Vegetative Turing Pattern Formation: A Historical Perspective

Bonni J. Kealy* and David J. Wollkind

Washington State University Department of Mathematics Pullman, WA 99164-3113

Joint Mathematics Meetings AMS-ALS Special Session on the Life and Legacy of Alan Turing January 2012

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Alan Turing

Born Alan Mathison Turing 23 June 1912 Maida Vale, London, England Died 7 June 1954 (aged 41) Wilmslow, Chesire, England

Figure: Turing at the time of his election to Fellowship of the Royal Society

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Alan Turing: World Class Distance Runner

Alan Turing achieved world-class Marathon standards. His best time of 2 hours, 46 minutes, 3 seconds, was only 11 minutes slower than the winner in the 1948 Olympic Games. In a 1948 cross-country race he finished ahead of Tom Richards who was to win the silver medal in the Olympics.

Figure: From The Times, 25 August 1947 Figure: Running in 1946

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Paper: The Chemical Basis of Morphogenesis

Figure: An example of a ‘dappled’ Figure: Philosophical Transactions of pattern as resulting from a type (a) the Royal Society of London. Series B, morphogen system. A marker of unit Biological Sciences, Vol. 237, No. 641. length is shown. (Aug. 14, 1952), pp. 37-72.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Examples

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Research Area

The vegetative pattern formation in arid flat environments modeled by an interaction-diffusion model system is investigated by nonlinear stability analyses applied to the model system. Previous Work: Other studies of the Turing-like patterns that occur in semi-arid and arid environments (Africa, Australia, the Americas, and Asia) Arid ecosystems are most prominent in self-organized patchiness Flat ground yields stationary irregular mosaics (primarily stripes) Patterns based on vegetation variation (grass, shrubs, trees), vegetation density, Figure: Lefever and rainfall, soil, and water infiltration Lejune (1997)

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Arid/Semi-Arid Regions of the World

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Interaction-Diffusion Model System

Our system is an extension of a pair of partial differential equations found in Klausmeier (1999). Let (X,Y ) be defined on an infinite two-dimensional domain. Define W = surface water, N = plant biomass, τ = time. ∂N = F (W, N) + D ∇2N ∂τ 1 2 ∂W = G(W, N) + D ∇2W ∂τ 2 2 where ∂2 ∂2 ∇2 = + 2 ∂X2 ∂Y 2

D1 and D2 are the diffusion coefficients for plants and water, respectively

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective For simplicity in our techniques, we define

F (W, N) = RJWN 2 − MN

G(W, N) = A − LW − RWN 2 Plants take up water at rate Rf(W )g(N)N, where f(W ) is the functional response of plants to water and g(N) describes how plants increase water infiltration For simplicity we take f(W ) = W and g(N) = N (linear) J is the yield of plant biomass per unit water consumed MN is the density-independent mortality and maintenance rate through which plant biomass is lost Water is supplied uniformly at rate A and is lost due to evaporation at rate LW

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Bare Ground Equilibrium Point

We find the equilibrium points of this system by considering F (We,Ne) = 0 and G(We,Ne) = 0, which yields two possible stable points: A N ≡ 0,W ≡ L corresponding to a bare ground or no vegetation situation that always exists and is always stable.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Homogeneous Vegetation Equilibrium Point

" #1/2 AJ  AJ 2  L  N ≡ N = + − e 2M 2M R M W ≡ We = RJNe corresponding to a situation of homogeneous vegetation that  AJ 2 L exists when ≥ and the stability of which is the 2M R primary focus of this research.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Nondimensionalizing

N W  L 1/2 n = , w = , t = Lτ, (x, y) = (X,Y ) Ne We D2 AR1/2J a = (nondimensional rate of precipitation) L3/2 M α = (nondimensional rate of plant loss) L D µ = 1 (relates diffusion coefficients) D2 a ν = (relates precipitation and plant loss) 2α R p β = N 2 = (ν + ν2 − 1)2 (plant density) L e Note: where ν ≥ 1 ⇒ β ≥ 1 ⇒ a ≥ 2α

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Nondimensionalized System

Our new system: ∂n = αwn2 − αn + µ∇2n ∂t ∂w = 1 + β(1 − wn2) − w + ∇2w ∂t where ∂2 ∂2 ∇2 = + ∂x2 ∂y2 Note: The equilibrium point for the nondimensionalized system is (1, 1).

