The mathematics of desertification: searching for early warning signals

Arjen Doelman (UL), Max Rietkerk (UU) Jens Rademacher (CWI) , Sjors van der Stelt (UvA)

ORThe dynamics of vegetation patterns

• Vegetation patterns & early warning signals.

• Pattern formation in reaction-diffusion models.

• The Busse balloon.

• The dynamics of modulated waves.

• Bifurcations with slowly varying parameters.

• Desertification scenarios.

• Claims/conjectures/comments/observations.

Structure of the talk

An intermezzo: the Hopf dance.

Desertification: • a fast (‘catastrophic’) process in which a patterned state suddenly collapses. • caused by a slow change in the ‘environment’.

A(n almost) irreversible process: the desert is ‘very stable’.

Early warning signals• The appearance of vegetation patterns is the first (very) early warning signal.• Patterned states are quite robust: they are stable under a large range of varying circumstances.• Nevertheless, they may suddenly collapse when a ‘threshold’ is crossed.

Tiger bushes/ Brousse tigrée

Questions• Can we predict the collapse?• Can we measure how close a system is to threshold?• Can we invent tools by which this ‘distance to threshold’ can be measured?

Crucial difference: this (type of) system(s) is not spatially extended, the models only involve temporal dynamics.

The most simple example: The ‘catastrophic collapse’ induced by a saddle node bifurcation.

_y= ¯ +y2

A very simple early warning signal: the transition is preceded by ‘a slowing down’ behavior.

ODEs ↔ PDEs

[M. Scheffer, J. Bascompte, W.A. Brock, V. Brovkin, S.R. Carpenter, V. Dakos, H. Held, E.H. van Nes, M. Rietkerk, G. Sugihara (2009), Early-warning signals for critical transitions, Reviews, Nature 461(3) 53-59]

YES

There are many PDE models in the ecological literature, all of reaction-diffusion type.

Central components:

• Biomass• Water• Groundwater

Translation into mathematical terms

Can one model vegetation dynamics?

[Meron, Klausmeier, Rietkerk, ...]

The Klausmeier & Gray-Scott (GS) models

See the talk of Sjors van der Stelt this afternoon

The dynamics of patterns in the GS equation

[J. Pearson (1993), Complex patterns in a simple system]

‘stripes’ ‘labyrinths’

‘spots’

There is a (very) comparable richness in types of vegetation patterns ...

It’s tempting to play a game of finding corresponding structures in the GS system and in actual vegetated areas ...

• Can we use our insights in GS to find ‘early warning signals’?

• Has this question perhaps already been considered (in disguise)?

More scientifically:

The GS equation perhaps is one of the most well-studied reaction-diffusion equations of the last decades.

[Pearson, Swinney et al. 1994]

chemical reaction

numerical simulation

It’s mostly famous’ for exhibiting ‘self-replication dynamics’

water

Let’s first consider one-dimensional patterns

In one space dimension:

Vegetation patterns correspond to singular spatially periodic solutions of the reaction-diffusion model.

biomass

First question:

For which parameter values can spatially periodic patterns be observed?

bifu

rcat

ion

para

met

er R

wave number k

Region in (wavenumber, parameter)-space in which stable spatially periodic patterns exist

onset

[Busse, 1978] (convection)

The Busse balloon

[D, Rademacher & van der Stelt, ’11]

Turing

‘Fall of patterns’ at k=0

A Busse balloon for the GS model

↔ yearly rainfall

Periodic patterns near k=0: singular localized pulses (of vegetation pattern kind)

Coexisting stable patterns (for the same parameter values)

• Near onset/the Turing bifurcation: ‘full analytical control’ through Ginzburg-Landau theory.

• A complete classification of the generic character of the boundary of the Busse balloon [Rademacher & Scheel, ’07].

• Near the ‘fall of patterns’: existence and stability of singular patterns [Doelman, Gardner & Kaper, ’01; van der Ploeg & D, ’05].

What do we know analytically?

See the talk of Sjors van der Stelt this afternoon

Note: Collapse takes place as ‘the system’ crosses through the boundary of the Busse balloon.

A spin-off: the Hopf dance, a novel fine-structure!

↔

A ‘dance’ of intertwining Hopf bifurcations.

