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.

Hans, thank you once again. The derivatives by coordinates "x" and "y" are first order while the Laplacian is second order. I am just writing Newmann boundary conditions different from NewmannValue. I wrote then explicitly. Because the reaction term mixes C1 and C2, I am not sure if NewmannValue would be appropriate in that case.

Hans, thank you for your time. Here is the block dedicated only to the reaction diffusion solution. Comments were added to the end of each line to be understood. The workflow is as follows: Definition of the region (rectangular region) Definition of the reaction functions for each specie C1 and C2 Definition of the Diffusion operator (Dt C1 - Laplacian[C1]==f[C1,C2]) eqn1=Derivative by time of C1 - Laplacian2D[C1]== reaction function f[C1,C2] eqn2=Deivative by time of C2 - Laplacian2D[C2]==reaction function g[C1,C2] NDSolve is called for solution.

Hans, thank you for your time. Here is the block dedicated only to the reaction diffusion solution. Comments were added to the end of each line to be understood. The workflow is as follows: Definition of the region (rectangular region) Definition of the reaction functions for each specie C1 and C2 Definition of the Diffusion operator (Dt C1 - Laplacian[C1]==f[C1,C2]) eqn1=Derivative by time of C1 - Laplacian2D[C1]== reaction function f[C1,C2] eqn2=Deivative by time of C2 - Laplacian2D[C2]==reaction function g[C1,C2] NDSolve is called for solution.

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