Review L15 Turing-Gierer-Meinhardt models Local excitation, global inhibition 2 2 2 2 22 x i Diak t i x a Da i a kr t a iii aaaa ? ? +?= ? ? ? ? +?+= ? ? γ γ a: concentration activator i: concentration inhibitor t: time x: position variables r a : basal activator synthesis rate k a , k i : rate constant for synthesis γ a ,γ i : decay rates D a , D i : diffusion constants constants (parameters) 1 2 2 2 2 22 x i Diak t i x a Da i a kr t a iii aaaa ? ? +?= ? ? ? ? +?+= ? ? γ γ choose dimensionless variable normalize 4 variables () 2 2 2 2 22 1 s I PIAQ τ I s A A I A R τ A ? ? +?= ? ? ? ? +?+= ? ? 2 )1( 1 += += RI RA homogeneous solution 0// =??=?? ts 2 homogeneous solution 0// =??=?? ts A 3 A s I I s stability of homogeneous solution ? ? ? ? ? ? ? ? ?+ + ? + ? = ? ? ? ? ? ? ? ? ? ?? QQR R R R R QQA I AR I AR )1(2 )1(1 1 2 1 2 2 2 2 trace < 0 det > 0 0 1 1 > < + ? Q Q R R ),('),( ),('),( ττ ττ sIIsI sAAsA += += inhomogeneous solution: 4 inhomogeneous solution 5 A s s A I I’(s,τ) ),('),( ),('),( ττ ττ sIIsI sAAsA += += I 2 2 2 2 2 ' '')1(2 ' ' ' )1( ' 1 1' s I PQIARQ I s A I R R A R RA ? ? +?+= ? ? ? ? + + ? + ? = ? ? τ τ ),('),( ),('),( ττ ττ sIIsI sAAsA += += )cos()( ? ),(' )cos()( ? ),(' l l s IsI s AsA ττ ττ = = trial solution: 6 7 )cos()( ? ),(' )cos()( ? ),(' l l s IsI s AsA ττ ττ = = A I s s A I I’(s,τ) ),('),( ),('),( ττ ττ sIIsI sAAsA += += I P QARQ d Id I R R A R R d Ad ?? )1(2 ? ? )1( ? 1 1 1 ? 2 22 ? ? ? ? ? ? +?+= + ? ? ? ? ? ? ? ? + ? = l l τ τ )cos()( ? ),(' )cos()( ? ),(' l l s IsI s AsA ττ ττ = = 0 1 1 1 0 1 21 1 1 22 22 < ? ? ? ? ? ? ? + ? ?+ > + + ? ? ? ? ? ? + ? ? ? ? ? ? ? + ? ? ll ll R RP Q R QRP Q R R stability inhomogeneous solution 1 1 + ? > R R P Q 8 1 1 + ? > R R Q homogeneous stability: inhomogeneous stability: 1 1 + ? > R R P Q s I I’(s,τ) I 9 s I I homogeneously stable: I relaxes back to previous value after small uniform disturbance s I I I’ relaxes back to after small spatial disturbance I inhomogeneously stable: 10