Theory of Turing Patterns on Time Varying Networks

The process of pattern formation for a multispecies model anchored on a time varying network is studied. A nonhomogeneous perturbation superposed to an homogeneous stable fixed point can be amplified following the Turing mechanism of instability, solely instigated by the network dynamics. By properly tuning the frequency of the imposed network evolution, one can make the examined system behave as its averaged counterpart, over a finite time window. This is the key observation to derive a closed analytical prediction for the onset of the instability in the time dependent framework. Continuously and piecewise constant periodic time varying networks are analyzed, setting the framework for the proposed approach. The extension to nonperiodic settings is also discussed.

Figure 1

Twin network. (a) $T$-periodic network built from two static networks, of adjacency matrices ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$. Each network in this illustrative example is made of $N=6$ nodes. In the network stored in matrix ${\mathbf{A}}_1$, symmetric edges are drawn between the pairs (1,2), (3,4), and (5,6). The second network, embodied in matrix ${\mathbf{A}}_2$, links nodes (6,1), (2,3), and (4,5). For $t\in [0,\gamma T)$, the $T$-periodic network coincides with ${\mathbf{A}}_1$, $\mathbf{A}(t)={\mathbf{A}}_1$, while, in $[\gamma T,T)$, we set $\mathbf{A}(t)={\mathbf{A}}_2$. The time varying network is then obtained by iterating the process in time. (b) The ensuing time averaged network $⟨\mathbf{A}⟩=\gamma {\mathbf{A}}_1+(1-\gamma ){\mathbf{A}}_2$. (c) Dispersion relation ($\max \mathrm{Re}{\lambda }_{\alpha }$ vs $-{\mathrm{\Lambda }}_{\alpha }$) or the averaged network (the red circles), for each static twin network (the black stars) and for the continuous support case (the blue curve). Here, the networks are generated as discussed above, but now $N=50$. (d) Patterns in the averaged network. Nodes are blue if they present an excess of concentration with respect to the homogeneous equilibrium solution ($[u_i(\infty )-\overline{u}]\ge 0.1$) and red otherwise ($[u_i(\infty )-\overline{u}]\le -0.1$). The outer drawing represents the entries of $\overrightarrow{v}$, the eigenvector of the Jacobian matrix $\mathcal{J}$ associated with the eigenvalues that yields the largest value of the dispersion relation. The black ring represents the zeroth level; red and yellow areas are associated with positive entries of $\overrightarrow{v}$, while blue and light blue regions refer to negative values. The reaction model is the Brusselator with $b=8$, $c=10$, $D_u=3$, and $D_v=10$. The homogeneous equilibrium is $\overline{u}=1$ and $\overline{v}=0.8$. The remaining parameters are set to $\gamma =0.3$, $T=1$, $D_u=3$, and $D_v=10$.

Figure 2

The critical threshold ${\epsilon }^{*}$. (a) Pattern amplitude, $S(\epsilon )$, as a function of $\epsilon$ normalized to the amplitude of the pattern for the averaged network $⟨\mathbf{A}⟩=\gamma {\mathbf{A}}_1+(1-\gamma ){\mathbf{A}}_2$, for a $T$-periodic twin network made of $N=50$ nodes. (Insets) Asymptotic patterns ($u$ variable) for $\epsilon =0.1<{\epsilon }^{*}$; blue (red) nodes represent a high (low) concentration of $u$ with respect to the homogeneous equilibrium. For $\epsilon =0.3>{\epsilon }^{*}$, the system converges toward the homogeneous equilibrium; nodes are plotted in yellow when the concentration of species $u$ is close to the homogeneous value. The employed reaction model is again the Brusselator, and parameters are set as in Fig. 1. (b) ${\epsilon }^{*}$ vs $N$. The black dotted lines are drawn after Eq. (6), while the red symbols (circles for $N/2$ even and squares for $N/2$ odd) are numerically computed. (Inset) The maximum of the dispersion relation as a function of the network size (blue circles for $N/2$ even and black down triangles for $N/2$ odd).