nonlinear dynamics of vegetation patterns in...

Ecological BackgroundThe Mathematical Model

Travelling Wave EquationsPattern Stability

Conclusions

Nonlinear Dynamics of Vegetation Patterns inSemi-Deserts

Jonathan A. Sherratt

Department of MathematicsHeriot-Watt University

University of Dundee26 May 2011

This talk can be downloaded from my web sitewww.ma.hw.ac.uk/∼jas

Jonathan A. Sherratt www.ma.hw.ac.uk/∼jas Nonlinear Dynamics of Vegetation Patterns in Semi-Deserts

Conclusions

Vegetation Pattern FormationMechanisms for Vegetation PatterningTwo Key Ecological Questions

Outline

1 Ecological Background

2 The Mathematical Model

3 Travelling Wave Equations

4 Pattern Stability

5 Conclusions

Conclusions

Vegetation Pattern Formation

Bushy vegetation in Niger Mitchell grass in Australia(Western New South Wales)

Banded vegetation patterns are found on gentle slopes insemi-arid areas of Africa, Australia and Mexico

First identified by aerial photos in 1950s

Plants vary from grasses to shrubs and trees

Conclusions

Mechanisms for Vegetation Patterning

Basic mechanism: competition for water

Conclusions

Possible detailed mechanism: water flow downhill causesstripes

Conclusions

The stripes move uphill (very slowly)

Conclusions

Two Key Ecological Questions

How does the spacing of the vegetation bands depend onrainfall, herbivory and slope?

At what rainfall level is there a transition to desert?

Conclusions

Mathematical Model of KlausmeierTypical Solution of the ModelHomogeneous Steady StatesApproximate Conditions for PatterningShortcomings of Linear Stability Analysis

Outline

4 Pattern Stability

5 Conclusions

Conclusions

Mathematical Model of Klausmeier

Rate of change = Rainfall – Evaporation – Uptake by + Flowof water plants downhill

Rate of change = Growth, proportional – Mortality + Randomplant biomass to water uptake dispersal

∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2

Conclusions

∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2

The nonlinearity in wu2 arises because the presence of rootsincreases water infiltration into the soil.

Conclusions

∂w/∂t = A − w − wu2 + ν∂w/∂x

∂u/∂t = wu2 − Bu + ∂2u/∂x2

Parameters: A: rainfall B: plant loss ν: slope

Conclusions

Typical Solution of the Model

Conclusions

Homogeneous Steady States

For all parameter values, there is a stable “desert” steadystate u = 0, w = A

Conclusions

When A ≥ 2B, there are also two non-trivial steady states,one of which is unstable to homogeneous perturbations

Conclusions

When A ≥ 2B, there are also two non-trivial steady states,one of which is unstable to homogeneous perturbations

Patterns develop when the other steady state (us, ws) isunstable to inhomogeneous perturbations

Conclusions

Approximate Conditions for Patterning

Look for solutions (u, w) = (us, ws) + (u0, w0) exp{ikx + λt}

The dispersion relation Re[λ(k)] is algebraically complicated

Conclusions

An approximate condition for pattern formation is

A < ν1/2 B5/4/ 81/4

Conclusions

An approximate condition for pattern formation is

2B < A < ν1/2 B5/4/ 81/4

One can niavely assume that existence of (us, ws) gives asecond condition

Conclusions

An Illustration of Conditions for Patterning

The dots show parameters forwhich there are growinglinear modes.

Conclusions

An Illustration of Conditions for Patterning

Numerical simulations showpatterns in both the dottedand green regions ofparameter space.

Conclusions

Predicting Pattern Wavelength

Pattern wavelength is the most accessible property ofvegetation stripes in the field, via aerial photography.Wavelength can be predicted from the linear analysis.

Conclusions

Predicting Pattern Wavelength

Pattern wavelength is the most accessible property ofvegetation stripes in the field, via aerial photography.Wavelength can be predicted from the linear analysis.

However this predictiondoesn’t fit the patternsseen in numericalsimulations.

Conclusions

Shortcomings of Linear Stability Analysis

Linear stability analysis fails in two ways:

It significantly over-estimates the minimum rainfall level forpatterns.

Close to the maximum rainfall level for patterns, itincorrectly predicts a variation in pattern wavelength withrainfall.

Conclusions

Travelling Wave EquationsBifurcation Diagram for Travelling Wave EquationsWhen do Patterns Form?Pattern Formation for Low Rainfall

Outline

4 Pattern Stability

5 Conclusions

Conclusions

Travelling Wave Equations

The patterns move at constant shape and speed⇒ u(x , t) = U(z), w(x , t) = W (z), z = x − ct

d2U/dz2 + c dU/dz + WU2 − BU = 0

(ν + c)dW/dz + A − W − WU2 = 0

The patterns are periodic (limit cycle) solutions of theseequations

Conclusions

Bifurcation Diagram for Travelling Wave Equations

Conclusions

When do Patterns Form?

Conclusions

Pattern Formation for Low Rainfall

Patterns are also seen forparameters in the greenregion.

