StripePatternSelectionbyAdvectiveReaction-DiffusionSystems:ResilienceofBandedVegetationonSlopes

Eric Siero1, Arjen Doelman1, Maarten Eppinga2, Jens Rademacher3, Max Rietkerk2, Koen Siteur21Leiden University, 2Utrecht University, 3University of Bremen

IntroductionIn water-limited arid ecosystems, where water distribution and infil-tration play a vital role, various models have been set up to explainvegetation patterning. On sloped terrains observations of vegetationaligned in bands along contours are widespread. On this poster weconsider the appearance, stability and bifurcations of 2D stripe orband patterns in an arid ecosystem model.

Model & previous workWe study an extension of themodel proposed by Klausmeier[1] akin to the Gray-Scott model,a two-component (water, veg-etation) system of reaction-advection-diffusion equations onthe plane R2:

wt =d∆w+2cwx + a− w − wn2

nt = ∆n −mn+ wn2,

where ∆ = ∂∂x2 +∂

∂y2 .We choose d = 500, the parame-

ter c measures advection of waterdownhill, a ≥ 0 models rainfalland m = 0.45 is an effective deathrate. The terms ±wn2 model wa-ter uptake and −w evaporation.From weakly nonlinear analysisin 1D we know that a Turing(-Hopf) bifurcation occurs for a =aT , below which stripe patternsexist [2]. By numerical continu-ation the boundary of the stripepattern stability region, Busse bal-loon, can be computed [3].

Research questionsMotivated by studies in one space dimension [2,4], whose 1D patternsideally represent stripe patterns by trivially extending them into asecond space dimension, we pose the following questions:• Which 1D stable patterns extend to 2D stable stripe patterns?• What 2D destabilization mechanisms appear?• What is the influence of the advection c on stripe patterns?

Stripe pattern formationIn [5] we prove a ‘Squire theo-rem’ for a class of two-componentreaction-advection-diffusion sys-tems that includes our model,showing onset of pattern forma-tion in 2D is due to 1D instabilitiesin the direction of advection, nat-urally leading to stripe patterns.In conjunction we prove mono-tone increase of the locus of theTuring-Hopf instability aT in c.

- 0.5

0

0.5k - 0.5

0

0.5

{

- 0.1

0

0.1

ReH ΛL

For 2c = 182.5. Growth rate Re(λ) ofperturbations of the homogeneous statewith (x, y)-wavenumbers k resp. ` afterTuring-Hopf, showing dominance of per-turbations with ` = 0.

Stripe pattern destabilizationFor c = 0 we prove transverseinstability of long-wavelengthstripe patterns through an Evansfunction approach in [5].For general c, 2D perturba-tions (w̃, ñ) of a stripe pat-tern with x-wavenumber κ (ina comoving frame) are decom-posed in products w̃(ξ, y; γ, `) =w̃1(ξ; γ)w̃2(y; `), and similarly forñ. Here γ is on the unit circle,` ∈ R and

w̃1

(ξ +

2π

κ; γ

)=γw̃1(ξ; γ)

w̃2(y; `) =ei`y.

Special cases are:

Stripe-stripe transition. Casewhere ` = 0. Sideband instabil-ity: curvature of the spectrum at-tached to the origin changes sign.

Stripe-rectangle breakup. Casewhere γ = 1. Transverse desta-bilization with same phase inneighboring stripes.

Stripe-hexagon breakup. Casewhere γ = −1. Transverse desta-bilization with opposite phase inneighboring stripes.

For 2c = 182.5. From left to right: stripe-hexagon bifurcation diagram with branchof homogeneous equilibrium (red), bifurcating stripe patterns (blue), subcritically bi-furcating hexagon pattern (black); stripe pattern; destabilizing perturbation at stripe-hexagon bifurcation; hexagon pattern at a ≈ 1.2. Solution plots extend periodically inboth directions.

2D Busse balloons of stripe patterns

c = 0 2c = 182.5

2c = 365 2c = 500

Union of all colored regions: stripe patterns that are 1D stable. Blue and green: stabilityagainst stripe-rectangle resp. stripe-hexagon breakup. Full 2D stable stripe patternsare colored dark-green (teal). During simulations with decreasing a (da/dt = −10−5)the wavenumber is indicated with pink if the system resides in a stripe pattern state,purple arrows in between indicate transient dynamics or residence in 2D states.

Simulation with decreasing rainfall

a = 4.5 a = 4.4 a = 2.5 a = 1.5 a = 1 a = 0.72

a = 0.71 a = 0.7 a = 0.65 a = 0.45 a = 0.3 a = 0.12

For 2c = 365. When a(t) is decreased (da/dt = −10−5) below Turing-Hopf, a stripepattern forms. The system then first undergoes three stripe-stripe transitions, beforebreakup occurs. At a ≈ 0.7 it briefly returns to a stripe pattern before breaking upagain, eventually reaching the bare desert state. With periodic boundary conditions.

Main conclusionThe ecological resilience, measured in the amount of variance in rain-fall a that is allowed before a stripe pattern wavenumber becomesunstable, is larger on steeper slopes (so for larger c).

AcknowledgmentSupported by the Netherlands Organisation for Scientific Research,within the Complexity program and by a NDNS+ travel grant.

References[1] C. Klausmeier. Regular and Irregular Patterns in Semiarid Vegetation. In Science 284 (1999)[2] S. van der Stelt, A. Doelman, G. Hek, J. Rademacher. Rise and Fall of Periodic Patterns for a

Generalized Klausmeier-Gray-Scott Model. In J. Nonlinear Sci 23 (2013)[3] J. Rademacher, B. Sandstede, A. Scheel. Computing absolute and essential spectra using con-

tinuation. In Physica D 229 (2007)[4] K. Siteur, E. Siero, M.B. Eppinga, J.D.M. Rademacher, A. Doelman, M. Rietkerk. Beyond Turing:

the response of patterned ecosystems to environmental change. In Ecological Complexity 20(2014)

[5] E. Siero, A. Doelman, M.B. Eppinga, J.D.M. Rademacher, M. Rietkerk, K. Siteur. Stripe patternselection in advective reaction-diffusion systems: resilience of banded vegetation on slopes.Submitted.