An activator-inhibitor reaction diffusion system in dimensionless form is given by

$$u_{t}=u_{x x}+\frac{u^{2}}{v}-b u, \quad v_{t}=d v_{x x}+u^{2}-v$$

where $b$ and $d$ are positive constants. Which is the activitor and which the inhibitor? Determine the positive steady states and show, by an examination of the eigenvalues in a linear stability analysis of the spatially uniform situation, that the reaction kinetics is stable if $b<1$.

Determine the conditions for the steady state to be driven unstable by diffusion. Show that the parameter domain for diffusion-driven instability is given by $0<b<1$, $b d>3+2 \sqrt{2}$, and sketch the $(b, d)$ parameter space in which diffusion-driven instability occurs. Further show that at the bifurcation to such an instability the critical wave number $k_{c}$ is given by $k_{c}^{2}=(1+\sqrt{2}) / d$.