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 r2N @τ 1 2 @W = G(W; N) + D r2W @τ 2 2 where @2 @2 r2 = + 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) = RJW N 2 − MN G(W; N) = A − LW − RW N 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 + µr2n @t @w = 1 + β(1 − wn2) − w + r2w @t where @2 @2 r2 = + @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 + " n cos(qx)eσt + O("2) Let 1 11 1 σt 2 w(x; t) = 1 + "1w11cos(qx)e + O("1) 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 p 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.