What can pattern formation theory tell us about desertification and restoration of degraded landscapes? Ehud Meron Department of Solar Energy & Environmental Physics and Physics Department Ben-Gurion University of the Negev Spatio-Temporal Dynamics in Ecology 8-12 December 2014, Leiden Desertification - an irreversible decrease in biological productivity induced by climatic variations and anthropogenic disturbances 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 Claim: The concepts and tools of pattern formation theory are crucial for understanding desertification and restoration Desertification in the northern Negev Collaborators Colleagues: Golan Bel (BGU) Aric Hagberg (LANL) Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Students: Yuval Zelnik Yair Mau (now postdoc at Duke U) Lev Haim (now at Soroka University Medical Center) Shai Kinast (Now at NCRN) Outline Desertification in spatially extended ecosystems: 1. Pattern formation aspects: local disturbances induce front dynamics gradual desertification 2. The Namibian fairy-circle ecosystem Reversing desertification by water harvesting: 1. A spatial resonance problem 2. Restoring in stripe vs. in rhombic patterns Conclusion Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Vegetation states of different productivity: 1. Feedbacks inducing pattern-forming instabilities 2. Mathematical modeling 3. Uniform and nonuniform states along the rainfall gradient Vegtetation states: pattern-forming feedbacks Three different transport mechanisms: + Local vegetation growth Water transport towards growing vegetation + Kinast, Zelnik, Bel, Meron, PRL 2014 Water uptake and conduction by laterally extended roots Water transport helps local vegetation growth but inhibits growth in the patch surroundings mechanism for pattern formation Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Vegetation pattern formation results from instabilities driven by positive feedbacks Vegtetation states: mathematical modeling Gilad, Hardenberg, Provenzale, Shachak, Meron PRL 2004, JTB 2007 = 1 − / − + 2 Biomass b bx, y, t = ℎ − − + 2 Soil-water content w wx, y, t ℎ = − ℎ − ⋅ = −2ℎ ℎ (ℎ + ) Surface-water height ℎ = ℎ(, , ) Three water (, ) = ∫transport , ′ , mechanisms: ′ , ′ ′ Overland flow (, ) = ∫ ′, , , ′ Conduction by roots ′ 2 − 1 ′ Soil-water diffusion , , = 2 − 2 0 (,) Root augmentation as plant grows = 0−1 /|=0 - root to shoot ratio ≈ 0 (1 + ) c= 1 no contrast (, ) + / (, ) = (, ) + Walker 1980; Rietkerk et al. AN 2002 h Infiltration contrast between vegetation patch and bare soil c>>1 I /c 0 high contrast b Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud A model that captures all three feedbacks (in dimensionless form): Vegtetation statess: basic vegetation states bare soil uniform spot stripe gap pattern pattern pattern vegetation Localized structures – building blocks for extended patterns Spots in Zambia biomass Five basic states along the rainfall gradient: Bare soil 1 Precipitation p Stripes in Niger Gaps in Senegal Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Model results Gilad, Hardenberg, Provenzale, Shachak, Meron PRL 2004, JTB 2007 Bifurcation diagram Desertification: what pattern formation theory can tell us? productive state Does not capture an important aspect disturbances are likely to be local: b Rather than a global shift to the alternative stable state, local domains of the alternative state can form. unproductive state Three aspects of front dynamics: 1. Dynamics of a single front 2. Front interactions 3. Front instabilities pf p pc General results for uniform states: Single fronts - propagate in general c productive 0 b unproductive space pm p Subsequent dynamics – transition-zone or front dynamics Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud The common view of desertification: Desertification: what pattern formation theory can tell us? p pm p pm < Unproductive domains merged productive Gradual process Space unproductive pf Time t p pm Simulation of a n activator-inhibitor model (FHN) with fast inhibitor diffusion Desertification can be incipient Desertification can be gradual and occur before the tipping point! – asymptotic state still includes (Bel, Hagberg, Meron, Theor. Ecol. 2012) productive domains Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Front interactions Unproductive domains have merged because of attractive interactions. Repulsive interactions can lead to asymptotic patterns Desertification: what pattern formation theory can tell us? Back to single front dynamics but for bistability of uniform and patterned states: Hagberg & Meron PRL 1994; Chaos 1994; Nonlinearity 1994; PRL 1997. Uniform vegetation Periodic pattern Bare soil Paja brava grass patterns in Bolivia Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Front instabilities Desertification: the Namibian fairy-circle ecosystem 0 Pomeau, Physica D 1986; Knobloch, Nonlinearity 2008 p Concrete system: Namibian Fairy Circle (NFC) ecosystem Sandy soil, confined root zones model equations simplify to: = 1 − / − + 2 = − − + 2 ≈ 1 + 2 , ≈ 1 + 2 Fairy circles = gap patterns Tlidi, Lefever, Vladimirov LNP 2008; Getzin K. Wiegand, T. Wiegand, Yizhaq, von Hardenberg & Meron, Ecography 2015, Zelnik, Meron & Bel submitted Soil-water content in FC higher than in vegetation matrix Cramer and Barger PLoS ONE 2013; Juergens Science 2013 Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud c Single fronts can be stationary in a parameter range Desertification: the Namibian fairy-circle ecosystem Within the bistability range of uniform vegetation and periodic pattern – many more solution branches of hybrid states: space Homoclinic snaking space (Edgar Knobloch) 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 2014. Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Choosing parameters that fit the NFC ecosystem we find the bifurcation diagram (Zelnik, Meron, Bel, submitted): Desertification: the Namibian fairy-circle ecosystem Periodic pattern Any indications of such processes in the NFC ecosystem? Birth of FC Death of FC Instances of hybrid-state transitions Tschinkel, PLoS ONE 2012 Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud This suggests another form of gradual desertification in a fluctuating environment - temporal escapes outside the snaking range where fronts are not pinned (Gandhi, Knobloch & Beaume 2015). Front propagation then leads to the creation of Uniform additional gaps and to a cascade of hybrid state vegetation transitions to lower-productivity states. Desertification: the Namibian fairy-circle ecosystem Zelnik, Meron, Bel, submitted Observations (Namibia) Birth of FC = front propagation outside snaking range ? Model simulations 2004 image as init. cond. within snaking range Drought outside Drought is over, the snaking back to snaking range 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. Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Drought in 2007: 41mm/y vs. 100-300mm/y in other years Reversing desertification - a spatial resonance problem 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. Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud The common approach: water harvesting by ground modulations, e.g. periodic stripe-like embankments that capture runoff and along which the vegetation is planted. Restoration by water harvesting can fail Question: is this the best restoration practice? Are there other practices more resilient to environmental fluctuations? Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Restoration in the northern Negev by the JNF-KKL Simplified model Approximate the root kernel by a delta function and set = 0: = 1 − / − + 2 = ℎ − − + 2 ℎ = − ℎ − ⋅ = −2ℎ ℎ (ℎ + ) Instead of modulating the topography function = (, ) to simulate embankments, modulate the infiltration rate. Represents periodic crust removal to form bands of higher infiltration rates – a mimetic forcing, mimics what the natural vegetation does anyway when exists. For simplicity we choose harmonic modulations and assume = constant = + 1+ [( )+1] 2 + Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Assume plant species with laterally confined roots and small root-to-shoot ratios pattern formation induced by the infiltration feedback Restoring in 2d resonant patterns Instead Yes! restore embankment vegetation 1:1 resonant 1d pattern 2:1 resonant 2d pattern The 2d patterns are 2:1 resonant in the sense that the pattern’s periodicity in the forcing direction is exactly twice the forcing periodicity, in a range of forcing periods. Claim: The 2d resonant patterns are more resilient to rainfall fluctuations Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Is there a better alternative approach to the 1:1 practice? 2d resonant solutions of the model equations Pattern’s periodicity in the forcing direction = 2 × forcing periodicity Exist in a wide forcing wavenumber range that overlaps with 1:1 resonant stripes. (Mau, Hagberg & Meron PRL 2012) Describe rhombic patterns that consist of three resonating modes: two oblique modes, exp ± ⋅ ± = ± , = and a stripe mode, exp − ⋅ 0 2 , |± | = 0 , 0 − selected pattern wavenumber in unmodulated system The modes satisfy the resonance condition: + +− − = 0, which makes rhombic patterns ≈ −0 + + + ⋅ + − − ⋅ + . . robust. Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud 2:1 resonant solutions (biomass): Collapse of restored 1:1 stripes Stripe pattern Rhombic pattern Collapse to bare soil Bare soil (Mau, Haim, Meron, submitted) p Responses of restored stripes to precipitation downshifts can involve ecosystem collapse. Responses of restored rhombic patterns to precipitation uphifts involve a smooth transition to stripe patterns. Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud In what sense rhombic patterns are more resilient? Amplitude equations kx kx 2 kx ( + ) ( − ) − : ≈ 0 + + + − + . . and deriving amplitude equations for , + , − . 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. 2 + − Rhombic pattern 2 Use the amplitude equations to study the mechanism of the collapse process. Stripe pattern + + 2 Indeed they give a similar bifurcation diagram Bare soil p Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Get a better understanding by studying the interaction between the stripe and oblique modes in the vicinity of the 1:1 resonance where ≈ 0 (Lev Haim, PHD thesis): Dynamics in phase space 2 + − 2 + + 2 The mechanism of collapse – Rhombic pattern - R = |+ | = − = Stripe pattern - S the disappearance of unstable stripe solutions Bare soil - B p R R S B S Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Project the amplitude dynamics onto the phase plane spanned by Conclusion Bistability of uniform states Repulsive front interactions incipient shifts productive productive unproductive pf p pm Bistability of uniform and patterned states less productive p 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. 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. Less is more: less intervention, less areal coverage, more resilience. Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud Desertification is not necessarily abrupt – can occur gradually by front propagation. Conclusion 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. Nonlinear Physics of Ecosystems Ehud Meron Introduction I Overview II Pattern formation theory III Applications to Ecology Ben Gurion University, Ehud Meron - www.bgu.ac.il/~ehud The concepts and tools of pattern formation theory can be crucial for the understanding of spatially extended ecosystems.