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective One-Dimensional Analysis

n(x, t) = 1 + ε n cos(qx)eσt + O(ε2) Let 1 11 1 σt 2 w(x, t) = 1 + ε1w11cos(qx)e + O(ε1) where q ≥ 0 is the wavenumber σ is the growth rate of the linear perturbation quantities 2 2 n11 and w11 satisfy n11 + w11 6= 0 Then

σ2 +[(1+µ)q2 +β +1−α]σ +µq4 +[(β +1)µ−α]q2 +α(β −1) = 0

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Linear Stability Analysis

To guarantee the onset of a diffusive instability from a uniform steady state that is stable to linear homogeneous perturbations, we require β + 1 − α > 0 and α(β − 1) > 0 and (q2 + 1)(α − µq2) β < β (q2; α, µ) = 0 µq2 + α Critical Values:

−α + p2α(α − µ) (3α − µ) − 2p2α(α − µ) q2 = , β = c µ c µ

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Instability Graph

β

2 ()qc, β c •

Unstable

q2 ()0,1 ()α µ− 2,1 Note: β > 1, α > 2µ

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective (3α − µ) − 2p2α(α − µ) Plot β = c µ

β 2

1.9

1.8

1.7

1.6

1.5

1.4 Unstable 1.3

1.2

1.1

1 α 2 3 4 5 6 7 8 9 10 x 10 −3 Note: β > 1, α > 2µ, µ = 0.001

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective 1  1/2 −1/2 Plot a = 2α cosh 2 ln(βc) = α(βc + βc )

a 120 µ = 0.001

100

80

60

40

Unstable 20

0 0 0.5 1 1.5 2 2.5 3 3.5 4 α

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Stuart-Watson Analysis     n(x, t) njk Defining v(x, t) = and vjk = we perform a w(x, t) wjk weakly nonlinear stability analysis by seeking a real Stuart-Watson type solution of the form

2 v(x, t) ∼ v00 + A1(t)v11 cos(qcx) + A1(t)[v20 + v22 cos(2qcx)] 3 + A1(t)[v31 cos(qcx) + v33 cos(3qcx)]

where n00 = w00 = 1 and the amplitude function A1(t) satisfies the Landau equation dA 1 ∼ σA − a A3 dt 1 1 1 Substituting this and expanding the interaction terms in a Taylor

series about A1 ≡ 0, we obtain a sequence of vector systems. We find that the O(A1) system is equivalent to the linear stability problem 2 3 with q ≡ qc. Using the appropriate O(A1), O(A1), and O(A1) systems, we obtain a representation for Landau constant a1. Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Nonlinear Stability Analysis

a a 1 0.8 1 0.02 µ = 0.001 µ = 0.001 0.7

0.6 0.01

0.5

0.4 0 α0 = 0.0101 0.3

0.2 −0.01

0.1

0 α −0.02 α 0 0.5 1 1.5 2 2.5 3 3.5 4 0.002 0.006 0.01 0.014 0.018 √ 2 10 2 − 7 −(qc + 1)αr31 a1 ∼ α (red ‘*’) vs. a1 = (blue) 36 β + 1 − α + (1 + µ)q2 c β=βc 2 r31 = 3n11w11/4 + (n11 + w11)(2n20 + n22) + n11(2w20 + w22)

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Nonlinear Stability Analysis

