I have the Sel'kov reaction diffusion model for glycolysis as follows:
\begin{eqnarray} u_t=D_uu_{xx}-u+av+u^2v\\ v_t=D_vv_{xx}+b-av-u^2v \end{eqnarray}
How can I obtain the values for $D_u$ and $D_v$ for a pattern formation via Turing criteria. The last equations model's glycolysis in 1D. In two dimensions:
\begin{eqnarray} u_t=D_u(u_{xx}+u_{yy})-u+av+u^2v\\ v_t=D_v(v_{xx}+v_{yy})+b-av-u^2v \end{eqnarray}
Using the same Turing criteria, how can I obtain the exact analitic forms for $D_u$ and $D_v$ diffusive constants. I'm using the Turing criteria as follows:
\begin{eqnarray} p(\sigma)=\mid J-\sigma I-\lambda D\mid=0\\ Re(p(\sigma))<0 \end{eqnarray}
where $J$ is the Jacobian matrix evaluated in the EP $(b,\frac{b}{a+b^2})$, $Re(p(\sigma))$ es the real part of the equation and $\lambda$ is the eigenvalue for the linealization and vectorization to obtain a lineal PDE wich is $\lambda=\frac{n^2\pi^2}{L^2}$ and $D$ is the diffusive matrix as follows $D=\begin{pmatrix} D_u & 0\\ 0 & D_v \end{pmatrix}$.
I've trying to obtain the relation for $D_u$ and $D_v$ and I can't do it, not in the $1D$ case or the $2D$ case. Please somebody helps me out.
Your system is more widely known as the Gray-Scott model; a nice overview and description of the Turing instability analysis can be found here.
To find out more about the relationship between the real part of the eigenvalues of the Jacobian and its trace and determinant, see my answer to this question.
I believe that, with these two ingredients, you'll be able to obtain the required bounds on $D_u$ and $D_v$.