The homoclinic (k=0) ‘oasis’ pattern is the last to destabilize

(Ni’s conjecture)

AN INTERMEZZO

Two types of Hopf bifurcations?

in phase

out of phase

Why only these two?

Spectral analysis

¤i(s)

¤j (s)

Note: § 1 endpoints correspond to H§ 1 Hopf bifurcations.

+1

-1

» 1=k

Im(¸)

¤h(s)

The long wavelength limit ( k ~ 0)

A novel general insight in the ‘fall of patterns’

In a general class of reaction-diffusion models:

• The homoclinic ‘oasis’ pattern is the last pattern to become unstable (↔Ni’s conjecture).

• The Hopf dance: near the destabilization of the homoclinic pattern, the Busse balloon has a ‘fine structure’ of two intertwining curves of Hopf bifurcations.

[D, Rademacher & van der Stelt, ’11]

END INTERMEZZO

There is no uniquely determined ‘attractor’ (as in ODEs).

The dynamics within a (continuous) family of stable

patterns.

Back to desertification/the catastrophic collapse of patterns

What is still missing?

I. Which pattern does nature choose?

,

The dynamics of modulated ‘waves’

[van Harten, ’95; D, Sandstede, Scheel & Schneider, ’09]

Various ‘phase equations’ can be obtained:

• The Burgers equation.

• The Korteweg-de Vries equation.

• The Ginzburg-Landau equation.

• The Kuramoto-Sivashinsky equation.

• …

Near a side band type boundary of the Busse balloon,

,

What is still missing?

II. The dynamics induced by slowly varying parameters.

Answer: It may (for instance) significantly delay the bifurcation.

from [Baer, Erneux, Rinzel, ’89]

¹x = ¹x("t)

unstablestable

O(1) delay!

Hopf bifurcation curve

This type of issues has not (at all!) been studied in PDEs

Desertification & early warning signals

Various different scenarios

A typical Busseballoon,destabilization possibleby:² thesidebandmechanism,² a saddlenodebifurcation,² a Hopf bifurcation,² co-dimension 2 corners.

à A=A("t)

Turing

(i) – the Hollywood scenario: A decreases (relatively) fast

• The vegetation pattern cannot adapt (although the destabilization may be delayed).• The vegetation patterns collapse into the desert state with the same wave number as given by the Turing bifurcation.

Unlikely, has not been not observed

(ii-iv) – Sliding along the boundary of the Busse balloon

A decreases so slow that the ‘internal dynamics’ within the family of spatially periodic patterns can adapt.

By decreasing its (average) wave number k the pattern slides back into the Busse balloon.

By this sliding mechanism, k decreases to realistic values

Claims and/or conjectures

•The adaptation process is governed by the local character of the (boundary of the) Busse balloon.

(side band: a Kuramoto-Sivashinsky equation with non-constant coefficients (?))

•The ‘flexibility of the internal dynamics’ can be determined by a phase equation analysis.

And thus:

• There will be various destabilization mechanisms.• These can be recognized by the character of the internal dynamics.• This provides explicit early warning signals.

Some final comments/observations

• Numerical simulations will play a crucial role in the analysis.• A priori, significant aspects of the analysis will have to be formal.

- Phase equations: about 30 years between first derivations & rigorous proofs.- Delayed bifurcations in ODEs: proper tool (‘geometric blow-up’) only developed 10-15 years after the first papers appeared in the literature.

• This is all only in one spatial dimension!- There’s not even a proper Ginzburg-Landau theory in 2D. - Similar conceptional ‘challenges’ for phase equations in 2D. - …

• However: 1-D insight will lead to 2D conjectures …

THE BELLY DANCE

The spectral branch is only to leading order a straight line/an interval.

In general it will be (slightly) bent.

This may yield small regions of ‘internal Hopf destabilizations’ and the corners in the boundary of the BB will disappear ↔ the orientation of the belly.

belly

A more typical Busse balloon?

Or more generic (?): sometimes a co-dimension 2 intersection, sometimes an `internal Hopf bridge’?

This is however not the case. In the class of considered model system, a BELLY DANCE takes place.

The belly always points away from the Im-axis near the ‘corner’ at which +1 and -1 cross at the same time.

WHY??

The theory includes in essence ‘all explicit models in the literature’ (!?). HOWEVER, if one looks carefully it’s clear these models are in fact very special.

→ Back to the drawing board …