Conclusions

Minimum Rainfall for Patterns

Conclusions

The Eigenvalue ProblemNumerical Calculation of Eigenvalue SpectrumStability in a Parameter PlaneHysteresis

Outline

4 Pattern Stability

5 Conclusions

Conclusions

The Eigenvalue Problem

PDE model: ut = uzz + cuz + f (u, w)

wt = νwz + cvz + g(u, w)

Periodic wave satisfies: 0 = Uzz + cUz + f (U, W )

0 = νWz + cWz + g(U, W )

Consider u(z, t) = U(z) + eλtu(z) with |u| ≪ |U|

w(z, t) = W (z) + eλtw(z) with |w | ≪ |W |

⇒ Eigenfunction eqn: λu = uzz + cuz + fu(U, W )u + fw (U, W )w

λw = νwz + cwz + gu(U, W )u + gw (U, W )w

Boundary conditions: u(0) = u(L)eiγ (0 ≤ γ < 2π)

w(0) = w(L)eiγ (0 ≤ γ < 2π)

Conclusions

0 = νWz + cWz + g(U, W )

w(0) = w(L)eiγ (0 ≤ γ < 2π)

Conclusions

0 = νWz + cWz + g(U, W )

w(0) = w(L)eiγ (0 ≤ γ < 2π)

Conclusions

0 = νWz + cWz + g(U, W )

w(0) = w(L)eiγ (0 ≤ γ < 2π)

Conclusions

0 = νWz + cWz + g(U, W )

w(0) = w(L)eiγ (0 ≤ γ < 2π)

Conclusions

Eigenfunction eqn: λu = uzz + cuz + fu(U, W )u + fw (U, W )w

Here 0 < z < L, with (u, w)(0) = (u, w)(L)eiγ (0 ≤ γ < 2π)

Conclusions

Re(λ) < 0

Conclusions

Re(λ) > 0

Conclusions

Numerical Calculation of Eigenvalue Spectrum(based on Jens Rademacher, Bjorn Sandstede, Arnd Scheel Physica D 229 166-183, 2007)

1 solve numerically for the periodic waveby continuation in c from a Hopf bifnpoint in the travelling wave eqns

0 = Uzz + cUz + f (U, W )

0 = νWz + cWz + g(U, W ) (z = x − ct)

Conclusions

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

λu = uzz + cuz + fu(U, W )u + fw (U, W )w , u(0) = u(L)eiγ

λw = νwz + cwz + gu(U, W )u + gw (U, W )w , w(0) = w(L)eiγ

Conclusions

3 continue the eigenfunction equationsnumerically in γ, starting from each ofthe periodic eigenvalues

λu = uzz + cuz + fu(U, W )u + fw (U, W )w , u(0) = u(L)eiγ

λw = νwz + cwz + gu(U, W )u + gw (U, W )w , w(0) = w(L)eiγ

Conclusions

3 continue the eigenfunction equationsnumerically in γ, starting from each ofthe periodic eigenvalues

This gives the eigenvalue spectrum, and hence (in)stability

Conclusions

STABLE

Eckhausinstability

UNSTABLE

This gives the eigenvalue spectrum, and hence (in)stability

Conclusions

Stability in a Parameter Plane

By following this procedure at each point on a grid in parameterspace, regions of stability/instability can be determined.

In fact, stable/unstable boundaries can be computed accuratelyby numerical continuation of the point at which

Reλ = Imλ = γ = ∂2Reλ/∂γ2 = 0

(Eckhaus instability point)

Conclusions

Stability in a Parameter Plane

Conclusions

Pattern Stability: The Key Result

Key Result

Many of the possible patterns areunstable and thus will never be seen.

However, for a wide range of rainfalllevels, there are multiple stablepatterns.

Conclusions

Hysteresis

The existence of multiple stablepatterns raises the possibility ofhysteresis

We consider slow variations in therainfall parameter A

Parameters correspond to grass,and the rainfall range corresponds to130–930 mm/year

Conclusions

Hysteresis

<< Mode 5 >> <<<<< Mode 1 >>>>> < Mode 3 >

Conclusions

Hysteresis

<< Mode 5 >> <<<<< Mode 1 >>>>> < Mode 3 >

Wavelength vs Rainfall

Conclusions

Predictions of Pattern WavelengthReferences

Outline

4 Pattern Stability

5 Conclusions

Conclusions

Predictions of Pattern Wavelength

In general, pattern wavelength depends on initialconditions

When vegetation stripes arise from homogeneousvegetation via a decrease in rainfall, pattern wavelengthwill remain at its bifurcating value.

Conclusions

Predictions of Pattern Wavelength

In general, pattern wavelength depends on initialconditions

When vegetation stripes arise from homogeneousvegetation via a decrease in rainfall, pattern wavelengthwill remain at its bifurcating value.

Wavelength =

Conclusions

References

J.A. Sherratt: An analysis of vegetation stripe formation in semi-aridlandscapes. J. Math. Biol. 51, 183-197 (2005).

J.A. Sherratt, G.J. Lord: Nonlinear dynamics and pattern bifurcationsin a model for vegetation stripes in semi-aridenvironments. Theor. Pop. Biol. 71, 1-11 (2007).

J.A. Sherratt: Pattern solutions of the Klausmeier model for bandedvegetation in semi-arid environments I. Nonlinearity 23,2657-2675 (2010).

J.A. Sherratt: Pattern solutions of the Klausmeier model for bandedvegetation in semi-arid environments II. Patterns withthe largest possible propagation speeds. Submitted.

J.A. Sherratt: Pattern solutions of the Klausmeier model for bandedvegetation in semi-arid environments III. The transitionbetween homoclinic solutions. Submitted.

Conclusions

List of Frames

Vegetation Pattern Formation

Two Key Ecological Questions

2 The Mathematical ModelMathematical Model of KlausmeierTypical Solution of the ModelHomogeneous Steady States

Shortcomings of Linear Stability Analysis

Travelling Wave Equations

When do Patterns Form?Pattern Formation for Low Rainfall

4 Pattern StabilityThe Eigenvalue Problem

Numerical Calculation of Eigenvalue Spectrum

Stability in a Parameter PlaneHysteresis

5 ConclusionsPredictions of Pattern Wavelength

References

nonlinear dynamics of vegetation patterns in...

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