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MA256 Spatial Patterns Tutorial Problem 1: Consider the one-dimensional di↵usion equation on the whole of R . @u @2u =D 2 (1) @t @x with initial conditions u(x, 0) = x x is the Dirac delta function, which is zero except at x = 0 and whose integral is unity The solution of this initial value problem is given by ✓ ◆ 1 x2 u(x, t) = p exp (2) 4Dt 4⇡Dt Show by di↵erentiation that the function u(x, t) (eq 2) satisfies (eq 1) Hint: Some algebra can be avoided by using the fact that log u = 1 log(4⇡Dt) 2 x2 4Dt Problem 2: Consider the dimensionless Bruesselator model which can give rise to spatial patterns through a di↵usion-driven Turing instability. @u @t @v @t = = @2u @2u 2 = a (b + 1) u + u v + D u @x2 @x2 2 2 @ v @ v g (u, v) + Dv 2 = bu u2 v + Dv 2 @x @x f (u, v) + Du (3) (4) with parameters a, b, Du , and Dv all positive. a) Which term describes this model best: Activator-inhibitor model, enhanced degradation model, or substrate depletion model? b) ⇤ ⇤ Determine the steady ✓ @f state ◆(u , v ) of equations (3,4) and compute @f @v the Jacobian J = @u at (u⇤ , v ⇤ ). Considering its trace tr (J) @g @g @u @v and determinant det (J), what condition must b meet to obtain a stable equilibrium? 1 c) The existence of a stable fixed point in the absence of di↵usion as shown in b) is one condition for a di↵usion driven Turing instability. Furthermore we require Du p @g @f + Dv > 2 Du Dv ⇤ det (J) > 0 @v @u (5) u Using equation (5) and = D Dv , determine bcrit above which you expect stable spatial patterns to occur. d) Setting a = 2, Du = 4, and Dv = 16, obtain a numerical for qvalue @g @f Du @v +Dv @u bcrit and compute the corresponding wave number kcrit = 2Du Dv and the corresponding wavelength crit . e) Assume parameters as in d) and that b is slightly above bcrit so that an initial random pertubation grows into a stable spatial pattern. No-flux boundary conditions shall apply. For a domain of size L = 3 2 crit sketch all possible solutions for the spatial profiles you can expect for u (x) and v (x) . Label the x-axis in units of ⇡. On the y-axis show the numerical value of the steady state values u⇤ and v ⇤ which can be taken as 2. 2