Transient Instability and Patterns of Reactivity in Diffusive-Chemotaxis Soil Carbon Dynamics

TL;DR

This study shows that pattern formation in a diffusive-chemotaxis soil carbon model can arise from transient instabilities driven by non-normality, even when the homogeneous state is asymptotically stable. By analyzing the MOMOS model, the authors connect reactivity and finite-time amplification to the emergence of stable spatial patterns in two dimensions near the boundary of a subcritical Turing bifurcation, where bistability between homogeneous and patterned states exists. They introduce quantitative measures of transient growth, identify parameter regimes yielding stable reactive patterns, and demonstrate that 2D geometry is essential for bistability and pattern persistence. The findings have implications for modeling microbial hotspot formation and more broadly for cross-diffusion systems, suggesting that transient dynamics can expand the space of observable patterns beyond classical Turing instabilities.

Abstract

We study pattern formation in a chemotaxis model of bacteria and soil carbon dynamics as an example system where transient dynamics can give rise to pattern formation outside of Turing unstable regimes. We use a detailed analysis of the reactivity of the non-spatial and spatial dynamics, stability analyses, and numerical continuation to uncover detailed aspects of this system's pattern-forming potential. In addition to patterning in Turing unstable parameter regimes, reactivity of the spatial system can itself lead to a range of parameters where a spatially uniform state is asymptotically stable, but exhibits transient growth that can induce pattern formation. We show that this occurs in the bistable region of a subcritical Turing bifurcation. Intriguingly, such bistable regions appear in two spatial dimensions, but not in a one-dimensional domain, suggesting important interplays between geometry, transient growth, and the emergence of multistable patterns. We discuss the implications of our analysis for the bacterial soil organic carbon system, as well as for reaction-transport modeling more generally.

Figures (10)

• Figure 1: (a) Bifurcation diagram for the MOMOS model, \ref{['eq:model']}, with parameters set to $D_u = D_v = 0.6$, $k_1 = 0.4$, $k_2 = 0.6$, and $c = 0.8$. Data points (white crosses) indicate parameter pairs where patterns of reactivity, arising due to non-normality, are observed at the parameters $(q, \beta) = (0.0196639, 0.474095), (0.0804361, 1.23535), (0.061122, 1.01668)$, and $(0.0433, 0.806)$. (b) A heterogeneity norm from systematic simulations across one of these data points for varying size of the initial data, $\eta$. The red vertical line indicates the Turing bifurcation point, where everything to the left of this line is in the Turing stable parameter region.
• Figure 2: Patterns of reactivity arising due to non-normality in the MOMOS model, \ref{['eq:model']}, for $q=0.0433$, $\beta=0.806$. The other parameters are set to $D_u = D_v = 0.6$, $k_1 = 0.4$, $k_2 = 0.6$, and $c = 0.8$.
• Figure 3: On the left, the polynomials $h(k^2)$ and $\tilde{h}(k^2)$ are plotted as functions of $k^2$. For Case 1, $\tilde{h}(k^2)$ has a negative quadratic coefficient with positive discriminant, resulting in reactivity of $J_{k}$ for $k^2 > \tilde{k}_m = 0.1173$. At $k^2 = 0.7812$, corresponding to the minimum of $h(k^2)$, the largest negative eigenvalue of $J_{k}$ approaches zero. On the right, the measure of non-normality $\delta(k^2)$ is shown for values of $k^2 \geq \tilde{k}_m$, demonstrating that $\delta(k^2)$ decreases as $k^2$ increases. This confirms that $J_{k}$ is non-normal, with non-normality becoming more pronounced at higher $k^2$.
• Figure 4: Time evolution of the amplification envelope $\rho_{k}(t)$, its theoretical estimate $\chi(t)$, and its maximum value $\chi^*$ for wavenumbers $0.2 \leq k^2 \leq 1$. The plots demonstrate a close agreement between $\rho_{k}(t)$ and $\chi(t)$, confirming the predictive capability of $\chi(t)$ for transient dynamics. The amplification becomes more sustained as $k^2$ increases, reaching an estimated maximum of $\chi^* = 1.7108$ at $k^2=1$. At $k^2 = 0.7812$, corresponding to the minimum of $h(k^2)$, the return time—defined as the time for $\rho_{k}(t)$ to decay back to its initial value—is maximized. Beyond this point, larger $k^2$ values result in a more rapid decline of $\rho_{k}(t)$, indicating faster stabilization of the system.
• Figure 5: Time evolution of $\rho_{k}(t)$, $\chi(t)$, and $\chi^*$ for higher wavenumbers $10 \leq k^2 \leq 10^5$. As $k^2$ increases, the return time decreases significantly, showing the rapid stabilization of the system at larger wavenumbers. The transient amplification becomes more pronounced, with $\rho_{k}(t)$ achieving higher peak values. However, the maximum amplification stabilizes at $\chi^* = 2.1242$ for sufficiently large $k^2$, suggesting the existence of a maximum possible amplification value.

Theorems & Definitions (4)

• Proposition 1
• proof
• Proposition 2
• proof