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 Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Applications of Wavetrains Objective of Talk

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

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Applications of Wavetrains Objective of Talk

Applications of Wavetrains

This is a wavetrain: a periodic function of x − ct

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Applications of Wavetrains Objective of Talk

Applications of Wavetrains

This is a wavetrain: a periodic function of x − ct Wavetrains occur in a wide variety of applications

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Applications of Wavetrains Objective of Talk

Objective of Talk

Objective of talk: to present a step-by-step guide to the numerical study of wavetrain solutions of partial differential equations Case study: a model for mussel bed patterns

蚌 = mussel

Software: all of my calculations and figures are done using the free software package WAVETRAIN www.ma.hw.ac.uk/wavetrain

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Applications of Wavetrains Objective of Talk

Outline

1

Case Study: Mussel Bed Patterns in the Wadden Sea

2

Calculation of Wavetrain Existence

3

Calculation of Wavetrain Stability

4

Calculation of Stability Boundaries in Parameter Space

5

Conclusions

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Mussel Bed Patterns Typical Pattern Solution Travelling Wave Equations

Mussel Bed Patterns

In the Wadden Sea, mussel beds self-organise into striped patterns

蚌 = mussel

Aerial photo of a mussel bed

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Mussel Bed Patterns Typical Pattern Solution Travelling Wave Equations

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 ∂a/∂t =

advection by tide

• β ∂a/∂x +

transfer to/ from upper water layers

• α(1 − a) −

eaten by mussels

• am

∂m/∂t = ∂2m/∂x2

• random

movement

+ δam

• birth

− γ m/(1 + m)

• dislodgement

by waves

蚌 = mussel

藻類 = algae

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Mussel Bed Patterns Typical Pattern Solution Travelling Wave Equations

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 ∂a/∂t =

advection by tide

• β ∂a/∂x + f(a, m)

∂m/∂t = ∂2m/∂x2

• random

movement

+ g(a, m)

蚌 = mussel

藻類 = algae

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Mussel Bed Patterns Typical Pattern Solution Travelling Wave Equations

Typical Pattern Solution

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Mussel Bed Patterns Typical Pattern Solution Travelling Wave Equations

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

• =

d2 m/dz2 + c d m/dz + g

• a,

m

• =

f( a, m) = α(1 − a) − a m g( a, m) = δ a m − γ m/(1 + m) 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

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations Step 2: Calculate the Locus of Homoclinic Solutions

Outline

1

Case Study: Mussel Bed Patterns in the Wadden Sea

2

Calculation of Wavetrain Existence

3

Calculation of Wavetrain Stability

4

Calculation of Stability Boundaries in Parameter Space

5

Conclusions

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations Step 2: Calculate the Locus of Homoclinic Solutions

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

• =

d2 m/dz2 + c d m/dz + g

• a,

m

• =

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

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations Step 2: Calculate the Locus of Homoclinic Solutions

Step 1: Calculate the Locus of Hopf Bifurcations

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations Step 2: Calculate the Locus of Homoclinic Solutions

Step 1: Calculate the Locus of Hopf Bifurcations

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations Step 2: Calculate the Locus of Homoclinic Solutions

Step 2: Calculate the Locus of Homoclinic Solutions

The homoclinic locus is approximated by the locus of wavetrains

• f a fixed, very long period

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Step 1: Calculate the Locus of Hopf Bifurcations Step 2: Calculate the Locus of Homoclinic Solutions

Step 2: Calculate the Locus of Homoclinic Solutions

The homoclinic locus is approximated by the locus of wavetrains

• f a fixed, very long period

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

Outline

1

Case Study: Mussel Bed Patterns in the Wadden Sea

2

Calculation of Wavetrain Existence

3

Calculation of Wavetrain Stability

4

Calculation of Stability Boundaries in Parameter Space

5

Conclusions

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

The Eigenvalue Problem

Partial differential eqns: at = β az + caz + f(a, m) mt = mzz + cmz + g(a, m) Wavetrain satisfies: 0 = β az + c az + f( a, m) 0 = mzz + c mz + g( a, m) Consider a(z, t) = a(z) + eλta(z) with |a| ≪ | a| m(z, t) = m(z) + eλtm(z) with |m| ≪ | m| ⇒ Eigenfunction eqn: λa = β az + caz + fa( a, m)a + fm( a, m)m λm = mzz + cmz + ga( a, m)a + gm( a, m)m Boundary conditions: a(0) = a(L)eiγ (0 ≤ γ < 2π) m(0) = m(L)eiγ (0 ≤ γ < 2π)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

