the dynamics of vegetation patterning in semi-arid ...jas/talks/edinburgh_jan07.pdfvegetation...

Ecological BackgroundThe Mathematical Model

Linear AnalysisTravelling Wave Equations

Bifurcations in the PDEsConclusions

The Dynamics of Vegetation Patterning inSemi-Arid Environments

Jonathan A. Sherratt

Department of MathematicsHeriot-Watt University

University of Edinburgh, 19 January 2007

Jonathan A. Sherratt The Dynamics of Vegetation Patterning in Semi-Arid Environments

In collaboration withGabriel Lord

Outline

1 Ecological Background

2 The Mathematical Model

3 Linear Analysis

4 Travelling Wave Equations

5 Bifurcations in the PDEs

6 Conclusions

Vegetation Pattern FormationMore Pictures of Vegetation PatternsVegetation Pattern Formation (contd)Mechanisms for Vegetation Patterning

Outline

3 Linear Analysis

6 Conclusions

Vegetation Pattern Formation

Vegetation patterns are found in semi-arid areas of Africa,Australia and Mexico (rainfall 100-700 mm/year)First identified by aerial photos in 1950sPlants vary from grasses to shrubs and trees

More Pictures of Vegetation Patterns

Labyrinth of bushyvegetation in Niger

Striped pattern ofbushy vegetationin Niger

Labyrinth of grassin Israel

Vegetation Pattern Formation (contd)

On flat ground, irregularmosaics of vegetationare typical

On slopes, the patternsare stripes, parallel tocontours (“Tiger bush”)

Mechanisms for Vegetation Patterning

Basic mechanism: competition for water

Possible detailed mechanism: water flow downhill causesstripes

This mechanism suggests that the stripes would moveuphill; this remains controversial.

Mathematical Model of KlausmeierTypical Solution of the Model

Outline

3 Linear Analysis

6 Conclusions

Mathematical Model of Klausmeier

Rate of change = Rainfall – Evaporation – Uptake by + Flowof water plants downhill

Rate of change = Growth, proportional – Mortality +Randomplant biomass to water uptake dispersal

∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2

Mathematical Model of Klausmeier

Rate of change = Rainfall – Evaporation – Uptake by + Flowof water plants downhill

Rate of change = Growth, proportional – Mortality +Randomplant biomass to water uptake dispersal

∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2

The nonlinearity in wu2 arises because the presence of rootsincreases water infiltration into the soil.

Typical Solution of the Model

Homogeneous Steady StatesApproximate Conditions for PatterningAn Illustration of Conditions for PatterningPredicting Pattern Wavelength

Outline

3 Linear Analysis

6 Conclusions

Homogeneous Steady States

For all parameter values, there is a stable “desert” steadystate u = 0, w = A.

When A ≥ 2B, there are also two non-trivial steady states

uu =2B

A +√

A2 − 4B2wu =

A +√

A2 − 4B2

us =2B

A −√

A2 − 4B2ws =

A −√

A2 − 4B2

When A ≥ 2B, there are also two non-trivial steady states

uu =2B

A +√

A2 − 4B2wu =

A +√

A2 − 4B2

2unstable

us =2B

A −√

A2 − 4B2ws =

A −√

A2 − 4B2

2stable to homogpertns for B < 2

Patterns develop when (us, ws) is unstable toinhomogeneous perturbations

Approximate Conditions for Patterning

Look for solutions (u, w) = (us, ws) + (u0, w0) exp{ikx + λt}

The dispersion relation Re[λ(k)] is algebraically complicated

Simplification using ν � 1 implies that for pattern formationA < ν1/2 B5/4/ 81/4

Simplification using ν � 1 implies that for pattern formation2B < A < ν1/2 B5/4/ 81/4

One can niavely assume that existence of (us, ws) gives asecond condition

An Illustration of Conditions for Patterning

Predicting Pattern Wavelength

Pattern wavelength is the most accessible property ofvegetation stripes in the field, via aerial photography.Wavelength can be predicted from the linear analysis

However this predictiondoesn’t fit the patternsseen in numericalsimulations.

