arXiv subjects

Angela Monti

Publications and source records attributed to Angela Monti.

8 recordsLinked to original sources

arXiv2026

Pattern formation: reactivity is not necessary for chemotaxis--driven instabilities

A classical result by Neubert, Caswell and Murray states that reactivity of a spatially homogeneous equilibrium is a necessary condition for diffusion-driven (Turing) instability. In this work, we investigate whether the same conclusion remains valid in the presence of chemotaxis. We consider a general reaction--diffusion system coupled with a chemotactic flux and establish necessary conditions for asymptotic instability. We show that the classical requirement of reactivity can be relaxed when the chemotactic contribution is sufficiently strong. In particular, while reactivity remains necessary for Turing instability, it is not a necessary condition for chemotaxis-driven instability. From a computational viewpoint, we extend the matrix-oriented formulation developed for reaction--diffusion systems to the more general class of reaction--diffusion--chemotaxis models. The chemotactic transport term is discretized in a form compatible with the matrix-oriented approximation of the diffusion operator, yielding an efficient numerical framework for the simulation of chemotaxis-driven pattern formation. A geometric interpretation of the instability region is presented, highlighting the distinct roles played by diffusion and chemotaxis. The theoretical and numerical developments are illustrated through two representative examples: a chemotaxis-extended Schnakenberg model, showing how chemotaxis modifies classical Turing patterns, and a predator--prey model, demonstrating that chemotaxis alone can induce pattern formation in the absence of both reactivity and diffusion-driven instability. These results reveal a fundamental difference between diffusion-driven and chemotaxis-driven mechanisms of spatial self-organization and provide new theoretical and computational insights into the role of non-symmetric transport processes in biological pattern formation.

math.DS↗

arXiv2026

Turing-region preservation in matrix-oriented splitting methods for reaction-diffusion systems

We develop matrix-oriented formulations of first-order splitting integrators for two-species reaction-diffusion systems on 2D domains, using the Gierer-Meinhardt system as a benchmark for discrete Turing instability. Exploiting the differential matrix equation associated with tensor-product spatial discretizations, we obtain the families IE-S and EX-S, where the diffusive flow is treated by implicit Euler or exactly, respectively, and the reaction substep is approximated by explicit, symplectic, adjoint-symplectic, Poisson, and explicit-variant local maps. Starting from the continuous diffusion-driven instability threshold, expressed through the modal relation $J_\mu = J^* - \mu D$, we derive fully discrete modal amplification matrices and their Jury conditions. These conditions separate the continuous Turing mechanism, carried by the first Jury condition, from discrete effects carried by the second. Specializing the analysis to the Gierer-Meinhardt model, we exhibit two opposite pathologies. First, in a continuous Turing-stable regime, IMEX may generate a stable spurious pattern through a violation of the second Jury condition. Second, in a real continuous Turing regime, the adjoint-symplectic family may be spuriously stable and suppress the pattern that IMEX correctly detects. For IMEX, whose first Jury condition reproduces exactly the sign of the continuous Turing polynomial, we give an explicit time-step condition guaranteeing preservation of the continuous Turing region, and show that controlling the second Jury condition is needed only in the Turing-stable regime. These examples show that each integrator induces its own discrete Turing region, which should be compared with the continuous one before interpreting numerical patterns; we frame this requirement as preserving a qualitative property of the continuous problem, in the spirit of structure-preserving numerical integration.

math.NA↗

arXiv2025

Exponential Consensus through Z-Control in High-Order Multi-Agent Systems

In this work, we introduce a Z-control strategy for multi-agent systems of arbitrary order, aimed at driving the agents toward consensus in the highest-order observable state. The proposed framework supports both direct and indirect control schemes, making it applicable in scenarios where high-order derivatives such as acceleration cannot be directly manipulated. Theoretical analysis ensures exponential convergence while preserving the average dynamics, and a hierarchy of control laws is derived accordingly. Numerical experiments up to third-order models, including opinion dynamics and Cucker-Smale flocking systems, demonstrate the robustness and flexibility of Z-control under varying interaction regimes and control intensities.

math.OC↗

arXiv2025

Hierarchical clustering and dimensional reduction for optimal control of large-scale agent-based models

Agent-based models (ABMs) provide a powerful framework to describe complex systems composed of interacting entities, capable of producing emergent collective behaviours such as consensus formation or clustering. However, the increasing dimensionality of these models -- in terms of both the number of agents and the size of their state space -- poses significant computational challenges, particularly in the context of optimal control. In this work, we propose a scalable control frame work for large-scale ABMs based on a twofold model order reduction strategy: agent clustering and projection-based reduction via Proper Orthogonal Decomposition (POD). These techniques are integrated into a feedback loop that enables the design and application of optimal control laws over a reduced-order representation of the system. To illustrate the effectiveness of the approach, we consider the opinion dynamics model, a prototyp ical first-order ABM where agents interact through state-dependent influence functions. We show that our method significantly improves control efficiency, even in scenarios where direct control fails due to model complexity. Beyond its methodological contributions, this work also highlights the rel evance of opinion dynamics models in environmental contexts -- for example, modeling the diffusion of pro-environmental attitudes or decision-making processes in sustainable policy adoption -- where controlling consensus formation plays a crucial role.

math.OC↗

arXiv2025

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

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.

math.NA↗

arXiv2024

Patterns in soil organic carbon dynamics: integrating microbial activity, chemotaxis and data-driven approaches

Models of soil organic carbon (SOC) frequently overlook the effects of spatial dimensions and microbiological activities. In this paper, we focus on two reaction-diffusion chemotaxis models for SOC dynamics, both supporting chemotaxis-driven instability and exhibiting a variety of spatial patterns as stripes, spots and hexagons when the microbial chemotactic sensitivity is above a critical threshold. We use symplectic techniques to numerically approximate chemotaxis-driven spatial patterns and explore the effectiveness of the piecewice dynamic mode decomposition (pDMD) to reconstruct them. Our findings show that pDMD is effective at precisely recreating chemotaxis-driven spatial patterns, therefore broadening the range of application of the method to classes of solutions different than Turing patterns. By validating its efficacy across a wider range of models, this research lays the groundwork for applying pDMD to experimental spatiotemporal data, advancing predictions crucial for soil microbial ecology and agricultural sustainability.

math.NA↗

arXiv2023

Piecewise DMD for oscillatory and Turing spatio-temporal dynamics

Dynamic Mode Decomposition (DMD) is an equation-free method that aims at reconstructing the best linear fit from temporal datasets. In this paper, we show that DMD does not provide accurate approximation for datasets describing oscillatory dynamics, like spiral waves and relaxation oscillations, or spatio-temporal Turing instability. Inspired from the classical "divide and conquer" approach, we propose a piecewise version of DMD (pDMD) to overcome this problem. The main idea is to split the original dataset in N submatrices and then apply the exact (randomized) DMD method in each subset of the obtained partition. We describe the pDMD algorithm in detail and we introduce some error indicators to evaluate its performance when N is increased. Numerical experiments show that very accurate reconstructions are obtained by pDMD for datasets arising from time snapshots of some reaction-diffusion PDE systems, like the FitzHugh-Nagumo model, the lambda-omega system and the DIB morpho-chemical system for battery modeling.

math.NA↗

arXiv2022

Adaptive POD-DEIM correction for Turing pattern approximation in reaction-diffusion PDE systems

We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction-diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, first of all, we propose a POD-DEIM technique with a correction term that includes missing information in the reduced models. To improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD systems, i.e. FitzHugh-Nagumo, Schnackenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.

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