Figure 5
Turing patterns for the Mimura-Murray model on a multiplex 
, with G complete network.
Left panel: Turing Patterns (for the species u) in the multiplex 
(nG = 5), where H is the network obtained by using the Watts-Strogatz algorithm23 with nH = 50, pH = 0.04 and average degree 
. Blue nodes correspond to nodes where the asymptotic concentration of species u is larger than the homogeneous values 
(
); red nodes correspond to nodes where the asymptotic concentration of species u is lower than the homogeneous values 
(
). To help the reader only 40% of links among different layers have been drawn. Notice that the patterns results in a segregation between activator rich - activator poor layers. This is an interesting self-organized stationary solution of the reaction diffusion model, that we will describe in details elsewhere. Middle panel: 
is plotted versus 
. nG denotes the nodes of the complete network G. Once again, H is a Watts-Strogatz23 network composed by nH = 50 nodes, with a probability to rewire a link equal to pH = 0.04 and average degree 
. Turing patterns can develop if and only if 
for some 
. For the complete network 
and 
for α > 1 with multiplicity nG − 1. Hence, 
is negative if and only if q− < −nG < q+. For our choice of the parameters (see below), one finds q−~7.74 and q+~3.55. Turing patterns can hence develop on G and thus on the multiplex 
, if and only if −7 ≤ nG ≤ −4. This is confirmed by inspection of the annexed insets, where the asymptotic concentration of species u is reported against an integer which runs over the nodes, for different choices of nG. This result follows a numerical integration of the relevant reaction-diffusion equations. The horizontal solid lines represent the unperturbed homogeneous fixed point. Here, the diffusion coefficients are 
, 
, 
and 
. The initial condition is set as explained in the caption of Fig. 2. Right panel: absence of Turing Patterns (for the species u) in the multiplex 
(nG = 3), where H is again a network obtained using the Watts-Strogatz algorithm with nH = 50, pH = 0.04 and average degree 
. Yellow nodes correspond to nodes where the asymptotic concentration of species u is equal to the homogeneous values, 
, for all i. To help the reader only 40% of links among different layers have been drawn.