The Eigenvalue Problem

Partial differential eqns: at = β az + caz + f(a, m) mt = mzz + cmz + g(a, m) Wavetrain satisfies: 0 = β az + c az + f( a, m) 0 = mzz + c mz + g( a, m) Consider a(z, t) = a(z) + eλta(z) with |a| ≪ | a| m(z, t) = m(z) + eλtm(z) with |m| ≪ | m| ⇒ Eigenfunction eqn: λa = β az + caz + fa( a, m)a + fm( a, m)m λm = mzz + cmz + ga( a, m)a + gm( a, m)m Boundary conditions: a(0) = a(L)eiγ (0 ≤ γ < 2π) m(0) = m(L)eiγ (0 ≤ γ < 2π)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

The Eigenvalue Problem

Partial differential eqns: at = β az + caz + f(a, m) mt = mzz + cmz + g(a, m) Wavetrain satisfies: 0 = β az + c az + f( a, m) 0 = mzz + c mz + g( a, m) Consider a(z, t) = a(z) + eλta(z) with |a| ≪ | a| m(z, t) = m(z) + eλtm(z) with |m| ≪ | m| ⇒ Eigenfunction eqn: λa = β az + caz + fa( a, m)a + fm( a, m)m λm = mzz + cmz + ga( a, m)a + gm( a, m)m Boundary conditions: a(0) = a(L)eiγ (0 ≤ γ < 2π) m(0) = m(L)eiγ (0 ≤ γ < 2π)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

The Eigenvalue Problem

Partial differential eqns: at = β az + caz + f(a, m) mt = mzz + cmz + g(a, m) Wavetrain satisfies: 0 = β az + c az + f( a, m) 0 = mzz + c mz + g( a, m) Consider a(z, t) = a(z) + eλta(z) with |a| ≪ | a| m(z, t) = m(z) + eλtm(z) with |m| ≪ | m| ⇒ Eigenfunction eqn: λa = β az + caz + fa( a, m)a + fm( a, m)m λm = mzz + cmz + ga( a, m)a + gm( a, m)m Boundary conditions: a(0) = a(L)eiγ (0 ≤ γ < 2π) m(0) = m(L)eiγ (0 ≤ γ < 2π)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

The Eigenvalue Problem

Partial differential eqns: at = β az + caz + f(a, m) mt = mzz + cmz + g(a, m) Wavetrain satisfies: 0 = β az + c az + f( a, m) 0 = mzz + c mz + g( a, m) Consider a(z, t) = a(z) + eλta(z) with |a| ≪ | a| m(z, t) = m(z) + eλtm(z) with |m| ≪ | m| ⇒ Eigenfunction eqn: λa = β az + caz + fa( a, m)a + fm( a, m)m λm = mzz + cmz + ga( a, m)a + gm( a, m)m Boundary conditions: a(0) = a(L)eiγ (0 ≤ γ < 2π) m(0) = m(L)eiγ (0 ≤ γ < 2π)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

The Eigenvalue Problem

Eigenfunction eqn: λa = β azz + caz + fa( a, m)a + fm( a, m)m λm = mzz + cmz + ga( a, m)a + gm( a, m)m Here 0 < z < L, with (a, m)(0) = (a, m)(L)eiγ (0 ≤ γ < 2π) Re(λ) < 0

→

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

The Eigenvalue Problem

Eigenfunction eqn: λa = β azz + caz + fa( a, m)a + fm( a, m)m λm = mzz + cmz + ga( a, m)a + gm( a, m)m Here 0 < z < L, with (a, m)(0) = (a, m)(L)eiγ (0 ≤ γ < 2π) Re(λ) > 0

→

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

Numerical Calculation of Eigenvalue Spectrum

(based on Jens Rademacher, Bjorn Sandstede, Arnd Scheel Physica D 229 166-183, 2007) 1

solve numerically for the wavetrain by continuation in c from a Hopf bifn point in the travelling wave eqns = β az + c az + f( a, m) =

