Hi Hans: I found in the definition of the Laplacian in the second equation the index for the variable C it was C1 when it should be C2, cause the message of derivative of higher order. Second, I changed the way I wrote the Newmann boundary conditions at the frontier points. I fixed all of these and it is running now, however giving some errors of stiffness and that I should run with MinStepSize lower in order to capture the rapid variations. I did MinStepSize -> 0.005, takes a while for computing, it is not giving me yet what I am suppose to see, because the numerical error still persist, negative values for the concentration for example and exponential explosion of the numerical values, that does not make any sense. On the positive side, I have to say that the nonlinearity concern due to one of the functions is dissipating a bit.

miu = 0.0008; q = 0.55; eps1 = 0.04; kappa = 1; tmax = 100; (* time interval length *)

Lx = 1; (* x-side of the rectangle *)

Ly = 1; (* y-side of the rectangle *)

frontier = Rectangle[{-Lx, -Ly}, {Lx, Ly}]; C1ini = 1; C2ini = 1; freact[p, s] := p(1 - p) - ((p - miu)/(p + miu))q* s; (* reaction kinetics for eqn involving C1 *)

greact[p, s] := p - s; \ (* reaction kinetics for eqn involving C2 *)

eqn1 = D[C1[t, x, y], t] - D[C1[t, x, y], x, x] - D[C1[t, x, y], y, y] == freact[C1[t, x, y], C2[t, x, y]]/ eps1; (* diffusion-reaction eqn 1 *)

eqn2 = D[C2[t, x, y], t] - kappa(D[C1[t, x, y], x, x] + D[C1[t, x, y], y, y]) == greact[C1[t, x, y], C2[t, x, y]]; ( diffusion-reaction eqn 2 *)

InitialConditions = {C1[0, x, y] == C1ini, C2[0, x, y] == C2ini}; (* initial homogeneous concentration in the petri-dish *)

BoundaryConditions = {(D[C1[t, x, y], x] /. x -> -Lx) == (D[C2[t, x, y], x] /. x -> -Lx) == (D[C1[t, x, y], x] /. x -> Lx) == (D[C2[t, x, y], x] /. x -> Lx) == 0, (D[C1[t, x, y], y] /. y -> -Ly) == (D[C2[t, x, y], y] /. y -> -Ly) == (D[C1[t, x, y], y] /. y -> Ly) == (D[C2[t, x, y], y] /. y -> Ly) == 0}; \ (absence of flux through the boundaries ) solution = NDSolve[Join[{eqn1, eqn2}, InitialConditions, BoundaryConditions], {C1, C2}, {t, 0, tmax}, {x, -Lx, Lx}, {y, -Ly, Ly}]

Would the fact that freact[C1,C2] is a nonlinear function, create the problem for NDSolve? This is an example of partially nonlinear diffusion equation.