10 C=1.1 Facilitation Competition Aridity stress

# of functional groups (fg) Community level: a model for several functional groups Ben Gurion University, Ehud Meron - Two functional groups: b 1 - woody, b 2 - herbaceous Uniform woody Uniform herbaceous Spots b 1 b 2

Inter-specific interactions along a rainfall gradient: Community level: Competition vs. facilitation Ben Gurion University, Ehud Meron - Mechanism: Infiltration remains high, but uptake drops down because of smaller woody patch. I 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. b1b1 b2b2 Woody- herbaceous Community level: Competition vs. facilitation Ben Gurion University, Ehud Meron - Inter-specific interactions and pattern transitions:

Current form of model cannot provide information about species assemblage properties such as species diversity. Large communities Ben Gurion University, Ehud Meron - Extend the space over which biomass variables are defined to include a trait subspace and use this trait space to distinguish among different species within a functional group. physical subspacetrait subspace 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  RichnessHeight  AbundancePosition  Composition ξ B Small plants, long rootsBig plants, short roots Deriving community-level properties Ben Gurion University, Ehud Meron -

Stationary pulse solutions at increasing precipitation rates: Species diversity along a rainfall gradient Ben Gurion University, Ehud Meron - 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 Richness (herbs) Precipitation Herbs only In the presence of Woody patches Can woody patches buffer species diversity loss?

Conclusion Ben Gurion University, Ehud Meron - 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. 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 Gurion University, Ehud Meron -

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 Gurion University, Ehud Meron J. Von Hardenberg, E. Meron, M. Shachak, Y. Zarmi, “Diversity of Vegetation Patterns and Desertification” Phys. Rev. Lett. 89, (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, (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, (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 E. Sheffer E., Yizhaq H., Gilad E., Shachak M. and & Meron E., “Why do plants in resource deprived environments form rings?” Ecological Complexity 4, (2007). 10.E. Meron, H. Yizhaq and E. Gilad E., “Localized structures in dryland vegetation: forms and functions”, Chaos 17, (2007) 11.Kletter A., von Hardenberg J., Meron E., Provenzale A., "Patterned vegetation and rainfall intermittency", J. Theoretical Biology 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, (2008). References Ben Gurion University, Ehud Meron -

Biological soil crusts Karnieli Soil crust Areal photographs Egypt-Israel border Ben Gurion University, Ehud Meron -

Desertification induced by drought Remains of a spot pattern of Noaea mucronata in the northern Negev Moshe Shachak (2009)