Figure 2
Turing patterns for the Mimura-Murray model on the Cartesian product GWH.
Panel (a): G exhibits Turing patterns, the asymptotic concentration of species u (circles, blue online) varies from node to node and differs from the corresponding equilibrium value 
(horizontal line). The diffusion constants are set so that Turing patterns cannot develop on graph H (data not shown). Panel (b): the Mimura-Murray model defined on the cartesian product 
displays Turing patterns. Once again the asymptotic concentration of species u (circles, blue online) varies from node to node and differs from the predicted equilibrium solution (horizontal black line). Panel (c): level sets of the function 
. Notice the (dark blue online) zone in the top left corner of the figure (enlarged in the inset) where the function takes negative values. The ensemble of vertical (red online) circles represents the eigenvalues of LG. As anticipated, 
, which points to the existence of Turing instability on subspace G. White diamonds identify the pairs 
for which 
: patterns are hence supported on the Cartesian product network 
. On the other hand, for 
, the function 
is positive. One cannot find 
for which 
, in agreement with our initial working assumption: patterns cannot grow on graph H, when taken isolated. Here, G and H are Watts-Strogatz23 networks composed respectively of nG = 30 and nH = 50 nodes. Their associated links rewiring probabilities are taken to pG = 0.01 and pH = 0.04 and the average degree are given by 
and 
. The diffusion coefficients are set to the values 
, 
, 
and 
. The initial condition is a perturbation of the homogeneous fixed point (
). Such externally imposed perturbation is node dependent, hence inhomogeneous, drawn from a uniform distribution and scaled with an amplitude factor δ = 0.005.