Note: Biologically meaningful values: αtree = 0.045 and αgrass = 0.45 Satisfies constraints: α > α0(µ) for 0 < µ ≤ 0.001 Hence the zero of a1 is irrelevant and we will consider a1 positive. Thus, the amplitude function A1(t) undergoes a standard supercritical pitchfork bifurcation at β = βc. For β > βc, the undisturbed state A1 = 0 is stable, yielding a uniform homogeneous vegetative pattern n(x, t) ∼ 1. 1/2 For 1 < β < βc, A1 = Ae = (σc/a1) > 0 is stable, yielding a periodic one-dimensional vegetative pattern consisting of stationary parallel stripes n(x, t) ∼ ne(x) = 1 + Ae cos(2πx/λc) of characteristic wavelength λc = 2π/qc.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective One-Dimensional Pattern Formation Results

a 20

18 Homogeneous 16 Vegetation 14 Stripes 12

10

Rainfall 8

6

4

2 Bare Ground

0 α 0 0.5 1 1.5 2 2.5 3 3.5 4 Plant Loss  1  1/2 −1/2 a = 2α cosh 2 ln(βc) = α(βc + βc ), a = 2α, µ = 0.001

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Wavelength

α p 2 p β = βc(α) ⇒ = 3β − 1 + 2 2β(β − 1) ⇒ q = β − 1 + 2β(β − 1) µ c

 1/2 2π ∗ D2 2 λc = , λc = λc = 95.5λc, for D2 = 100m /d, L = 4/yr qc L a λ = 0.50 λ = 0.75 20 c c Homogeneous λ =1.00 Vegetation c 16

λc =1.25 Stripes λ =1.57 12 c λc =1.75

Rainfall 8

4 Bare Ground

0 α 1 2 3 4 Plant Loss Note: The line a = 2α corresponds to λc → ∞.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Comparison: Lefever and Lejune (1997)

n

1+ Ae

n∼ ne

1 x

A n =1 − e 2

1− Ae

λ λ 2λ 0 λc c c c λ 4 3 2 3 c

 2πx  σ Lefever and Lejune (1997) 2 ne = 1 + Ae cos , Ae = λc a1 ∗ a α λc λc 1 0.325 1.57 150 1.386 0.450 1.57 150

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Two-Dimensional Nonlinear Stability Analysis

To determine the solvability conditions, introducing a transformation, we work with a reduced form of the hexagonal-planform solution √ n(x, y, t) − 1 ∼ A1(t) cos (qcx) + B1(t) cos (qcx/2) cos ( 3qcy/2)

with an analogous expansion for w(x, y, t), where

dA 1 ∼ σA − a B2 − A (a A2 + a B2) dt 1 0 1 1 1 1 2 1 dB 1 ∼ σB − 4a A B − B [2a A2 + (a + 2a )B2/4] dt 1 0 1 1 1 2 1 1 2 1 In the same manner as the one-dimensional expansions, we find representations for Landau constants a0 and a2.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Orbital Stability Behavior

To catalogue and summarize the orbital stability behavior of the critical points, we employ the quantities

2 2 2 σ−1 = −4a0/(a1 + 4a2), σ1 = 16a1a0/(2a2 − a1) ,

2 2 σ2 = 32(a1 + a2)a0/(2a2 − a1) There exist equivalence classes I, II, and III, of the critical points whose existence are dependent on the assumption that a1, a1 + 4a2 > 0 and whose stability behavior depends on the signs of a0 and 2a2 − a1, with the further assumption that a1 + a2 > 0.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Behavior of the Landau Constants

a′ 5 a1+ 4 a 2

4

3

2 a1+ a 2

2a2− a 1 1 a2 a1 a 0 0

−1 α 0 0.5 1 1.5 2 2.5 3 3.5 4

a0 (black), a1 (red), a2 (cyan), a1 + a2 (blue), a1 + 4a2 (green), 2a2 − a1 (magenta) 2 Note: |a0| << (a1 + 4a2)

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Orbital Stability Behavior

The orbital stability conditions for the critical points can be posed in terms of σ and are summarized in the following table:

a0 2a2 − a1 Stable Structures − + −, 0 III for σ > σ−1 − + + III for σ−1 < σ < σ2, II for σ > σ1 0 − III± for σ > 0 0 + II for σ > 0 + − + III for σ−1 < σ < σ2, II for σ > σ1 + − −, 0 III for σ > σ−1

Since a0, 2a2 − a1 > 0, we restrict our attention to the highlighted second row.