• mzz + c

mz + g( a, m) (z = x − ct)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

Numerical Calculation of Eigenvalue Spectrum

(based on Jens Rademacher, Bjorn Sandstede, Arnd Scheel Physica D 229 166-183, 2007) 1

solve numerically for the wavetrain by continuation in c from a Hopf bifn point in the travelling wave eqns

2

for γ = 0, discretise the eigenfunction equations in space, giving a (large) matrix eigenvalue problem λa = β az + caz + fa( a, m)a + fm( a, m)m, a(0) = a(L)eiγ λm = mzz + cmz + ga( a, m)a + gm( a, m)m, m(0) = m(L)eiγ

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

Numerical Calculation of Eigenvalue Spectrum

(based on Jens Rademacher, Bjorn Sandstede, Arnd Scheel Physica D 229 166-183, 2007) 1

solve numerically for the wavetrain by continuation in c from a Hopf bifn point in the travelling wave eqns

2

for γ = 0, discretise the eigenfunction equations in space, giving a (large) matrix eigenvalue problem

3

continue the eigenfunction equations numerically in γ, starting from each of the periodic eigenvalues λa = β az + caz + fa( a, m)a + fm( a, m)m, a(0) = a(L)eiγ λm = mzz + cmz + ga( a, m)a + gm( a, m)m, m(0) = m(L)eiγ

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

Numerical Calculation of Eigenvalue Spectrum

(based on Jens Rademacher, Bjorn Sandstede, Arnd Scheel Physica D 229 166-183, 2007) 1

solve numerically for the wavetrain by continuation in c from a Hopf bifn point in the travelling wave eqns

2

for γ = 0, discretise the eigenfunction equations in space, giving a (large) matrix eigenvalue problem

3

continue the eigenfunction equations numerically in γ, starting from each of the periodic eigenvalues This gives the eigenvalue spectrum, and hence (in)stability

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

Step 3: Calculate Stability over a Parameter Grid

• =stable
• =unstable

=no

wavetrain

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Outline

1

Case Study: Mussel Bed Patterns in the Wadden Sea

2

Calculation of Wavetrain Existence

3

Calculation of Wavetrain Stability

4

Calculation of Stability Boundaries in Parameter Space

5

Conclusions

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Stability Changes of Eckhaus Type

• =stable
• =unstable

=no

wavetrain

Stability change of Eckhaus type The curvature of the spectrum at the origin changes sign

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Eckhaus Stability Boundaries

At stability changes of Eckhaus type, ∂2Reλ/∂γ2 = 0

δ=0.155 c=0.15 δ=0.155 c=0.08

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Eckhaus Stability Boundaries

At stability changes of Eckhaus type, ∂2Reλ/∂γ2 = 0

δ=0.155 c=0.15 δ=0.155 c=0.08

1

Fix λ = 0 and δ Vary c to find the point at which ∂2Reλ/∂γ2 = 0

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Eckhaus Stability Boundaries

At stability changes of Eckhaus type, ∂2Reλ/∂γ2 = 0

δ=0.155 c=0.15 δ=0.155 c=0.08

1

Fix λ = 0 and δ Vary c to find the point at which ∂2Reλ/∂γ2 = 0 This requires numerical continuation of the eigenfunction equation and its first and second derivatives with respect to γ

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Eckhaus Stability Boundaries

At stability changes of Eckhaus type, ∂2Reλ/∂γ2 = 0

δ=0.155 c=0.132

1

Fix λ = 0 and δ Vary c to find the point at which ∂2Reλ/∂γ2 = 0 This requires numerical continuation of the eigenfunction equation and its first and second derivatives with respect to γ

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Eckhaus Stability Boundaries

At stability changes of Eckhaus type, ∂2Reλ/∂γ2 = 0

δ=0.155 c=0.132

1

Fix λ = 0 and δ Vary c to find the point at which ∂2Reλ/∂γ2 = 0

2

Fix λ = ∂2Reλ/∂γ2 = 0 Vary δ and c This requires numerical continuation of the eigenfunction equation and its first and second derivatives with respect to γ

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Step 4: Calculate Stability Boundary (Eckhaus Type)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Stability Changes of Hopf Type

