Computing simultaneous Partial differential equation using FIPY in python

I was trying to solve pde using fipy, since I'm new to it; I'm unable to debug error in my code. It is not giving any spatial pattern but just different monochromatic square(s) over a period of time.The equation which I'm trying to solve are-

(∂u(x,y,t))/∂t=G_1 (u,v)+d_11 ∇^2 u+d_12 ∇^2 v and

(∂v(x,y,t))/∂t=G_2 (u,v)+d_21 ∇^2 u+d_22 ∇^2 v

from fipy import *

nx=ny=200
dx=dy=0.25
L=dx*nx
dt=0.01
E=1.0
A=0.91881 #Alpha
B=0.0327 #Beta
D=0.05 #Delta
G=0.15 #Gamma
d11=0.1
d12=0.01
d21=0.5
d22=1.5

mesh =Grid2D(dx=dx,dy=dy,nx=nx,ny=ny)
u=CellVariable(name='u Variable',mesh=mesh) #pray
v=CellVariable(name='v Variable',mesh=mesh) #predator

u.setValue(GaussianNoiseVariable(mesh=mesh,mean=0.18,variance=0.01))
v.setValue(GaussianNoiseVariable(mesh=mesh,mean=0.6,variance=0.01))

eq_u=(TransientTerm(coeff=1,var=u)==u*(1-u)-(u*v*E)/(u+A*v)+ImplicitDiffusionTerm(coeff=d11,var=u)+ImplicitDiffusionTerm(coeff=d12,var=v))
eq_v=(TransientTerm(coeff=1,var=v)==B*v*(1-v)+(u*v*G)/(u+A*v)-D*v+ImplicitDiffusionTerm(coeff=d21,var=u)+ImplicitDiffusionTerm(coeff=d22,var=v))

#creating viewer
if __name__ == "__main__":
    viewer_u=Viewer(vars=u,datamin=0.16,datamax=0.18) 
    viewer_u.plot()
    viewer_v=Viewer(vars=v,datamin=0.0,datamax=0.4)
    viewer_v.plot()

#solving
steps=250
for step in range(steps):
    eq_u.solve(var=u,dt=dt)
    eq_v.solve(var=v,dt=dt)
    if __name__ == "__main__":
        viewer_u.plot()
        viewer_v.plot()

Solution

These equations are ODEs, not PDEs. FiPy may be able to solve them, but there are other tools much better optimized for solving ODEs, such as SciPy's odeint and scikits.odes.
I am not sure exactly what you mean by "It is not giving any pattern but just different monochromatic square(s) over a period of time." The random initial condition does not appear to evolve, but if I increase dt to 1.0, then V evolves to a uniform value of about 0.5. U does not appear to evolve for me, but that's a display glitch of some kind. V evolves to 0.598 and U evolves to 0.179. There is no pattern in space (if that's what you were expecting) because these equations have no spatial dependence; they only have partial derivatives in time, so there is absolutely nothing to cause the variables to flux from one cell to another.
You appear to get away with it here, but your variables should always be defined on the same mesh, rather than u_mesh for U and v_mesh for V.
No-flux boundary conditions are the default in FiPy, so there's no reason to specify U.faceGrad.constrain, etc.
- Furthermore, there are no fluxes in these equations (because they're ODEs), so setting boundary fluxes isn't meaningful.
Normally I would recommend making your equations more implicit, coupling them together, and sweeping, but none of this seems to have much effect on their evolution in this case.