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Vegetative Pattern Formations for Hexagonal Planform Analysis

− II:Vegetative Stripes III+ :Vegetative Spots III :Vegetative Gaps Figures: Boonkorkuea et al. (2010)

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Hexagonal Planform Nonlinear Stability Analysis

β β 700 σ −1 βc β 600 σ1

500

βσ 2 400

300

200

100

0

−100 α 0 0.5 1 1.5 2 2.5 3 3.5 4

βσ−1 (red), βσ1 (blue), βσ2 (green), βc (black) 2 [α+µ−(1−µ)σc] where β = βσc (α; µ) = √ µ[3α−µ+(1−µ)σc+2 2α{α−µ+(1−µ)σc}]

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Two-Dimensional Pattern Formation Results

1/2 −1/2 a = α(β + β ) where aσ−1 (red), a a a c σ1 20 aσ1 (blue), aσ2 (green), ac (black), with aσ 18 −1 a = 2α (magenta) a 16 σ2 14 Note: aσ−1 , ac, and aσ1 are visibly 12 coincident. Hence, for biologically 10

8 meaningful parameter values, we shall

6 a = 2α take aσ−1 = ac = aσ1 . 4

2

0 α 0 0.5 1 1.5 2 2.5 3 3.5 4

Aridity Classification Scheme a range Stable Pattern for α = 0.045 and µ = 0.001 a > ac Homogeneous Dry-subhumid (a > 0.1442): Homogeneous Semiarid (0.1198 < a < 0.1442): Gaps and Stripes aσ2 < a < aσc Gaps and Stripes Arid (0.0900 < a < 0.1198): Stripes 2α < a < aσ2 Stripes 0 < a < 2α Bare Ground Hyperarid (0 < a < 0.0900): Bare Ground

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Examples of Labyrinth Tiger Bush

Tree Labryrinth (stripe regime) Leopard Bush (Spots)

Irregular Shrub Mosaic (stripe and gap overlap Tiger Bush regime) (migrating stripes)

Grass Labyrinth Shrub net or web (stripe regime) (gap regime)

Figures: Rietkerk et al. (2004) Bushy Patterns: Niger (A)-(B), Israel (C), French Guiana (E) Peatlands: Western Siberia (F)-(G) Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Chlorine Dioxide-Iodine-Malonic Acid (CDIMA) Reaction

− + MA + I2 → IMA + I + H , − − 1 ClO2 + I → ClO2 + 2 I2, − − + − ClO2 + 4I + 4H → Cl + 2I2 + 2H2O, − − S + I2 + I SI3 .

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Chemical Patterns

Honeycomb Hexagons

Stripes Stripes

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective System ∂x µ ∂y = F (x, y; α) + ∇2x, = β(1 + K)G(x, y) + ∇2y, ∂t 1 + K 2 ∂t 2

where

F (x, y; α) = 5α − x − 4xy/(1 + x2),G(x, y) = x − xy/(1 + x2),

2 P2 2 2 ∇2 = i=1 ∂ /∂ri

with

F (x0, y0; α) = G(x0, y0) = 0

2 ⇒ x0 = x0(α) = α, y0 = y0(α) = 1 + α .

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Patterns

Nets Stripes Spots

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective Striped and Hexagonal Patterns in α-β Space

Figure: µ = 1, K = 100, α ∼ [MA], β ∼ [I2], K ∼ [S]

Bonni J. Kealy* and David J. Wollkind Washington State University Vegetative Turing Pattern Formation: A Historical Perspective