• =stable
• =unstable

=no

wavetrain

Stability change of Hopf type The spectrum crosses the imaginary axis away from the origin

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Hopf Stability Boundaries

At stability changes of Hopf type, Reλ = ∂Reλ/∂γ = 0

δ=0.155 c=0.485 δ=0.155 c=0.48

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Hopf Stability Boundaries

At stability changes of Hopf type, Reλ = ∂Reλ/∂γ = 0

δ=0.155 c=0.485 δ=0.155 c=0.48

1

Fix δ and c Vary γ to find the point at which ∂Reλ/∂γ = 0

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Hopf Stability Boundaries

At stability changes of Hopf type, Reλ = ∂Reλ/∂γ = 0

δ=0.155 c=0.485 δ=0.155 c=0.48

1

Fix δ and c Vary γ to find the point at which ∂Reλ/∂γ = 0

2

Fix δ and ∂Reλ/∂γ = 0 Vary c to find the point at which Reλ = 0

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Hopf Stability Boundaries

At stability changes of Hopf type, Reλ = ∂Reλ/∂γ = 0

δ=0.155 c=0.132

1

Fix δ and c Vary γ to find the point at which ∂Reλ/∂γ = 0

2

Fix δ and ∂Reλ/∂γ = 0 Vary c to find the point at which Reλ = 0

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Numerical Calculation of Hopf Stability Boundaries

At stability changes of Hopf type, Reλ = ∂Reλ/∂γ = 0

δ=0.155 c=0.132

1

Fix δ and c Vary γ to find the point at which ∂Reλ/∂γ = 0

2

Fix δ and ∂Reλ/∂γ = 0 Vary c to find the point at which Reλ = 0

3

Fix Reλ = ∂Reλ/∂γ = 0 Vary δ and c

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Step 5: Calculate Stability Boundary (Hopf Type)

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

Step 5: Calculate Stability Boundary (Hopf Type)

This plot is a complete account of wavetrain existence and stability

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Summary and Conclusions Example of Application: Hysteresis Publications

Outline

1

Case Study: Mussel Bed Patterns in the Wadden Sea

2

Calculation of Wavetrain Existence

3

Calculation of Wavetrain Stability

4

Calculation of Stability Boundaries in Parameter Space

5

Conclusions

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Summary and Conclusions Example of Application: Hysteresis Publications

Summary and Conclusions

Numerical continuation provides a comprehensive approach to the calculation of wavetrain existence and stability All of the calculations that I have described (and the plots) are implemented in the free software package WAVETRAIN www.ma.hw.ac.uk/wavetrain Calculations of this type provide an essential background for understanding results from numerical simulations

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Summary and Conclusions Example of Application: Hysteresis Publications

Example of Application: Hysteresis

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Summary and Conclusions Example of Application: Hysteresis Publications

Publications

J.A. Sherratt: Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations. Appl. Math. Comp. 218: 4684-4694 (2012). J.A. Sherratt: Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential

• equations. Submitted.

J.A. Sherratt: History-dependent patterns of whole ecosystems. Submitted.

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability

Case Study: Mussel Bed Patterns in the Wadden Sea Calculation of Wavetrain Existence Calculation of Wavetrain Stability Calculation of Stability Boundaries in Parameter Space Conclusions Summary and Conclusions Example of Application: Hysteresis Publications

List of Frames

Applications of Wavetrains Objective of Talk

1

Case Study: Mussel Bed Patterns in the Wadden Sea Mussel Bed Patterns Typical Pattern Solution Travelling Wave Equations

2

Calculation of Wavetrain Existence Step 1: Calculate the Locus of Hopf Bifurcations Step 2: Calculate the Locus of Homoclinic Solutions

3

Calculation of Wavetrain Stability The Eigenvalue Problem Numerical Calculation of Eigenvalue Spectrum Step 3: Calculate Stability over a Parameter Grid

4

Calculation of Stability Boundaries in Parameter Space Stability Changes of Eckhaus Type Step 4: Calculate Stability Boundary (Eckhaus Type) Stability Changes of Hopf Type Step 5: Calculate Stability Boundary (Hopf Type)

5

Conclusions Summary and Conclusions Example of Application: Hysteresis Publications Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Calculation of Periodic Travelling Wave Stability