The nonlinear physics of dryland landscapes

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

2. Motivation 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: 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.

3. Motivation Added value of mathematical modeling: Laboratory and field experiments are limited by duration, spatial extent and by uncontrollable environmental factors. Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Mathematical models circumvent these limitations and allow: • Identifying asymptotic behaviors (rather than transients). • Isolating factors and elucidating mechanisms of ecological processes • Studying various scenarios of ecosystem dynamics • Proposing and testing management practices 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

4. Outline Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud • 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

5. Background: Vegetation patterns Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud A worldwide phenomenon observed in arid and semi-arid regions, 50–750 mm rainfall(Valentin et al. 1999) 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 Salt formation in the Atacama desert(Marcus Hauser)

6. Background:Biomass-water and below-aboveground feedbacks (2) Root augmentation (1) Infiltration Precipitation Precipitation Soil crusts reduce infiltration Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud infiltration Positive feedback Positive feedback Biomass Soil water Biomass Water uptake Quantified by a root augmentation parameter  Quantified by an infiltration contrast parameter c Water infiltration Root extension Both feedbacks can induce vegetation patterns because they involve water transport  help patch growth but inhibit growth in the patch surroundings

7. Population level: a spatially explicit model h  Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud c= 1 no contrast Root augmentation as plant grows ~ root to shoot ratio c>>1 high contrast Infiltration contrast between vegetation patch and bare soil I 0 b 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)] Biomass Soil-water content Surface-water height

8. Population level: Vegetation states along a rainfall gradient Constant slope: Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Same uniform states Pattern states: Spots, bands  slope Mechanism of migration: Precipitation Slope Downhill ~ 1 cm/yr infiltration Plane topography: Uniform states: Bare-soil state (b = 0) Fully vegetated state (b0) Plain topography Pattern states: Spots, stripes, gaps

9. Population level: Vegetation states along a rainfall gradient S Precipitation range where both bare-soil and spots are stable B Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud • Implications: • Stable localized structures (120) Squill חצב Mathematically – homoclinic snaking (Knobloch, Nonlinearity 2008). Equivalent to localized structures in nonlinear optics, fluid dynamics, etc. http://desert.bgu.ac.il Multistability of states: spot pattern Max(b) Bistability range for any other consecutive pair of states: spots & stripes, stripes & gaps, gaps & uniform vegetaqtion • Spatially mixed patterns (240, 360)

10. Population level: Vegetation states along a rainfall gradient Desertification - an irreversible decrease in biological productivity induced by a climate change. Desertification in the northern Negev Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud • State transitions S Spot pattern A dynamical-system view of desertification B 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, …

11. Population level: Observations of vegetation patterns Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Stripes of Paspalum vaginatum Spots Stripes Gaps

12. Population level: Observations of vegetation patterns Mixed gaps and stripes Spots Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Mixed spotsand stripes Rietkerk Barbier

13. Population level: Soil-water patterns Strong infiltration Weak augmentation Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Infiltration contrast Aridity stress 14m 3.5m 3.5m Strong augmentation Weak infiltration Effects of the biomass-water feedbacks: Facilitation Root augmentation (water uptake) C=10 C=1.1 Competition

14. Community level: a model for several functional groups b1 b2 b1 b2 Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Spots Uniform woody b1 b2 Uniform herbaceous b1 b2 # of functional groups (fg) Two functional groups: b1 - woody, b2 - herbaceous

15. Community level: Competition vs. facilitation Mechanism: Infiltration remains high, but uptake drops down because of smaller woody patch. Facilitation Competition Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud 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 Woody-herbaceous system: I Competition  facilitation 0 b Inter-specific interactions along a rainfall gradient: Woody species alone: Ameliorates its micro-environment as aridity increases. Woody patches can buffer species diversity loss as aridity increases

16. Community level: Competition vs. facilitation Downhill b1 Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud b2 Mechanism: spots “see” bare areas uphill twice as long as bands and infiltrate more runoff. Woody-herbaceous Inter-specific interactions and pattern transitions: Woody alone 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.

17. Large communities Species A Species B physical subspace trait subspace Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud A simple system: • A single functional group with one-dimensional trait space  Big plants, short roots Small plants, long roots • Homogeneous system (no spatial patterns) Current form of model cannot provide information about species assemblage properties such as species diversity. 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.

18. Deriving community-level properties Width  Richness Height  Abundance Position  Composition Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Pulse solutions: provide information on species assemblage properties: B ξ Big plants, short roots Small plants, long roots

19. Species diversity along a rainfall gradient Precipitation rate Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Derive diversity-resource relations Richness (herbs) In the presence of Woody patches Herbs only Can woody patches buffer species diversity loss? Precipitation Stationary pulse solutions at increasing precipitation rates: As precipitation rate increases: • Species diversity (width) increases • Abundance (height) increases • Average composition moves to lower ξ values, i.e. to species investing more in above-ground biomass and less in roots.

20. Conclusion 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 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!

21. Acknowledgement Efrat Sheffer Moshe shachak Moran Segoli Antonello Provenzale Assaf Kletter Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Jonathan Nathan Hezi Yizhaq Jost von Hardenberg Erez Gilad

22. Funded by: Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Israel Science Foundation James S. McDonnel Foundation (Complex Systems program) The Center for Complexity Science Israel Ministry of Science (Eshcol program) References • J. Von Hardenberg, E. Meron, M. Shachak, Y. Zarmi, “Diversity of Vegetation Patterns and Desertification” Phys. Rev. Lett.89, 198101 (2001). • E. Meron, E. Gilad, J. Von Hardenberg, M. Shachak, Y. Zarmi, “Vegetation Patterns Along a Rainfall Gradient”, Chaos Solitons and Fractals 19, 367 (2004). • 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). • H. Yizhaq, E. Gilad, E. Meron, “Banded vegetation: Biological Productivity and Resilience”, Physica A356, 139 (2005). • 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.

23. References Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud • 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). • E. Gilad, M. Shachak, E. Meron, “Dynamics and spatial organization of plant communities in water limited systems” , Theo. Pop. Biol. 72, 214-230 (2007). • 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. • 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). • E. Meron, H. Yizhaq and E. Gilad E., “Localized structures in dryland vegetation: forms and functions”, Chaos17, 037109 (2007) • Kletter A., von Hardenberg J., Meron E., Provenzale A., "Patterned vegetation and rainfall intermittency", J. Theoretical Biology 2008. • 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).

25. Biological soil crusts Soil crust Areal photographs Egypt-Israel border Karnieli Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud

26. Desertification induced by drought Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Remains of a spot pattern of Noaea mucronata in the northern Negev Moshe Shachak (2009)