Ehud Meron Blaustein Institutes for Desert Research

& Physics Department

German Physical Society meeting, Regensburg 2016

From patterns to function in living systems: dryland ecosystems as a case study

Ben-Gurion University of the Negev

With students:

Yuval Zelnik, Omer Tzuk, Yair Mau, Lev Haim

and colleagues:

Hezi Yizhaq, Jost von Hardenberg, Golan Bel, Aric Hagberg, Moshe Shachak, Stephan Getzin

Ben G

urion Unive

rsity, Ehud

Meron - w

ww

.bgu.ac.il/~

ehud

New qualities may appear when the number of system’s constituents (molecules, grains, cells, organisms) becomes large, e.g. phase transitions,

In ecosystems too:

or symmetry-breaking spatial patterns, which may appear in any nonlinear spatially extended system

Liu et al. Nature Communication 2014

Deblauwe et al., Global Ecol. Biogeogr. 2008

How does the abiotic environment affect ecosystem function?

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urion Unive

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Meron - w

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Patterns are intriguing, but in living systems they also have functional values – quite often they are means to cope with resource stresses.

Community level

Biodiversity change

Landscape level Pattern formation

(spatial self-organization)

Ecosystem response

Abiotic environment

Ecosystem function

Organism level

Phenotype change

Complex response at several organization levels – how can we study it?

How does the abiotic environment affect ecosystem function?

Ben G

urion Unive

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Meron - w

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ehud

Back to Anderson’s paper: fundamental laws should be sought at any level of complexity. The elementary entities of science X obey the laws of science Y of the next lower complexity level:

Science X = landscape ecology

Science Y = organism physiology, soil physics, ecohydrology

2. How can we scale up the information contained in these principles to information at the landscape level?

1. What fundamental principles at the level of a single organism determine emergent phenomena at the landscape level, such as vegetation pattern formation?

Two questions:

Address these questions in the context of dryland ecosystems.

Dryland landscapes

Ben G

urion Unive

rsity, Ehud

Meron - w

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ehud

Drylands (water limited systems) occupy 2/5 of the terrestrial earth surface. Home to over 1/3 of the world population → understanding drylands is important

Outstanding problems can be addressed: desertification and biodiversity loss.

Show a wide variety of pattern formation phenomena

Outline

1. Fundamental principles: pattern-forming biomass-water feedbacks at the single plant scale

2. Scaling up to landscape scale by mathematical modeling: a platform of mathematical models

3. Emergence of landscape-scale patterns and morphological pattern changes along the rainfall gradient

4. Ecosystem-function aspects of vegetation pattern formation: desertification and restoration.

Several transport mechanisms:

Water uptake and conduction by laterally

extended roots

Local vegetation

growth

Water transport towards growing

vegetation

+

+

Large-scale patterns often induced by instabilities of uniform states driven by positive feedbacks

Model these small-scale feedbacks and study the emergence of large-scale patterns.

Fundamental principles: pattern-forming feedbacks

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urion Unive

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The water transport has 2 effects:

1. Provides an extra source of water to vegetation patches 2. Inhibits vegetation growth in the patch surroundings Allows survival at drier conditions but in a form of a patchy landscape

Ben G

urion Unive

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Meron - w

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Scaling up by mathematical modeling: discrete or continuum?

1. A single plant can change its biomass continuously by orders of magnitude (demographic noise is not significant)

Describe a plant population by a continuous biomass areal density, irrespective of how many individuals contribute to it.

2. Plants disperse long-lived seeds that can germinate once the conditions are favorable again (no extinction or absorbing state)

Continuum PDE models are advantageous over discrete agent-based or CA models in lending themselves to the powerful tools of pattern formation theory.

Is the PDE approach appropriate for describing dryland plant populations which are often small?

Yes, for two reasons:

Smallest entity is not a plant but rather a small area element and the processes occurring there

Scaling up by mathematical modeling: model equations

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Infiltration contrast between vegetation patch and bare soil

I

c/0

b

c= 1 no contrast

c>>1 high contrast

Walker 1980

𝐼(𝒙, 𝑡) = 𝛼𝑏(𝒙, 𝑡) + 𝑞/𝑐

𝑏(𝒙, 𝑡) + 𝑞

𝐺𝑏 = 𝜈∫ 𝑔 𝒙, 𝒙′, 𝑡 𝑤 𝒙′, 𝑡 𝑑𝒙′

𝐺𝑤 = 𝛾∫ 𝑔 𝒙′, 𝒙, 𝑡 𝑏 𝒙′, 𝑡 𝑑𝒙′

𝑔 𝒙, 𝒙′, 𝑡 = 1

𝜋𝑠02 𝑒𝑥𝑝 −

𝒙−𝒙′2

𝑠 𝑏(𝒙,𝑡) 2

Root augmentation as plant grows 𝜂 = 𝑠0

−1𝑑𝑠/𝑑𝑏|𝑏=0 - root to shoot ratio

𝑠 𝑏 ≈ 𝑠0(1 + 𝜂𝑏)

