Spatial scale separation and emergent patterns in coupled diffusive-nondiffusive systems

TL;DR

The paper extends classical pattern formation by analyzing reaction–diffusion–ODE systems that include nondiffusive components, establishing both the existence of far-from-equilibrium patterns and comprehensive conditions for diffusion-driven instability (DDI). It identifies two main mechanisms: (i) branch-switching patterns arising from a two-branch nondiffusive nullcline, which can produce discontinuities and persist independently of diffusion, and (ii) DDI arising from the interplay of nondiffusive and slow/fast diffusive components, including domain-size and diffusion-rate effects. The authors provide a detailed analysis of the minimal 1+2 component case and apply the theory to a receptor-based model, supported by numerical bifurcation and simulations that reveal discontinuous, aperiodic, or spike-like structures. This work broadens the theoretical landscape of pattern formation, showing how spatial scale separation and coupling between diffusive and nondiffusive dynamics yield rich patterns beyond classical Turing theory with potential biological relevance.

Abstract

This paper investigates pattern formation in reaction-diffusion systems with both diffusive and nondiffusive components, establishing the existence of far-from-equilibrium patterns and providing necessary and sufficient conditions for diffusion-driven instability (DDI). In particular, we prove the existence of far-from-equilibrium patterns exhibiting branch-switching and discontinuities in the nondiffusive components, which cannot occur in classical reaction-diffusion equations. While previous work has linked DDI to instability in the purely nondiffusive subsystem -- thereby destabilizing all regular Turing patterns -- we show that DDI can also arise from subsystems involving nondiffusive and slow-diffusive components. This leads to simple sufficient conditions for DDI in systems with arbitrary numbers of components. Further, we fully classify all possible sources of DDI in the case of two diffusive and one nondiffusive component, illustrating our results with a receptor-based model supported by numerical bifurcation analysis and simulations. These findings extend the theoretical foundations of pattern formation, demonstrating how coupling between diffusive and nondiffusive dynamics can generate patterns beyond the reach of the classical reaction-diffusion framework.

Figures (5)

• Figure 1: Cross sections of the six-dimensional parameter space, $\mathcal{R}$, in the $(m_i, \mu_j)$--planes, $i, j = 1, 2, 3$. In each panel, the regions $\mathcal{R}_1,\dots,\mathcal{R}_4$ are superimposed. The overlap, enclosed by a black curve, corresponds to parameters satisfying all constraints simultaneously. The black dot marks the feasible point $p_\star$.
• Figure 2: Bifurcation diagram for system \ref{['eq:toysys']} in the $(D_v, D_w)$-plane. The shaded region $\Gamma$ is the Turing unstable set, given by the union of all mode-specific regions $\Gamma_j$. Parameters: $\mu_1=1.00$, $\mu_2=1.00$, $\mu_3=0.60$, $m_1=2.50$, $m_2=9.68$, $m_3=7.00$.
• Figure 3: Patterned solutions to system \ref{['eq:toysys']}. Each row corresponds to one component of $X=(u,v,w)$; each column shows the outcome for a different diffusion pair $(D_v,D_w)$. In all cases, only a single spatial mode is unstable. Parameters: $\mu_1=1.00$, $\mu_2=1.00$, $\mu_3=0.60$, $m_1=2.50$, $m_2=9.68$, $m_3=7.00$. Init. Cond.: $u_0 = \bar{u} + \xi$, $v_0 = \bar{v} + \xi$, $w_0 = \bar{w} + \xi$, $\xi = \frac{x}{10} \sin(10\pi x)$.
• Figure 4: Patterns from simulations with $D_v = 0.006$ and $D_w = 0.017$, for which only the eigenmode $\lambda_4$ is unstable. Different initial conditions select patterns associated with different eigenmodes. Parameters: $\mu_1=1.00$, $\mu_2=1.00$, $\mu_3=0.60$, $m_1=2.50$, $m_2=9.68$, $m_3=7.00$.
• Figure 5: Simulation of System \ref{['eq:toysys']} with initial condition (dotted gray) taken from the pattern in Figure \ref{['fig:sim_patterns']} ($\lambda_4$ unstable) and modified by introducing an artificial jump to zero as indicated in the subfigures. The resulting stationary profile (black) exhibits jump discontinuities. Parameters: $\mu_1=1.00$, $\mu_2=1.00$, $\mu_3=0.60$, $m_1=2.50$, $m_2=9.68$, $m_3=7.00$.

Theorems & Definitions (12)

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