Figure 4
Turing patterns for the Mimura-Murray model on a degenerate multiplex support.
The multiplex is built as the Cartesian product 
. Here, G = •–•–• (namely, it is a linear chain with nearest neighbors connections) and 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 
. Panel (a): asymptotic distribution for the concentration of species u (circles, blue online) on each node of the Cartesian product 
. The recorded concentration varies from node to node and differs from the deputed equilibrium value 
(horizontal black line). For the selected parameters (see below), the Mimura-Murray model is Turing unstable on the linear chain G. Turing patterns cannot develop instead, when the reaction-diffusion model is made to evolve on the Watts—Strogatz network H alone. Panel (b): level sets of the function 
. In the top border of the picture (region enlarged in the inset, dark blue online) the function assumes negative values. The (red online) circle identifies the unstable eigenvalue of the Laplacian operator associated to the linear chain. As anticipated, it falls in the region 
, hence signaling the presence of Turing-like patterns on G. The white diamonds refer to the pairs 
for which 
, thus implying the existence of the instability on the Cartesian product 
. We remark that for 
, the function 
is positive definite: patterns cannot emerge when the diffusion of the interacting species is confined on graph H. Here, the diffusion coefficients are set to the representative values 
, 
, 
and 
. The initial condition is assigned as explained in the caption of Fig. 2.
