Pattern formation and Turing instability. Gurarie Topics: - Pattern formation through symmetry breaing and loss of stability - Activator-inhibitor systems with diffusion Turing proposed a mechanism for growth and development of patterns (morphogenesis) in biological systems (embryonic development, et al). According to Turing - Active genes stimulate production/activation of chemical agents (morphogenes) - Chemical reactions alone is too symmetric for pattern generation - But diffusion-driven instabilities create initial patterns and those can lead to further development Basic mathematical questions: Q1: can diffusion be stabilizing factor for reactive system? (Answer: yes in 1 and some other cases) Q: can diffusion destabilize a reactive system? Under what conditions? What are the resulting patterns? Eamples of symmetry breaing, and patterns - Physical systems: Phase transition solid-fluid-gas, fluid & gas have large continuous symmetries (all rotations, translations), solid a rigid (discrete) crystalline symmetry. - Mathematics: Hopf bifurcation from stable equilibrium (symmetric?) -> limit cycle (pattern?) Turing analysis involves i) Symmetric (e.g. spatially uniform) equilibria ii) iii) Bifurcations in various parameters, e.g. diffusivity, domain size Activation/inhibition type reactions
Stabilizing diffusion For single reactant u( t, ) there is no Turing instability. ( ) R [ ] ( ) n R ut = u + f u ; on or 0, l + BC, or multi- u = u+ f u ; on or Ω+ BC Indeed, a uniform ( - independent) OE u = f ( u) t has either stable m f ( u ) = < 0, or unstable (m>0) equilibrium. The diffusion will maintain a stable one, and it can stabilize the unstable one. To chec it we tae the linearized problem about v t, = u t, u u, for ( ) ( ) v = v + mv; t ( ) m= f u (1) Solution on R / R n ( ξ ) mt /4 t e v( t, ) = v0 ( ξ) dξ () 4π t For m>0, v( t, ) grows with t Figure 1: Point-source solution () on R for m>0
For m<0, all v( t, ) decay eponentially ( (, ) Finite interval [0,l] v t < Ce mt ). Use Neumann BC (no flu) and eigenfunction epansion π µ = µ 0; on [ 0, l ] + = l ; = 0,1,... 0, l= 0 π ( ) = cos l (3) The eigenvalues of BV problem (1) are ( ) unstable case get bifurcation values at ( π ) ( π ) µ = π / l + m. So stable case (m<0) remains stable, For / l = m; / l = m;... The resulting would be unstable mode (pattern) is the corresponding cos-fourier mode (3). Turing instability for double-diffusive systems A pair of reaction-diffusion species obeys a coupled system (, ) (, ) ut = u 1 + f u v vt = v + g uv (4) or its multi- version in (,y, ) space, with Neumann BC. So symmetric (-independent) (u,v) solve a pure reaction OS with equilibrium (, ) u v, and acobian fu f v = gu gv Then we get a linearized system for (, ) (, ) ; (, ) (, ) u t u t u v t v t v = = by u t = 1 u + au + bv a b ; with acobian = v t = v + cu + dv c d or in vector notation
1 u Ut = U UU ; ( t, ) + = v (5) Stability of system (5) depends on eigenvalues of a matri-differential operator 1 L= + (6) We denote by µ eigenvalues of scalar laplacian L 0 = or. + =, ( ) Eamples of eigenvalue problem for Laplacian: µ 0 { µ :? Free space RR ; Finite interval [ 0,l ] irichlet: Neuman: Periodic (n) Square, bo 0 < < a;0< y< b n 0 µ = < ( ) n =,,... R ; = 1 { µ ( / ) : 1,... = π l = { µ ( / ) : 0,1,... = π l = { µ ( / ) : = π l < < Product type: µ ( ) { µ ( ) ( ) m, = π / a + m/ b : m, = 1,... i { ( ) = e or { cos,sin π = sin l π = cos l { ( ) i / l = e π m or {cos, sin π πy = sin sin a b Polar dis (annulus) 0 < r < a; 0< θ < π product type (radial angular modes): µ = m ( z / a) m ( θ) = ( ), r, z r/ a e m m m ± imθ 1.
Having compute eigenvalues of L 0 we can pass to matri-operator (6). We search for eigenmodes of ( ) X L in the form scalar vector =, - eigenmode of Laplace L 0. It gives 1 λ I + µ X = 0 ; a b = c d The eigenvalue problem for "diagonalized" operator L is then reduced to matri-problem for a family of matrices 1 a µ 1 b B( µ, ) = µ = c d µ (7) Specifically, eigenvalues of L λ( L) λ ( µ, ) =eigen B( µ, ) where { µ vary over the entire spectrum of Laplacian. = (8) Turing instability analysis Basic question: given positive Laplace eigens{ µ can a stable chemistry (acobian ) producecan unstable matri B ( µ ) for specific diffusivities 1,, domain size l, or norm of (proportional to reaction rates in f,g). Stable Stable B-? tr a d 0 Table 1: stability test = + < µ ( ) det ad bc 0 trb= + + tr < 0 - always true! 1 = > det B = µ ( a + d ) µ + ( ad bc) = Q( µ ) 1 The answer depends on quadratic function Q ( µ ) -? Specifically, Case 1: Having symmetric (negative-definite), matri function B ( µ ) (7) is negativedefinite for all 0 µ >. So (, ) a b = b d a1 a λ µ remain negative, and diffusion leads to stabilization (no patterns).
Case : Scalar diffusivity 1 = =. Clearly, for stable and µ > 0, eigens B = eigens() - µ are more negative (stable). In general case (Turing) it depends on coefficients of quadratic function Q( µ ) det B( µ ) = in Table 1. Namely, (i) Q ( µ ) > 0 for all µ means stable B ( µ ), hence stable L; (ii) Q ( µ ) changing sign for some µ > 0, gives unstable B ( µ ), hence unstable mode for L. Turing conditions for unstable (negative) p ( µ ) are ( T1) a = a + d > 0 - negative slope Q 0 1 1 ( ) ( ) ( T ) a + d 4 ad bc - discriminant (9) 1 1 ( ) From (T1) and tr() <0 => coefficients (a,d) should have opposite signs, then (b,c) are also opposite. The only choice are Case 3: Activation-inhibition systems: (i) u grows, v decays; (ii) u inhibits v and v enhances u, or vice versa et B 5 0 + + + = ; or + M 1.1 15 10 M.9 5 M.7 5 10 15 0 5 30 ξ = / Figure : Cases of stable and unstable det(b) in terms of parameter 1 One can rewrite Turing condition (9) for a ± b = c d (a,b,c,d positive), as
a / ξ + dξ bc; with parameter ξ= / (10) 1 Combined set conditions (tr( )<0, det()>0, tr(b)<0, det(b)<0) gives a/ d 1; ξ = / 1 bc bc ad d It sets a limit (Fig.3) of ratio 1/ to get unstable Turing mode. Figure 3: Parameter range for 1 / to get Turing instability A positive (unstable) eigenvalue λ ( L) for some µ within the unstable interval of Fig. will give a spatial (eigenmode) part ( ) π ( e.g. cos l for π =, with µ = l ) to seed the follow up development of a spatial pattern for nonlinear reaction-diffusion (4), spawn by its symmetric unstable equilibrium (, ) u v. Bifurcation analysis in terms of ais. It can be saddle-node or Hopf bifurcation(?). i, depends on how parabola σ ( µ ) of Fig. will hit the positive µ