Review Turing-Gierer-Meinhardt models Local excitation, global inhibition 22r+hq a+D a +d a concentration activator concentration inhibitor variables time X posItion basal activator synthesis rate Ka, k;: rate constant for synthesis constants decay rates (parameters) D.D. diffusion constants
Review Turing-Gierer-Meinhardt models Local excitation, global inhibition 2 2 2 2 2 2 x i k a i D t i x a a D i a r k t a i i i a a a a ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ γ γ a: concentration activator i: concentration inhibitor t: time x: position variables ra: basal activator synthesis rate ka, ki: rate constant for synthesis γa,γi : decay rates Da, Di: diffusion constants constants (parameters) 1
aa tk rat =Ka i+ d choose normalize dimensionless 4 variables variable oA 1+R A+ Q +p only one fixed homogeneous point, since both solution a and >0 =R+1 0/as=0/at=0 =(R+1)2
2 2 2 2 2 2 x i k a i D t i x a a D i a r k t a i i i a a a a ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ γ γ choose dimensionless variable normalize 4 variables ( ) 2 2 2 2 2 2 1 s I Q A I P τ I s A A I A R τ A ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ only one fixed point, since both A and I >0 2 ( 1) 1 = + = + I R A R homogeneous solution ∂ / ∂s = ∂ / ∂t = 0 2
homogeneous solution 0/as=0/ot=0 A
homogeneous solution ∂ / ∂s = ∂ / ∂t = 0 A 3 A s I I s
stability of homogeneous solution eRA Ra R R trace <0 R+1(R+1)2 det >o 2AO Q」L2(R+1)Q or in general R-1 real part of eigenvalues>0 <2 R+1 Q>0 inhomogeneous A(S,r)=A+A(S,7) solution: /(S,7)=I+/(S)
stability of homogeneous solution ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − + − + − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − R Q Q R R R R A Q Q I RA I RA 2 ( 1 ) 1 ( 1 ) 1 2 1 2 2 2 2 trace < 0 det > 0 0 1 1 > < + − Q Q R R or in general real part of eigenvalues > 0 ( , ) '( , ) ( , ) '( , ) τ τ τ τ I s I I s A s A A s = + inhomogeneous = + solution: 4
inhomogeneous solution A(S,)=A+A(S,) A /(S,)=I+/(S,7) s,τ
inhomogeneous solution 5 A s s A I I’(s,τ) ( , ) '( , ) ( , ) '( , ) τ τ τ τ I s I I s A s A A s = + = + I