Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A....
Transcript of Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A....
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Self-Organisation in Arid Vegetation
Jonathan A. Sherratt
Department of Mathematicsand Maxwell Institute for Mathematical Sciences
Heriot-Watt University
May 2018
This talk can be downloaded from my web site
www.ma.hw.ac.uk/∼jas
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 2: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/2.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Outline
1 Vegetation Pattern Formation
2 History-Dependence in Vegetation Patterns
3 Inferring the Historical Origin of Patterned Vegetation
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 3: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/3.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Outline
1 Vegetation Pattern Formation
2 History-Dependence in Vegetation Patterns
3 Inferring the Historical Origin of Patterned Vegetation
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 4: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/4.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
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 Self-Organisation in Arid Vegetation
![Page 5: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/5.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
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 Self-Organisation in Arid Vegetation
![Page 6: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/6.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
RAINFALL
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 7: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/7.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
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 Self-Organisation in Arid Vegetation
![Page 8: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/8.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 9: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/9.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 10: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/10.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 11: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/11.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 12: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/12.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 13: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/13.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 14: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/14.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
Vegetation Patterns
Desert ecosystems provide a classic example of self-organised
pattern formation.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 15: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/15.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Vegetation Patterns
Banded Vegetation on Slopes
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 Self-Organisation in Arid Vegetation
![Page 16: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/16.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Outline
1 Vegetation Pattern Formation
2 History-Dependence in Vegetation Patterns
3 Inferring the Historical Origin of Patterned Vegetation
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 17: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/17.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
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 Self-Organisation in Arid Vegetation
![Page 18: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/18.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
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 Self-Organisation in Arid Vegetation
![Page 19: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/19.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Mathematical Model of Klausmeier
Water uptake=
Water density
× Plant density
×(
infiltrationrate
)
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 Self-Organisation in Arid Vegetation
![Page 20: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/20.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Typical Solution of the Model
0
6
12
2000
Distance uphill, x
Tim
e, t
750
0(a)
Note: pattern moves slowly uphill
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 21: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/21.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Pattern Existence and Stability
High rainfall ⇒ uniform vegetation
Low rainfall ⇒ no vegetation
Medium rainfall ⇒ patterns
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 22: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/22.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Pattern Existence and Stability
High rainfall ⇒ uniform vegetation
Low rainfall ⇒ no vegetation
Medium rainfall ⇒ patterns
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 23: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/23.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
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 Self-Organisation in Arid Vegetation
![Page 24: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/24.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 25: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/25.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 26: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/26.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 27: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/27.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 28: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/28.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 29: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/29.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 30: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/30.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 31: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/31.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 32: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/32.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 33: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/33.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 34: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/34.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 35: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/35.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 36: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/36.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 37: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/37.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 38: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/38.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 39: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/39.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 40: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/40.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 41: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/41.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 42: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/42.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 43: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/43.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 44: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/44.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 45: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/45.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 46: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/46.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 47: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/47.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 48: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/48.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 49: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/49.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 50: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/50.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 51: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/51.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 52: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/52.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 53: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/53.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 54: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/54.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 55: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/55.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 56: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/56.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 57: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/57.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 58: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/58.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 59: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/59.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 60: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/60.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 61: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/61.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 62: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/62.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 63: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/63.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 64: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/64.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 65: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/65.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 66: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/66.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 67: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/67.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 69: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/69.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 70: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/70.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 71: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/71.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 72: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/72.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 73: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/73.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 75: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/75.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 76: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/76.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 77: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/77.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 78: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/78.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 79: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/79.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 80: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/80.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Model prediction: as rainfall is varied within the range giving
patterns, abrupt changes in pattern wavelength occur.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
![Page 81: Self-Organisation in Arid Vegetationjas/talks/sherratt-edinburgh-may2018.pdf · Jonathan A. Sherratt Department of Mathematics and Maxwell Institute for Mathematical Sciences Heriot-Watt](https://reader031.fdocuments.net/reader031/viewer/2022022015/5b5bade77f8b9ac7498e8947/html5/thumbnails/81.jpg)
Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Variations in Rainfall: Hysteresis
Wavelength changes abruptly at the edge of the Busse Balloon.
(work of JAS, Koen Siteur, Eric Siero, Arjen Doelman, Max Rietkerk)
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
Calculation of the Busse Balloon
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Mathematical Model of Klausmeier
Typical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Outline
1 Vegetation Pattern Formation
2 History-Dependence in Vegetation Patterns
3 Inferring the Historical Origin of Patterned Vegetation
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Data on Wavelength vs Slope
I will show that the relationship between pattern wavelength
and slope provides valuable historical insights.
Data from Nevada, USA (Pelletier et al, J. Geophys. Res. 117: F04026, 2012)
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
The Origin of Vegetation Patterns
Vegetation patterns develop via
either degradation of uniform vegetation
or colonisation of bare ground
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Mathematical Model on a Slope
∂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
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Wavelength vs Slope for Degradation of Uniform
Vegetation
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Wavelength vs Slope for Degradation of Uniform
Vegetation
For realistic parameters, wavelength increases with slope
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Wavelength vs Slope for Colonisation
Wavelength decreases with slope
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
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 Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Example: The African Sahel
Patterned vegetation is widespread in the Sahel
Several studies of banded vegetation show wavelength ↓
as slope ↑
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Rainfall History in the Sahel
The Sahara and Sahel have been arid for about 5000
years, but the level of aridity has varied significantly.
The Sahel was relatively humid in the 16th and 17th
centuries.
Reasonable assumption: areas with vegetation patterns
today had uniform vegetation at the end of the 17th century.
Since wavelength decreases with slope, my results imply that
vegetation must have died out and then recolonised since the
end of the 17th century.
The most severe drought since 1700 was c. 1738-1756. So
today’s vegetation patterns result from recolonisation since
1760.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
References
J.A. Sherratt: History-dependent patterns of whole ecosystems.
Ecological Complexity 14, 8-20 (2013).
A.S. Dagbovie, J.A. Sherratt: Pattern selection and hysteresis in the
Rietkerk model for banded vegetation in semi-arid environments.
J. R. Soc. Interface 11: 20140465 (2014).
J.A. Sherratt: Using wavelength and slope to infer the historical origin of
semi-arid vegetation bands. PNAS USA 112: 4202-4207 (2015).
J.A. Sherratt: When does colonisation of a semi-arid hillslope generate
vegetation patterns? J. Math. Biol. 73: 199-226 (2016).
J.A. Sherratt: Using History to Predict the Future of Vegetation in the
African Sahel. Submitted.
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation
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Vegetation Pattern Formation
History-Dependence in Vegetation Patterns
Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
List of Frames
1 Vegetation Pattern Formation
Vegetation Patterns
Banded Vegetation on Slopes
2 History-Dependence in Vegetation Patterns
Mathematical Model of KlausmeierTypical Solution of the Model
Variations in Rainfall: Hysteresis
Calculation of the Busse Balloon
3 Inferring the Historical Origin of Patterned Vegetation
Banded Vegetation on Slopes
Mathematical Model on a Slope
Wavelength vs Slope for Degradation of Uniform Vegetation
Conclusions
Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Self-Organisation in Arid Vegetation