Finite propagation enhances Turing patterns in reaction-diffusion networked systems

TL;DR

This work extends the Cattaneo (finite-velocity) diffusion framework to reaction-diffusion systems on complex networks and derives a 4th-order dispersion polynomial $p_\alpha(\lambda)$ whose roots determine Turing-type instabilities via the Laplacian spectrum $\Lambda^{(\alpha)}$. Using Routh–Hurwitz stability analysis, the authors show that finite propagation speeds (inertial times $\tau_u,\tau_v$) significantly enlarge the parameter regions supporting stationary or wave-like Turing patterns, including regimes where the activator difuses faster than the inhibitor or where inhibitor-inhibitor interactions occur. They introduce the notion of inertia-driven instability and identify a $\tau_{\max}$ threshold (when $\tau_u=\tau_v$) beyond which the homogeneous state becomes unstable but not necessarily a classical Turing instability. These analytical results are complemented by numerical simulations of a networked FitzHugh–Nagumo model under the relativistic framework, revealing stationary and synchronised oscillatory patterns, including cases where dispersion alone fails to predict the long-time state. Overall, the paper provides a unified framework linking finite-velocity diffusion, network structure, and nonlinear dynamics to broaden the landscape of pattern formation in real-world systems.

Abstract

We hereby develop the theory of Turing instability for reaction-diffusion systems defined on complex networks assuming finite propagation. Extending to networked systems the framework introduced by Cattaneo in the 40's, we remove the unphysical assumption of infinite propagation velocity holding for reaction-diffusion systems, thus allowing to propose a novel view on the fine tuning issue and on existing experiments. We analytically prove that Turing instability, stationary or wave-like, emerges for a much broader set of conditions, e.g., once the activator diffuses faster than the inhibitor or even in the case of inhibitor-inhibitor systems, overcoming thus the classical Turing framework. Analytical results are compared to direct simulations made on the FitzHugh-Nagumo model, extended to the relativistic reaction-diffusion framework with a complex network as substrate for the dynamics.

Figures (9)

