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In this work we study the effect of density dependent nonlinear diffusion onpattern formation in the Lengyel--Epstein system. Via the linear stabilityanalysis we determine both the Turing and the Hopf instability boundaries andwe show how nonlinear diffusion intensifies the tendency to pattern formation;%favors the mechanism of pattern formation with respect to the classical lineardiffusion case; in particular, unlike the case of classical linear diffusion,the Turing instability can occur even when diffusion of the inhibitor issignificantly slower than activator's one. In the Turing pattern region weperform the WNL multiple scales analysis to derive the equations for theamplitude of the stationary pattern, both in the supercritical and in thesubcritical case. Moreover, we compute the complex Ginzburg-Landau equation inthe vicinity of the Hopf bifurcation point as it gives a slow spatio-temporalmodulation of the phase and amplitude of the homogeneous oscillatory solution.

Turing Instability and Pattern Formation in an Activator-Inhibitor System with Nonlinear Diffusion