postscript files/temp1/unstablespectrum essexjas/talks/migsaa-feb2017.pdf · 2017-02-10 ·...
Transcript of postscript files/temp1/unstablespectrum essexjas/talks/migsaa-feb2017.pdf · 2017-02-10 ·...
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