• Figure 1: Parameter region associated to the stability of the homogeneous solution for the FHN model. For a fixed value of $\gamma=4$, we study the stability of the homogeneous equilibrium $(u_i,v_i)=(0,0)$, $i=1,\dots,n$, as a function of $\beta$ and $\mu$: the black regions denote stability while white ones instability. Panel a) corresponds to the classical setting, i.e., $\tau_u=\tau_v=0$, the remaining panels are associated to positive values of the inertial times, $\tau_u=\tau_v=1$ (panel b)), $\tau_u=5$ and $\tau_v=1$ (panel c)) and $\tau_u=1$ and $\tau_v=5$ (panel d)). In all the panels the red line denotes the condition $\mathrm{tr}(J_0)=0$, while $\det(J_0)=0$ is represented by the yellow one; these two lines determine the boundary of the stability region in the classical setting. Such region is shrunk in the case of positive inertial times because of the additional constraints, Eq. \ref{['cond14']} (green line) and Eq. \ref{['cond15']} (blue one). The grey shaded region in panel b), coloured according to $\ln\textcolor{black}{\tau_{\mathrm{max}}}$, is associated to a stability of the homogenous equilibrium constrained to a bound on $\tau$, see Eq. \ref{['eq:cond14taubound']}, while in the black region any positive value of $\tau$ is admissible.
• Figure 2: Parameter region associated to Turing instability for the FHN model, $\tau_u=\tau_v$. For a fixed value of $\gamma=4$, we study the onset of Turing instability (black regions) close to the homogeneous equilibrium $(u_i,v_i)=(0,0)$, $i=1,\dots,n$, as a function of $\beta$ and $\mu$. Panel a) corresponds to the classical setting, i.e., $\tau_u=\tau_v=0$, the remaining panels are associated to positive values of the inertial times, $\tau_u=\tau_v=1$. In panels a) and b) the diffusivities have been set equal to $D_u=0.2$ and $D_v=2.2$, namely the inhibitor diffuses faster than the activator. Panel c) present a completely new setting where Turing instability can develop even for a slower inhibitor, $D_u=2.2$ and $D_v=0.2$. In all the panels the red line denotes the condition $\mathrm{tr}(J_0)=0$, while $\det(J_0)=0$ is represented by the yellow one. In panels a) and b), the dashed blue line represents the condition $D_v\partial_u f+D_u\partial_v g=0$ (Eq. \ref{['eq:inst11']}), while the dashed red line the condition $(D_u\partial_v g+D_v\partial_u f)^2-4{D_uD_v} \det(J_0)=0$ (Eq. \ref{['eq:inst12']}). Together with the blue line in panel b) corresponding to Eq. \ref{['cond15']}, these lines delimitate the parameter region allowing for Turing instability in the case $D_u < D_v$. In panel c), corresponding to $D_u > D_v$, a similar parameter region is bounded by the same blue line but also by the dashed black line, namely Eq. \ref{['eq:inst22']}. The grey shaded region in panels b) and c), coloured according to $\ln\textcolor{black}{\tau_{\mathrm{max}}}$, is associated to a stability of the homogeneous equilibrium constrained to a bound on $\tau$, see Eq. \ref{['eq:cond14taubound']}, while in the black region any positive value of $\tau$ is admissible.
• Figure 3: Dispersion relation and pattern for the FHN model, $\tau_u=\tau_v$. For fixed values of $\gamma=4$, $D_u=2.2$, $D_v=0.2$ and two couples $(\beta,\mu)$ we show in the main panels the dispersion relation, $\lambda_\alpha$ as a function of $\Lambda^{(\alpha)}$. Panel a) corresponds to the choice $\tau_u=\tau_v=1$ and $(\beta,\mu)=(0.8,1.0)$ (yellow star in the panel c) of Fig. \ref{['fig:mualphaFHNTPsametau']}), lying the Turing instability region and indeed the dispersion relation assumes positive values (red dots lying on the positive part of the blue curve). The homogeneous equilibrium is stable (the dispersion relation is negative for $\Lambda^{(1)}=0$), but it turns out to be unstable under heterogeneous perturbations and synchronised oscillatory patterns emerge (inset), indeed the critical root has positive imaginary part, $\rho_\alpha>0$ (conditions \ref{['eq:inst21']} and \ref{['eq:inst22']} are satisfied). In panel b) we fix $\tau_u=\tau_v=2.2$ and $(\beta,\mu)=(0.7,1.0)$ (red triangle in the panel c) of Fig. \ref{['fig:mualphaFHNTPsametau']}), still in the Turing region but conditioned to the value of $\textcolor{black}{\tau_{\mathrm{max}}}$. The behaviour is similar to the one reported in panel a) but now the homogeneous equilibrium is weakly stable, the dispersion relation is negative but very close to $0$ for $\Lambda^{(1)}=0$, indeed for these values of the parameters we have $\textcolor{black}{\tau_{\mathrm{max}}}\sim 2.31$. Again, an oscillatory behaviour is obtained (inset) associated to $\rho_\alpha>0$. In panel c) we used the same parameters $(\beta,\mu)$ but we increased $\tau_u=\tau_v=3.5>\textcolor{black}{\tau_{\mathrm{max}}}$ and indeed the homogeneous equilibrium is unstable, the dispersion relation is positive for $\Lambda^{(1)}=0$. Again synchronised oscillatory patterns emerge (inset), they are indistinguishable from the ones one could obtain with the parameters used in panels a) and b) but they are not the result of Turing instability.
• Figure 4: Parameter region associated to the inertia-driven instability for the FHN model. For a fixed value of $\gamma=4$, we study the onset of Turing instability (black regions) close to the homogeneous equilibrium $(u_i,v_i)=(0,0)$, $i=1,\dots,n$, as a function of $\beta$ and $\mu$ and driven by the inertial times, $\tau_u\neq \tau_v$. Indeed we assume $D_u \geq D_v$, resulting in a setting where classical Turing instability cannot emerge. Panel a) corresponds to the setting, $\tau_u=5$ and $\tau_v=1$, $D_u=2.2$ and $D_v=0.2$. In panel b) we use the same diffusivities while the inertial times are exchanged, i.e., $\tau_u=1$ and $\tau_v=5$. Panel c) reports result for $D_u=D_v=2.2$ and $\tau_u=1$ and $\tau_v=5$. In all the panels the red line denotes the condition $\mathrm{tr}(J_0)=0$, while $\det(J_0)=0$ is represented by the yellow one. The green line represents condition \ref{['cond14']}, while the magenta one represents condition \ref{['cond15']}; once present, the dashed black line stands for Eq. \ref{['eq:inst22']}.
• Figure 5: Dispersion relation and pattern for the FHN model in the case $\tau_u\neq \tau_v$. For a fixed value of $\gamma=4$ and two couples $(\beta,\mu)$ and $(\tau_u,\tau_v)$ we show the dispersion relation (panels a) and d), $\lambda_\alpha$ as a function $\Lambda^{(\alpha)}$, the imaginary part of the root with the largest real part, $\rho_\alpha$ (panels b) and e)), and the time evolution of the solutions $u_i(t)$ (panels c) and f)). The top three panels correspond to the choice $\tau_u=5$ and $\tau_v=1$, $(\beta,\mu)=(0.6,1.0)$, and $D_u=2.2$ and $D_v=0.2$, lying the inertia-driven instability region (see panel a) Fig. \ref{['fig:mualphaFHNTPdifftau1']}), indeed the conditions \ref{['eq:inst21']} and \ref{['eq:inst22']} hold true. While the bottom three panels are associated to $\tau_u=1$ and $\tau_v=5$, $(\beta,\mu)=(2.5,0.18)$ and with equal diffusivity $D_u=D_v=2.2$; these values are still in the inertia-driven region (see panel c) Fig. \ref{['fig:mualphaFHNTPdifftau1']}) and the conditions \ref{['eq:inst21']} and \ref{['eq:inst22']} are satisfied. In both cases the aspatial equilibrium is stable ($\lambda_1<0$), but it turns out to be unstable under heterogeneous perturbations and synchronised oscillatory patterns emerge. Being $\rho_\alpha>0$ we are in presence of a Turing-wave instability driven by the inertial times.

Theorems & Definitions (2)

• Remark 1: Connection with the relativistic reaction-diffusion system defined on a continuous substrate
• Remark 2: Kinetic linear systems