I'm considering the following reaction-diffusion system:
$ \frac{\partial u}{\partial t} = f(u,v)+ D_1 \frac{d^2 u}{dx^2} $
$ \frac{\partial v}{\partial t} = g(u,v)+ D_2 \frac{d^2 v}{dx^2} $
where f and g describe the reaction kinetics, $ D_1 $ and $ D_2 $ are positive constants.
I have been looking at the conditions for diffusion-driven instability which are as follows:
$ f_u + g_v < 0 $
$ f_u g_v - f_v g_u > 0 $
$ D_2 f_u - D_1 g_v > 0 $
$ (D_2 f_u - D_1 g_v)^2 > 4D_1 D_2 (f_u g_v - f_v g_u) $
My notes then say that under these conditions bifurcation to solutions oscillating in time as well as space (called Hopf bifurcation) is not possible.
I don't really understand this and really appreciate someone explaining why this is the case / proving it.
Thanks