Surface-water height

Soil-water content per unit ground area

Areal biomass density 𝜕𝑡𝑏 = 𝐺𝑏𝑏 1 − 𝑏/𝜅 − 𝑏 + 𝛻2𝑏

𝜕𝑡𝑤 = 𝐼ℎ − 𝐿𝑤 − 𝐺𝑤𝑤 + 𝛿𝑤𝛻2𝑤

𝜕𝑡ℎ = 𝑝 − 𝐼ℎ − 𝜵 ⋅ 𝑱 𝑱 = −2𝛿ℎℎ 𝜵(ℎ + 𝜁)

h

𝜁

A PDE model that captures all three feedbacks (dimensionless form): (Gilad, Hardenberg, Provenzale, Shachak, Meron PRL 2004, JTB 2007)

Other models exist (Lejeune et al., Klausmeier, Rietkerk et al.) - capture a single feedback only. Interplay between different feedbacks can be significant (Gilad, et al. PRL 2004, JTB 2007; Kinast et al. PRL 2014)

Emergence of large-scale patterns

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urion Unive

rsity, Ehud

Meron - w

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ehud

Uniform states: Bare-soil (b = 0) Uniform vegetation (b 0)

Bifurcation diagram (plane terrain)

Uniform vegetation

Bare soil

𝑝𝑐

Borgogno et al. Reviews of Geophysics (2009)

Spots (Zambia) Labyrinth (Niger) Gaps (Senegal)

Note, the periodic pattern solutions extend to lower precipitation values compared to uniform vegetation – consistent with their role of providing an extra source of water.

Spatially periodic states: uniform vegetation loses stability to growth of non-uniform perturbations Periodic pattern

Wavenumber k

Gro

wth

rat

e

0

𝑘𝑐 Morphological changes are driven by the need to increase the water contributing areas

Emergence of large-scale patterns

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urion Unive

rsity, Ehud

Meron - w

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ehud

Fairy circles - bare patches of sandy soil

Two characteristics of this ecosystem allow considerable model simplification: sandy soil and confined roots.

Result in elimination of the ℎ equation and in getting rid of the root integrals:

𝜕𝑡𝑏 = 𝐺𝑏𝑏 1 − 𝑏/𝜅 − 𝑏 + 𝛻2𝑏 𝜕𝑡𝑤 = 𝑝 − 𝐿𝑤 − 𝐺𝑤𝑤 + 𝛿𝑤𝛻

2𝑤

𝐺𝑏 ≈ 𝜈𝑤 1 + 𝜂𝑏 2 𝐺𝑤 ≈ 𝛾𝑏 1 + 𝜂𝑏 2

Out of all feedbacks only the soil-water diffusion feedback remains: → implies anti-phase biomass-water distributions → observed in field studies (Cramer and Barger PLOS ONE 2013)

→ supports the prediction that this feedback is the driving mechanism

Model predictions are hard to test directly because of long time scales. Look for indirect evidence

The Namibian fairy circles (FC):

(Getzin et al. Ecography 2015)

Emergence of large-scale patterns

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urion Unive

rsity, Ehud

Meron - w

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Model studies of the NFC ecosystem not only predict periodic gap patterns that agree with observed FC, they also predict localized patterns (Zelnik, Meron, Bel, PNAS 2015; see also Fernandez-Oto et al., Philos. Trans. A 2014):

Within the bistability range of uniform vegetation and periodic pattern – many more solution branches of hybrid states:

space space

Hybrid states are significant for ecosystem function → provide more freedom in responding to environmental changes → adverse effects of desertification

may mitigate the

Fronts are pinned (Pomeau, Physica D

1986) → homoclinic snaking (Kozyreff & Chapman, PRL 2006; Lloyd et al. SIADS 2008; … Knobloch, Nonlinearity 2009)

Emergence of large-scale patterns

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urion Unive

rsity, Ehud

Meron - w

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The Namibian FC are still under debate: biomass-water feedbacks or under-ground termite colonies in the FC (Juergens, Science 2013).

If FCs are generic gap patterns – should be found in other drylands.

Termite hypothesis does not account for the emergence of large-scale order of FC.

