Vegetation Patterns in Semi-Arid Environments Jonathan A. Sherratt - PowerPoint PPT Presentation

Ecological Background The Mathematical Model Linear Analysis Travelling Wave Equations Conclusions Vegetation Patterns in Semi-Arid Environments Jonathan A. Sherratt Department of Mathematics Heriot-Watt University CMPD3, Bordeaux, May/June 2010 This talk can be downloaded from my web site www.ma.hw.ac.uk/ ∼ jas www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Outline Ecological Background 1 The Mathematical Model 2 Linear Analysis 3 Travelling Wave Equations 4 5 Conclusions www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Vegetation Pattern Formation Mitchell grass in Australia Bushy vegetation in Niger (Western New South Wales) Banded vegetation patterns are found on gentle slopes in semi-arid areas of Africa, Australia and Mexico First identified by aerial photos in 1950s Plants vary from grasses to shrubs and trees www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes WATER FLOW WATER FLOW www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Mechanisms for Vegetation Patterning Basic mechanism: competition for water Possible detailed mechanism: water flow downhill causes stripes The stripes move uphill (very slowly) www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Vegetation Pattern Formation Linear Analysis Mechanisms for Vegetation Patterning Travelling Wave Equations Two Key Ecological Questions Conclusions Two Key Ecological Questions How does the spacing of the vegetation bands depend on rainfall, herbivory and slope? At what rainfall level is there a transition to desert? www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Mathematical Model of Klausmeier Linear Analysis Typical Solution of the Model Travelling Wave Equations Conclusions Outline Ecological Background 1 The Mathematical Model 2 Linear Analysis 3 Travelling Wave Equations 4 5 Conclusions www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments

Ecological Background The Mathematical Model Mathematical Model of Klausmeier Linear Analysis Typical Solution of the Model Travelling Wave Equations Conclusions Mathematical Model of Klausmeier Rate of change = Rainfall – Evaporation – Uptake by + Flow of water plants downhill + Random Rate of change = Growth, proportional – Mortality plant biomass to water uptake dispersal A − w − wu 2 + ν∂ w /∂ x ∂ w /∂ t = wu 2 − Bu + ∂ 2 u /∂ x 2 ∂ u /∂ t = www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Vegetation Patterns in Semi-Arid Environments