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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

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Ben Gurion University, Ehud Meron - 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)

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Outline Ben Gurion University, Ehud Meron - 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

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Vegtetation states: pattern-forming feedbacks Ben Gurion University, Ehud Meron - 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

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Vegtetation states: mathematical modeling Ben Gurion University, Ehud Meron - 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

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Vegtetation statess: basic vegetation states Ben Gurion University, Ehud Meron - 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

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Rather than a global shift to the alternative stable state, local domains of the alternative state can form. Ben Gurion University, Ehud Meron - 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

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Desertification: what pattern formation theory can tell us? Ben Gurion University, Ehud Meron - 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

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Desertification: what pattern formation theory can tell us? Ben Gurion University, Ehud Meron - Front instabilities Hagberg & Meron PRL 1994; Chaos 1994; Nonlinearity 1994; PRL 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

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Desertification: the Namibian fairy-circle ecosystem Ben Gurion University, Ehud Meron - 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

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Desertification: the Namibian fairy-circle ecosystem Ben Gurion University, Ehud Meron - 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 space

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Desertification: the Namibian fairy-circle ecosystem Ben Gurion University, Ehud Meron - 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).

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Desertification: the Namibian fairy-circle ecosystem Ben Gurion University, Ehud Meron - Observations (Namibia) Drought in 2007: 41mm/y vs mm/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

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Reversing desertification - a spatial resonance problem Ben Gurion University, Ehud Meron - 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.

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Restoration by water harvesting can fail Ben Gurion University, Ehud Meron - 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?

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Simplified model Ben Gurion University, Ehud Meron - 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.

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Restoring in 2d resonant patterns Ben Gurion University, Ehud Meron - 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!

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2d resonant solutions of the model equations Ben Gurion University, Ehud Meron - 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.

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Stripe pattern Rhombic pattern p Bare soil Collapse of restored 1:1 stripes Ben Gurion University, Ehud Meron - 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.

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Amplitude equations Ben Gurion University, Ehud Meron - 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

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p Dynamics in phase space Ben Gurion University, Ehud Meron - Rhombic pattern - R Stripe pattern - S Bare soil - B SS R R B The mechanism of collapse – the disappearance of unstable stripe solutions

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Conclusion Ben Gurion University, Ehud Meron - 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.

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Conclusion Ben Gurion University, Ehud Meron - 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

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