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Hello, everybody.
I'm trying to replicate plots from
Yang et al. 2002. Spatial resonances and superposition patterns in a reactiondiffusion model with interacting Turing modes. Phys. Rev. 88(20)
using Mathematica, to explore specifically under what parameter values the pattern of Figure 3c is stable.
I am a biologist by training, and the notation of PDEs is a bit beyond me (I am also not sure how to define the initial conditions [perturbations of steady state]), but what I have so far to replicate the plots is (in Mathematica 7):
(* Fixed parameter values used to generate Figure 3c *)
a = 3;
Dx1 = 1.31;
Dy1 = 9.87;
Dx2 = 34;
Dy2 = 344.8999;
b = 6;
\[Alpha] = 1;
(* from equation 1, equations Structured as: Laplacian + interaction + Brusselator *)
e1 = D[c1[x1, y1, x2, y2, t], t] ==
Dx1*(D[c1[x1, y1, t], x1, x1] +
D[c2[x1, y1], y1, y1]) + \[Alpha]*(c3[x2, y2, t] 
c1[x1, y1, t]) +
a  (b + 1)*c1[x1, y1, t] + c1[x1, y1, t]^2*c2[x1, y1, t];
e2 = D[c2[x1, y1, x2, y2, t]] ==
Dy1*(D[c1[x1, y1, t], x1, x1] + D[c2[x1, y1, t], y1, y1]) +
\[Alpha]*(c4[x2, y2, t]  c2[x1, y1, t]) +
b*c1[x1, y1, t]  c1[x1, y1, t]^2*c2[x1, y1, t];
e3 = D[c3[x1, y1, x2, y2, t], t] ==
Dx2*(D[c3[x2, y2, t], x2, x2] +
D[c4[x2, y2, t], y2, y2]) + \[Alpha]*(c1[x1, y1, t] 
c3[x2, y2, t]) +
a  (b + 1)*c3[x2, y2, t] + c3[x2, y2, t]^2*c4[x2, y2, t];
e4 = D[c4[x1, y1, x2, y2, t], t] ==
Dy2*(D[c3[x2, y2, t], x2, x2] +
D[c2[x1, y1, t], y1, y1]) + \[Alpha]*(c2[x1, y1, t] 
c4[x2, y2, t]) + b*c3[x2, y2, t] 
c3[x2, y2, t]^2*c4[x2, y2, t];
InteractingBrusselator = NDSolve[{e1, e2, e3, e4},
{c1, c2, c3, c4}, {x1, x2,y1,y2,t}]
I get several errors, including one that says the equation c1[x1,y1,t] is not a function of all the variables. However, c1 and c2 define the interactions in one layer, and c3 and c4 define interactions in another layer. It is not clear to me how to proceed.
I don't need time dynamics for the equations, just the stable pattern, which should look something like the attached images from the software Ready.
Any help on setting up the system for plotting would be greatly appreciated.
John
I'm trying to replicate plots from
Yang et al. 2002. Spatial resonances and superposition patterns in a reactiondiffusion model with interacting Turing modes. Phys. Rev. 88(20)
using Mathematica, to explore specifically under what parameter values the pattern of Figure 3c is stable.
I am a biologist by training, and the notation of PDEs is a bit beyond me (I am also not sure how to define the initial conditions [perturbations of steady state]), but what I have so far to replicate the plots is (in Mathematica 7):
(* Fixed parameter values used to generate Figure 3c *)
a = 3;
Dx1 = 1.31;
Dy1 = 9.87;
Dx2 = 34;
Dy2 = 344.8999;
b = 6;
\[Alpha] = 1;
(* from equation 1, equations Structured as: Laplacian + interaction + Brusselator *)
e1 = D[c1[x1, y1, x2, y2, t], t] ==
Dx1*(D[c1[x1, y1, t], x1, x1] +
D[c2[x1, y1], y1, y1]) + \[Alpha]*(c3[x2, y2, t] 
c1[x1, y1, t]) +
a  (b + 1)*c1[x1, y1, t] + c1[x1, y1, t]^2*c2[x1, y1, t];
e2 = D[c2[x1, y1, x2, y2, t]] ==
Dy1*(D[c1[x1, y1, t], x1, x1] + D[c2[x1, y1, t], y1, y1]) +
\[Alpha]*(c4[x2, y2, t]  c2[x1, y1, t]) +
b*c1[x1, y1, t]  c1[x1, y1, t]^2*c2[x1, y1, t];
e3 = D[c3[x1, y1, x2, y2, t], t] ==
Dx2*(D[c3[x2, y2, t], x2, x2] +
D[c4[x2, y2, t], y2, y2]) + \[Alpha]*(c1[x1, y1, t] 
c3[x2, y2, t]) +
a  (b + 1)*c3[x2, y2, t] + c3[x2, y2, t]^2*c4[x2, y2, t];
e4 = D[c4[x1, y1, x2, y2, t], t] ==
Dy2*(D[c3[x2, y2, t], x2, x2] +
D[c2[x1, y1, t], y1, y1]) + \[Alpha]*(c2[x1, y1, t] 
c4[x2, y2, t]) + b*c3[x2, y2, t] 
c3[x2, y2, t]^2*c4[x2, y2, t];
InteractingBrusselator = NDSolve[{e1, e2, e3, e4},
{c1, c2, c3, c4}, {x1, x2,y1,y2,t}]
I get several errors, including one that says the equation c1[x1,y1,t] is not a function of all the variables. However, c1 and c2 define the interactions in one layer, and c3 and c4 define interactions in another layer. It is not clear to me how to proceed.
I don't need time dynamics for the equations, just the stable pattern, which should look something like the attached images from the software Ready.
Any help on setting up the system for plotting would be greatly appreciated.
John
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