Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Outline

1 Ecological Background

2 Pattern Formation in a Mathematical Model

3 Pattern Existence and Stability

4 Predictions of Pattern Wavelength vs Slope

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Outline

1 Ecological Background

2 Pattern Formation in a Mathematical Model

3 Pattern Existence and Stability

4 Predictions of Pattern Wavelength vs Slope

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

W National Park, Niger

Average patch width is 50 m

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Data from Burkina Faso

Rietkerk et alPlant Ecology 148: 207-224, 2000

More plants⇒ more roots and organic matter in soil

⇒ more infiltration of rainwater

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

RAINFALL

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

OFFRUN

RAINFALL

LOW INFILTRATION

HIGH INFILTRATION HIGH INFILTRATION

OFFRUN

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Vegetation Patterns

Desert ecosystems provide a classic example of self-organised

pattern formation.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Banded Vegetation on Slopes

On slopes, run-off occurs in one direction only, giving striped

patterns parallel to the contours.

Bushy vegetation in Niger Mitchell grass in Australia(Western New South Wales)

Banded vegetation patterns are found on gentle slopes in

semi-arid areas of Africa, Australia, Mexico and S-W USA.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Banded Vegetation on Slopes

On slopes, run-off occurs in one direction only, giving striped

patterns parallel to the contours.

Bushy vegetation in Niger Mitchell grass in Australia(Western New South Wales)

Wavelength can be measured via remote sensing.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Data on Wavelength vs Slope

Data from sub-Saharan Africa and S-W USA shows that the

wavelength of banded vegetation patterns is negatively

correlated with slope.

Data from Nevada, USA (Pelletier et al, J. Geophys. Res. 117: F04026, 2012)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Vegetation Patterns

Banded Vegetation on Slopes

Data on Wavelength vs Slope

Data on Wavelength vs Slope

Data from sub-Saharan Africa and S-W USA shows that the

wavelength of banded vegetation patterns is negatively

correlated with slope.

Data from Nevada, USA (Pelletier et al, J. Geophys. Res. 117: F04026, 2012)

How does this compare with predictions of mathematical models?

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Outline

1 Ecological Background

2 Pattern Formation in a Mathematical Model

3 Pattern Existence and Stability

4 Predictions of Pattern Wavelength vs Slope

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Mathematical Model of Klausmeier

∂u/∂t =

plantgrowth︷︸︸︷

wu2−

plantloss︷︸︸︷

Bu +

plantdispersal︷ ︸︸ ︷

∂2u/∂x2

∂w/∂t = A︸︷︷︸

averagerainfall

− w︸︷︷︸

evaporation& drainage

− wu2︸︷︷︸

uptakeby plants

+ ν ∂w/∂x︸ ︷︷ ︸

flowdownhill

+D ∂2w/∂x2

︸ ︷︷ ︸

diffusionof water

(Klausmeier, Science 284: 1826-8, 1999)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Mathematical Model of Klausmeier

∂u/∂t =

plantgrowth︷︸︸︷

wu2−

plantloss︷︸︸︷

Bu +

plantdispersal︷ ︸︸ ︷

∂2u/∂x2

∂w/∂t = A︸︷︷︸

averagerainfall

− w︸︷︷︸

evaporation& drainage

− wu2︸︷︷︸

uptakeby plants

+ ν ∂w/∂x︸ ︷︷ ︸

flowdownhill

+D ∂2w/∂x2

︸ ︷︷ ︸

diffusionof water

The nonlinearity in water uptake occurs because the presence of

plants increases water infiltration into the soil.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Mathematical Model of Klausmeier

Water uptake=

Water density

× Plant density

×

(infiltration

rate

)

The nonlinearity in water uptake occurs because the presence of

plants increases water infiltration into the soil.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Mathematical Model of Klausmeier

Typical Solution of the Model

Typical Solution of the Model

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

Outline

1 Ecological Background

2 Pattern Formation in a Mathematical Model

3 Pattern Existence and Stability

4 Predictions of Pattern Wavelength vs Slope

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

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

D d2W/dz2 + (ν + c)dW/dz + A−W −WU2 = 0

The patterns are periodic (limit cycle) solutions of these

equations

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

Pattern Stability

Not all of the possible patterns are stable as solutions of the

model equations.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

Pattern Stability: Numerical Approach

The boundary between stable and unstable patterns can be

calculated by numerical continuation of the essential spectrum.

-0.4

-0.2

0

0.2

0.4

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

Im(e

igen

valu

e)

Re(eigenvalue)

-0.4

-0.2

0

0.2

0.4

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2

Im(e

igen

valu

e)

Re(eigenvalue)

Calculations of this type can be performed using the software

package WAVETRAIN (www.ma.hw.ac.uk/wavetrain).

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

Pattern Stability: Wavelength vs Rainfall

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

Pattern Stability: Wavelength vs Rainfall

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

Pattern Stability: The Key Result

Key Result

Many of the possible patterns are

unstable and thus will never be seen.

However, for a wide range of rainfall

levels, there are multiple stable

patterns.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

Travelling Wave Equations

Pattern Stability

Variations in Rainfall: Hysteresis

Variations in Rainfall: Hysteresis

The existence of multiple stable patterns suggests the

possibility of hysteresis.

Domain length 150, periodic bc’s

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

Outline

1 Ecological Background

2 Pattern Formation in a Mathematical Model

3 Pattern Existence and Stability

4 Predictions of Pattern Wavelength vs Slope

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

How to Predict Pattern Wavelength

Pattern wavelength

is history-dependent

We must focus on the

�

wwww

onset of patterning

Degradation of<

Colonisation>

uniform vegetation of bare ground

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

Wavelength vs Slope for Degradation of Uniform

Vegetation

0

6

12

2000

Distance uphill, x

Tim

e, t

750

0(a)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

Wavelength vs Slope for Degradation of Uniform

Vegetation

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

Wavelength vs Slope for Degradation of Uniform

Vegetation

For realistic parameters, wavelength increases with slope,

contrary to data

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

When Does Vegetation Colonise Bare Ground?

y

Time

Downhill←→ Uphill

Very low rainfall: an isolated vegetation patch dies out

Slightly larger rainfall: both edges move uphill

Larger rainfall: the patch expands both uphill and downhill

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

Wavelength vs Slope for Colonisation

0 7

(b)

Distance uphill, x Distance uphill, x750 750

Tim

e, t

Tim

e, t

(a)

250250

00

0 0

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

Wavelength vs Slope for Colonisation

Wavelength decreases with slope, in agreement with data

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

How to Predict Pattern Wavelength

Pattern wavelength

is history-dependent

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

How to Predict Pattern Wavelength

Pattern wavelength

is history-dependent

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

How to Predict Pattern Wavelength

Pattern wavelength

is history-dependent

=⇒

We must focus on the

onset of patterning−→

−→

Degradation of Colonisation

uniform vegetation of bare ground

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation

Ecological Background

Pattern Formation in a Mathematical Model

Pattern Existence and Stability

Predictions of Pattern Wavelength vs Slope

How to Predict Pattern Wavelength

Wavelength vs Slope for Degradation of Uniform Vegetation

When Does Vegetation Colonise Bare Ground?

Wavelength vs Slope for Colonisation

Conclusions

Conclusions

Wavelength is positively correlated with slope ⇒ vegetation

pattern originated by degradation of uniform

vegetation

Wavelength is negatively correlated with slope ⇒ vegetation

pattern originated by colonisation of bare ground

Main message: combined wavelength–slope data is much

more valuable than wavelength data alone.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Using Mathematical Models to Infer the Historical Origin of Vegetation