FCs in Australia (Getzin et al. PNAS 2016 in press, under embargo):

These findings favor the pattern-formation view of FC, but a comparison between the Namibian and Australian ecosystems provides a yet deeper support for the pattern-formation view

Uncorrelated to termites

Emergence of large-scale patterns

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urion Unive

rsity, Ehud

Meron - w

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ehud

The Australian FC look very similar to the Namibian FC, but the two ecosystems differ in the water transport mechanism:

𝜕𝑡𝑏 = 𝐺𝑏𝑏 1 − 𝑏/𝜅 − 𝑏 + 𝛻2𝑏

𝜕𝑡𝑤 = 𝐼ℎ − 𝐿𝑤𝑤 − 𝐺𝑤𝑤 + 𝛿𝑤𝛻2𝑤

𝜕𝑡ℎ = 𝑝 − 𝐼ℎ − 𝐿ℎ ℎ + 𝛿ℎ𝛻2ℎ2

𝐺𝑏 ≈ 𝜈𝑤 1 + 𝜂𝑏 2 𝐺𝑤 ≈ 𝛾𝑏 1 + 𝜂𝑏 2

The species differ too but have confined roots - water transport by laterally extended roots is excluded.

The ℎ equation must be kept, but the root integrals can be calculated:

The soil in the Australian FC is compact and hard → water transport by overland water flow rather than soil-water diffusion.

Uniform vegetation loses stability to periodic patterns in a finite-k instability induced by the infiltration feedback.

In the present case: a finite-k or nonuniform stationary instability.

𝑘𝑜

Emergence of large-scale patterns

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urion Unive

rsity, Ehud

Meron - w

ww

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ehud

The different mechanisms induce different distributions of biomass and soil water, in-phase and anti-phase:

Anti-phase In-phase

biomass

Soil water

Yet, they both result in nearly hexagonal gap patterns → ecological manifestation of a basic universality principle of pattern formation theory:

Different systems undergoing the same instability type show similar patterns

Leads to stripe patterns with wave-vectors on the circle 𝑘𝑥2 + 𝑘𝑦

2 = 𝑘02,

𝑘𝑦

𝑘𝑥

The observation of hexagonal patterns, despite the different mechanisms, is in accord with the universality principle – provides further support.

patterns (three wave-vectors satisfying 𝒌1 + 𝒌2 + 𝒌3 = 0)

and to hexagonal

Ecosystem-function aspects: desertification

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urion Unive

rsity, Ehud

Meron - w

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ehud

Desertification – transition from a productive state to a less productive state, induced by an environmental change, e.g. drought, or disturbance

unproductive state

productive state

b

pf p pc

Common view does not take into account two spatial aspects:

1. Disturbances are likely to be spatially confined → local state transitions

productive

unproductive

space

Uniform vegetation

Bare soil

2. One of the two alternative stable states is often spatially patterned → hybrid states may exist

Periodic pattern

Clear cutting in Blue Creek Watershed Courtesy of Bruse Castle - EPFW

𝑝𝑀

Gradual desertification by front dynamics

Ecosystem-function aspects: desertification

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urion Unive

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These are model predictions – any indication in real ecosystems?

Impact of hybrid states on desertification:

Consider transition from uniform vegetation to the less productive periodic-pattern state

Uniform vegetation

Periodic patterns

If hybrid states where absent: local disturbances → gradual desertification by front propagation.

space

Periodic patterns

Uniform vegetation

But if hybrid states do exist: local disturbances → convergence to the closets hybrid state → no desertification.

space

A drought that takes the system outside the hybrid-state range, where fronts do propagate, can induce a transition to a less productive hybrid state, but no further desertification until the next drought.

Conclusion: hybrid states can prevent continual desertification and thereby mitigate the adverse effects of disturbances and droughts.

Ecosystem-function aspects: desertification

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2004 2008 2010 2013

Observations (Namibia)

Drought in 2007: 41mm/y vs. 100-200mm/y in other years

Model simulations

IC – 2004 + long simulation at 102mm/y (within snaking range)

Drought – 1y simulation at 84mm/y (outside snaking range)

Drought is over - 4y simulation at 102mm/y (within snaking range)

Drought is over – 5y simulation at 102mm/y

Reports on FC birth and death – are they related to hybrid states?

Uniform vegetation

Hybrid states Periodic

pattern

Birth of FC = hybrid-state transition induced by drought

Birth of FC - a transition to a hybrid state of lower productivity – no further desertification unless another strong drought occurs.

Ecosystem-function aspects: restoration

Since the unmodulated system tends anyway to form patterns (wavenumber 𝑘0), this is a spatial resonance problem analogous to periodically forced oscillators.