Travelling Wave EquationsBifurcation Diagram for Travelling Wave ODEsWhen do Patterns Form?Pattern Formation for Low Rainfall

Outline

3 Linear Analysis

6 Conclusions

Travelling Wave Equations

The patterns move at constant shape and speed⇒ u(x , t) = U(z), w(x , t) = W (z), z = x − ct

d2U/dz2 + c dU/dz + WU2 − BU = 0

(ν + c)dW/dz + A − W − WU2 = 0

The patterns are periodic (limit cycle) solutions of these ODEs

Bifurcation Diagram for Travelling Wave ODEs

When do Patterns Form?

Pattern Formation for Low Rainfall

Discretizing the PDEsBifurcation Diagram for Discretized PDEsSpeed vs Rainfall for Discretized PDEsKey ResultHysteresis

Outline

3 Linear Analysis

6 Conclusions

Discretizing the PDEs

To investigate pattern stability, we must work with the modelPDEs. We discretize these in space and then use AUTO tostudy the resulting ODE system:

∂ui/∂t = wiu2i − Bui + (ui+1 − 2ui + 2ui-1)/∆x2

∂wi/∂t = A − wi − wiu2i + ν(wi+1 − wi)/∆x

(i = 1, . . . , N).

(i = 1, . . . , N).We use upwinding for the convective term.

(i = 1, . . . , N).We use upwinding for the convective term.Most of our work has used N = 40 and ∆x = 2.We assume periodic boundary conditions.

Bifurcation Diagram for Discretized PDEs

Speed vs Rainfall for Discretized PDEs

c vs A for PDEs

Speed vs Rainfall for Discretized PDEs

c vs A for PDEs c vs A for travelling wave PDEs

Key Result

For a wide range of rainfall levels,there are multiple stable patterns.

Hysteresis

The existence of multiple stablepatterns raises the possibility ofhysteresis

We consider slow variations in therainfall parameter A

Parameters correspond to grass,and the rainfall range corresponds to130–930 mm/year

Hysteresis

Predictions of Pattern WavelengthOther Potential Mechanisms for Vegetation PatternsMathematical Moral

Outline

3 Linear Analysis

6 Conclusions

Predictions of Pattern Wavelength

In general, pattern wavelength depends on initialconditions

When vegetation stripes arise from homogeneousvegetation via a decrease in rainfall, pattern wavelengthwill remain at its bifurcating value.

Wavelength =

Other Potential Mechanisms for Vegetation Patterns

Rietkirk Klausmeier model with diffusion of water in the soil

van de Koppel Klausmeier model with grazing

Maron two variable model (plant density and water in thesoil) with water transport based on porous mediatheory

Lejeune short range activation (shading) and long rangeinhibition (competition for water)

All of these models predict patterns. To discriminate betweenthem requires a detailed understanding of each model.

Mathematical MoralPredictions based only onlinear stability analysis aremisleading for this model

List of Frames

Vegetation Pattern Formation

Vegetation Pattern Formation (contd)

2 The Mathematical ModelMathematical Model of KlausmeierTypical Solution of the Model

3 Linear AnalysisHomogeneous Steady States

Travelling Wave Equations

When do Patterns Form?Pattern Formation for Low Rainfall

5 Bifurcations in the PDEsDiscretizing the PDEs

Speed vs Rainfall for Discretized PDEsKey ResultHysteresis

6 ConclusionsPredictions of Pattern Wavelength

Other Potential Mechanisms for Vegetation Patterns

Mathematical Moral

Pattern Selection

For a range of rainfall levels, there is more than one stablepattern. Which will be selected?We consider initial conditions that are small perturbationsof the coexistence steady state (us, vs).All such initial conditions give a pattern, but the wavelengthdepends on the initial perturbation

Pattern Selection

The wavelengthis close to thatpredicted bylinear stabilityanalysis

Pattern Selection on Larger Domains

The proximity of thewavelength to the mostlinearly unstable modecontinues as thedomain is enlarged

10 11 12 13 140

A=1.8, L=320, dx=2

Pattern Selection on Larger Domains

The proximity of thewavelength to the mostlinearly unstable modecontinues as thedomain is enlarged

10 11 12 13 140

A=1.8, L=320, dx=2

But it does not apply for other initialconditions, such as perturbationsabout (uu, wu)

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