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l17_notes - Review Turing-Gierer-Meinhardt models Local...

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Unformatted text preview: Review Turing-Gierer-Meinhardt models Local excitation, global inhibition 2 2 2 2 2 2 x i D i a k t i x a D a i a k r t a i i i a a a a + = + + = 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 a , i : decay rates D a , D i : diffusion constants constants (parameters) 1 2 2 2 2 2 2 x i D i a k t i x a D a i a k r t a i i i a a a a + = + + = choose dimensionless variable normalize 4 variables ( ) 2 2 2 2 2 2 1 s I P I A Q I s A A I A R A + = + + = only one fixed point, since both A and I >0 2 ) 1 ( 1 + = + = R I R A homogeneous solution / / = = t s 2 homogeneous solution / / = = t s A 3 A s I I s stability of homogeneous solution + + + = Q Q R R R R R Q Q A I A R I A R ) 1 ( 2 ) 1 ( 1 1 2 1 2 2 2 2 trace < 0 det > 0 1 1 > < + Q Q R R or in general real part of eigenvalues > 0 ) , ( ' ) , ( ) , ( ' ) , ( s I I s I s A A s A + = + = inhomogeneous solution: 4 inhomogeneous solution 5 A s s A I I(s, ) ) , ( ' ) , ( ) , ( ' ) , ( s I I s I s A A s A + = + = I 2 2 2 2 2 ' ' ' ) 1 ( 2 ' ' ' ) 1 ( ' 1 1 ' s I P QI A R Q I s A I R R A R R A + + = + + + = ) , ( ' ) , ( ) , ( ' ) , ( s I I s I s A A s A + = + = ) cos( ) ( ) , ( ' ) cos( ) ( ) , ( ' l l s I s I s A s A = = trial solution: 6 7 ) cos( ) ( ) , ( ' ) cos( ) ( ) , ( ' l l s I s I s A s A = = A I s s A I I(s, ) ) , ( ' ) , ( ) , ( ' ) , ( s I I s I s A A s A + = + = I P Q A R Q d I d I R R A R R d A d ) 1 ( 2 ) 1 ( 1 1 1 2 2 2 + + = + + = l l ) cos( ) ( ) , ( ' ) cos( ) ( ) , ( ' l l s I s I s A s A = = 1 1 1 1 2 1 1 1 2 2 2 2 < + + > + + + + l l l l R R P Q R QR P Q R R stability inhomogeneous solution 1 1 + > R R P Q 8 1 1 + > R R Q homogeneous stability: stability against spatial distrubance: 1 1 + > R R P Q s I I(s, ) I 9 if P < 1 (D i <D a ), systems is always stable, against any perturbation both spatial and temporal s I I homogeneously stable:...
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This note was uploaded on 11/11/2011 for the course BIO 7.344 taught by Professor Bobsauer during the Spring '08 term at MIT.

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l17_notes - Review Turing-Gierer-Meinhardt models Local...

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