Restoration of degraded landscapes:

Periodic stripe-like embankments that capture runoff and along which the vegetation is planted. Characterized by their wavenumber 𝑘𝑓.

fkk 2fkk

2:1 1:1

𝑘𝑓/𝑘0 1 2 3 4

Resonance tongues in 1d:

The common practice - vegetation band at each embankment, amounts to a 1:1 resonant stripe pattern

embankment

vegetation

(Mau, Hageberg, Meron, PRL 2012)

Ben G

urion Unive

rsity, Ehud

Meron - w

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.bgu.ac.il/~

ehud

Ecosystem-function aspects: restoration

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urion Unive

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Meron - w

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𝑘𝑓 - modulation wavenumber

𝑘0 - wavenumber of unmodulated system

2d spatial systems, however, can respond by forming 2d resonant patterns:

where 𝑘𝑥 = 𝑘𝑓 2 , and 𝑘𝑦 is such that the mode’s wavenumber lies on the circle

𝑘𝑥2 + 𝑘𝑦

2 = 𝑘02 .

The response involves the growth of two oblique modes,

exp 𝑖𝒌± ⋅ 𝒙 , 𝒌±= 𝑘𝑥𝒙 ± 𝑘𝑦𝒚

Near the 1:1 resonance they resonate with the stripe mode exp −𝑖𝑘𝑓𝑥 , 𝒌++𝒌_ + 𝒌𝑓 = 0

and lead to 2:1 rhombic patterns.

Because of the freedom to select 𝑘𝑦 so as to obtain the optimal wavenumber 𝑘0, the growth range of the oblique modes is wide and overlaps with that of the 1:1 stripe mode

(Mau, Hagberg, Meron, PRL 2012)

Ecosystem-function aspects: restoration

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𝐵𝑖𝑜𝑚𝑎𝑠𝑠 ≈ 𝐴𝑒−𝑖𝑘𝑓𝑥 + 𝑎[𝑒𝑖𝒌+⋅𝒙+𝑒𝑖𝒌−⋅𝒙] + 𝑐. 𝑐. ,

Study the interaction of stripe and oblique modes near the 1:1 resonance by approximating

deriving equations for the modes’ amplitudes, and using them to study the existence and stability ranges of stripe patterns 𝐴, 𝑎) = (𝐴𝑆, 0 and of rhombic patterns 𝐴, 𝑎 = (𝐴𝑅 , 𝑎𝑅). (Mau, Haim, Meron, PRE 2015)

What patterns should we expect to find in the vicinity of the 1:1 resonance: 1:1 stripes or 2:1 rhombic patterns, and how should it affect the 1:1 restoration practice?

𝑏

p

1:1 stripe pattern (S)

2:1 rhombic pattern (R)

Bare soil (B)

The rhombic pattern is more robust: destabilizes the stripe pattern and extends to lower 𝑝 values.

𝑏 =

𝐴 2 + 2 𝑎 2

Moreover, the stripe pattern has poor resilience to droughts

Ecosystem-function aspects: restoration

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urion Unive

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𝑏

p

1:1 stripe pattern (S)

2:1 rhombic pattern (R)

Bare soil (B)

|𝑎|

|𝐴|

|𝑎|

|𝐴|

Restoration in 1:1 stripes can result in collapse to bare soil Restore instead by partial plantation to create the oblique modes and induce convergence to the robust rhombic patterns (Mau, Lev, Meron, PRE 2015)

Human intervention and ecological integrity:

Restoration by partial plantation - an intervention that results in high ecological integrity because it creates the oblique modes which the system naturally tends to develop.

Study dynamics in phase space Note: unstable solns are important

Conclusion

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An emerging new research field at the interface between pattern formation and spatial ecology

New book

Ben G

urion Unive

rsity, Ehud

Meron - w

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.bgu.ac.il/~

ehud

Introduction

I Overview

II Pattern formation theory

III Applications to Ecology

PRL 2004: Gilad et al., Ecosystem Engineers: From Pattern Formation to Habitat Creation

PRL 2012: Mau et al., Spatial periodic forcing can displace patterns it is intended to control

PRL 2014: Kinast et al., Interplay between Turing Mechanisms can Increase Pattern Diversity

PRE 2015: Mau et al., Reversing desertification as a spatial resonance problem

PNAS 2015: Zelnik et al., Gradual Regime Shifts in Fairy Circles

PNAS 2016 in press: Getzin et al., Discovery of fairy circles in Australia supports self-organization theory

Introduces the concepts and tools of pattern

formation theory and demonstrates their utility in

ecological research using problems from spatial

ecology …

Physics Today, Hugo Fort (2015)

Contemporary Physics, K. Alan Shore (2015)

Key papers relevant to this talk:

Mathematical Biosciences 2016: Meron, Pattern formation – a missing link in the study of ecosystem …