I gave you the equations, and I need to have the figures by using matlab . Therefore, could you please give me the figures and matlab code that shows the figures . Thanks!

Problems of Diffusion 321 Finite pattern forming domains ow consider the general system (12.5.4) defined on a bounded do- J. To complete the formulation of the problem we require the case of one-space dimension we consider the domain to be the the elas well as initial conditions. That is, in addition to (12.5.4) we sme or(at), oa(e,t) satisty i(a,0) fi(x). (,0) f2(), h(x) and f2(x) boundary conditions here fnie) and fa a) are given concentrations. Also we impose the no fux (z,t)=0, = 0, 1, i = 1,2. (126.2) The reason for choosing zero flux boundary conditions is that we are primarily interested in self-organisation of patterns. No lux boundary conditions mean that there is to be no external input of morphogens. To develop an analysis of Turing driven instabilities in this case we proceed in precisely the same way as we did above for an infnite domain. The crucial point is that the perturbations di and d2, which solve the system (12.5.11). st now satisfy the conditions (r,t) 0, 0,1, i1,2 (12.6.3) A little thought shows that di, d2 must be of the form (12.5.13) where (12.6.4) for r 0,1. Consequently, rather than being arbitrary, the parameter k must take the discrete values k, n-1,..., where sin kl=0, i.e., kT1,2... n n (12.6.5) This observation is crucial; it is fundamental to deciding which patterns are selected. Proceeding precisely as before we find that the critical wave number ke is given by n22 1 (a11 a22 (12.6.6) which may or may not be satisfied for a given 1. However if one is allowed to vary I then it is possible to choose a mode characterised by n so that (12.6.6) 322 Differential Equations and Mathematical Biology is satisfied. In other words a critical pattern of a certain form is the size of the domain. More generally we see that spatial patterns evolve if we values of n so that H() 0. The is satisfed determines the modes of pattern selection. To see this suppose the domain size l is such that (12.6.8) is satisfied only for n- 1. The only unstable mode is oos and morphogen concentration number of integers n for which (1268) where ) is the positive root of (12.5.6). This unstable mode is the don. inant one which emerges as t increases. If we say that black corresponds to a concentration above the steady state cy,o and white corresponds to a concentration below c,o then we have the pattern shown in Figure 12.6.1. FIGURE 12.6.1: Morphogen pattern for n 1. Similarly if n = 2 is the only value of n for which the inequality (1268) Problems of Diffusion Ci 323 (2 3614 GURE 12.6.2 Morphogen pattern for n 2. holds then (12.6.9) becomes 4z2 2 (12.6.10) leading to the pattern shown in Figure 12.6.2. Which mode or combination of modes and hence patterns are selected de- pends on initial conditions (12.6.1) ) Now consider the two-dimensional domain 2 defined by 0S t10ys h, with rectangular boundary 812 on which no-flux boundary conditions are imposed Once again the theory developed above in the one-dimensional case is fol- lowed with only minor modifications. Most importantly we seek solutions of the linearised problem (12.5.11) of the form [2]-l:/"Ooskizosby. where the wave numbers ki and k2 are chosen so that dy and da satisfy the (12.6.11) boundary conditions: Odi =0, z=0, 1, 12.6.12) d =0, y=0,h, i 1,2. By the method of separation of variables, or otherwise, we find that nt (12.6.13) 324 Differential Equations and Mathematical Biology for integers m,n 1,2.. Now proceed precisely as before to see that if we define (12.6.14) then we again arrive at equations (12.5.16) and (12.5.22) with 2 simply re- placed by K2. The critical wave number K2 is given by 2 2 1(a12 (12.6.15) and modes characterised by m and n exist if they satisfy the inequalities h2 (12.6.16) 2Di D2 To illustrate possible modes, suppose the domain size is sufficiently large so that (12.6.16) holds for m 3, n 2. Then the pattern shown in Figure 12.6.3 is possible where the shaded areas indicate regions in which the morphogen concentration is above the steady state. (m-3, n 2) FIGURE 12.6.3: Morphogen pattern for m-3, n-2 The fundamental assumption of pattern formation via Turing diffusion driven instabilities is that the linearly unstable modes that grow exponen- tially in time will eventually be bounded by the nonlinear kinetic terms in (12.5.4). This is indeed the case. To prove mathematically that this happers one has to show that in the positive quadrant of morphogen space there is a