Effects of cross diffusion on Turing bifurcations in two-species reaction-transport systems

We study the Turing bifurcation in general two-species reaction-transport systems, where particle dispersal is governed by diffusion, including cross diffusion. By performing a linear stability analysis, we find conditions for the Turing instability and compare the results with the standard Turing conditions. We apply our results to two model systems, the Lengyel-Epstein model and the Brusselator, and find strong effects of cross diffusion on the Turing bifurcation.

Figure 1

${\theta }_{u,0}$ vs ${\phi }_u$ for the LE model with $a=50.0$ and $b=32.45$.

Figure 2

${\theta }_{0,v}$ vs ${\phi }_v$ for the LE model with $a=50.0$ and $b=32.45$.

Figure 3

${\theta }_{u,0}$ vs ${\phi }_u$ for the Brusselator with $a=5.0$ and $b=23.4$.

Figure 4

${\theta }_{0,v}$ vs ${\phi }_v$ for the Brusselator with $a=5.0$ and $b=23.4$.