Calculation of Periodic Travelling Wave Stability: A Users Guide - PowerPoint PPT Presentation

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Applications of Wavetrains Calculation of Wavetrain Stability Objective of Talk Calculation of Stability Boundaries in Parameter Space Conclusions Calculation of Periodic Travelling Wave Stability: A Users’ Guide Jonathan A. Sherratt Department of Mathematics & Maxwell Institute for Mathematical Sciences Heriot-Watt University ICAM 2012, Hong Kong, 28 May-1 June 2012 This talk can be downloaded from my web site www.ma.hw.ac.uk/ ∼ jas www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Applications of Wavetrains Calculation of Wavetrain Stability Objective of Talk Calculation of Stability Boundaries in Parameter Space Conclusions Applications of Wavetrains This is a wavetrain: a periodic function of x − ct www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Applications of Wavetrains Calculation of Wavetrain Stability Objective of Talk Calculation of Stability Boundaries in Parameter Space Conclusions Applications of Wavetrains This is a wavetrain: a periodic function of x − ct Wavetrains occur in a wide variety of applications www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Applications of Wavetrains Calculation of Wavetrain Stability Objective of Talk Calculation of Stability Boundaries in Parameter Space Conclusions Objective of Talk Automatic * Search Objective of talk: to present a step-by-step guide to the numerical study of wavetrain solutions of partial differential equations 蚌 = mussel Case study: a model for mussel bed patterns Software: all of my calculations and figures are done using the free software package WAVETRAIN www.ma.hw.ac.uk/wavetrain www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Applications of Wavetrains Calculation of Wavetrain Stability Objective of Talk Calculation of Stability Boundaries in Parameter Space Conclusions Outline Case Study: Mussel Bed Patterns in the Wadden Sea 1 Calculation of Wavetrain Existence 2 Calculation of Wavetrain Stability 3 Calculation of Stability Boundaries in Parameter Space 4 5 Conclusions www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Mussel Bed Patterns Calculation of Wavetrain Stability Typical Pattern Solution Calculation of Stability Boundaries in Parameter Space Travelling Wave Equations Conclusions Mussel Bed Patterns In the Wadden Sea, mussel beds self-organise into striped patterns Automatic * Search 蚌 = mussel Aerial photo of a mussel bed www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Mussel Bed Patterns Calculation of Wavetrain Stability Typical Pattern Solution Calculation of Stability Boundaries in Parameter Space Travelling Wave Equations Conclusions Mussel Bed Patterns In the Wadden Sea, mussel beds self-organise into striped patterns a ( x , t ) = density of algae m ( x , t ) = density of mussels transfer to/ advection eaten from upper Automatic * Search by by water layers tide mussels � �� � � �� � ���� ∂ a /∂ t β ∂ a /∂ x + α ( 1 − a ) − am = ∂ m /∂ t ∂ 2 m /∂ x 2 + δ am − γ m / ( 1 + m ) = 蚌 = mussel ���� � �� � � �� � birth random dislodgement 藻類 = algae movement by waves www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Mussel Bed Patterns Calculation of Wavetrain Stability Typical Pattern Solution Calculation of Stability Boundaries in Parameter Space Travelling Wave Equations Conclusions Mussel Bed Patterns Model of van de Koppel et al (Am Nat 165:E66, 2005) a ( x , t ) = density of algae m ( x , t ) = density of mussels advection by Automatic * Search tide � �� � ∂ a /∂ t β ∂ a /∂ x + f ( a , m ) = ∂ m /∂ t ∂ 2 m /∂ x 2 + g ( a , m ) = 蚌 = mussel � �� � random 藻類 = algae movement www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Mussel Bed Patterns Calculation of Wavetrain Stability Typical Pattern Solution Calculation of Stability Boundaries in Parameter Space Travelling Wave Equations Conclusions Typical Pattern Solution www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Mussel Bed Patterns Calculation of Wavetrain Stability Typical Pattern Solution Calculation of Stability Boundaries in Parameter Space Travelling Wave Equations Conclusions Travelling Wave Equations Wavetrains satisfy a ( x , t ) = � a ( z ) , m ( x , t ) = � m ( z ) , z = x − ct �� � ( β + c ) d � a / dz + f a , � m = 0 �� � d 2 � m / dz 2 + c d � m / dz + g a , � m = 0 f ( � a , � m ) a ) − � a � m α ( 1 − � = g ( � a , � m ) a � m − γ � m / ( 1 + � m ) δ � = The key environmental parameter is δ ↔ algae supply rate Objective: calculation of the regions of the δ – c plane in which wavetrains exist, and in which they are stable www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Step 1: Calculate the Locus of Hopf Bifurcations Calculation of Wavetrain Stability Step 2: Calculate the Locus of Homoclinic Solutions Calculation of Stability Boundaries in Parameter Space Conclusions Outline Case Study: Mussel Bed Patterns in the Wadden Sea 1 Calculation of Wavetrain Existence 2 Calculation of Wavetrain Stability 3 Calculation of Stability Boundaries in Parameter Space 4 5 Conclusions www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Step 1: Calculate the Locus of Hopf Bifurcations Calculation of Wavetrain Stability Step 2: Calculate the Locus of Homoclinic Solutions Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations Wavetrains are limit cycle solutions of the travelling wave equations �� � ( β + c ) d � a / dz + f a , � m = 0 �� � m / dz 2 + c d � d 2 � m / dz + g a , � m = 0 Wavetrains lie on a solution branch that starts at a Hopf bifurcation point (in most cases) Step 1: Calculate the locus of Hopf bifurcations in the δ – c plane www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Step 1: Calculate the Locus of Hopf Bifurcations Calculation of Wavetrain Stability Step 2: Calculate the Locus of Homoclinic Solutions Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Step 1: Calculate the Locus of Hopf Bifurcations Calculation of Wavetrain Stability Step 2: Calculate the Locus of Homoclinic Solutions Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Step 1: Calculate the Locus of Hopf Bifurcations Calculation of Wavetrain Stability Step 2: Calculate the Locus of Homoclinic Solutions Calculation of Stability Boundaries in Parameter Space Conclusions Step 2: Calculate the Locus of Homoclinic Solutions The homoclinic locus is approximated by the locus of wavetrains of a fixed, very long period www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Step 1: Calculate the Locus of Hopf Bifurcations Calculation of Wavetrain Stability Step 2: Calculate the Locus of Homoclinic Solutions Calculation of Stability Boundaries in Parameter Space Conclusions Step 2: Calculate the Locus of Homoclinic Solutions The homoclinic locus is approximated by the locus of wavetrains of a fixed, very long period www.ma.hw.ac.uk/ ∼ jas Jonathan A. Sherratt Calculation of Periodic Travelling Wave Stability