[Paper Review] How inertia affects autotoxicity-mediated vegetation dynamics: from close-to to far-from-equilibrium patterns

Giancarlo Consolo, Carmela Currò+4 more|arXiv (Cornell University)|Mar 10, 2026

Ecosystem dynamics and resilience0 citations

TL;DR

The paper extends the Klausmeier model with inertia and autotoxicity to study vegetation patterns on sloped arid terrains, detailing how inertia shapes onset, wave instabilities, traveling pulses, and hysteresis through analytical and numerical methods.

ABSTRACT

In this work, the influence of inertial effects on the formation and evolution of vegetation patterns on sloped arid terrains is investigated from the onset of instability to far-from-equilibrium. Analyses are carried out in a hyperbolic extension of the one-dimensional Klausmeier model, where autotoxicity effects are also taken into account. As the system moves away from the wave bifurcation threshold, two classes of solutions arise: small-amplitude periodic migrating bands near onset and large-amplitude travelling pulses in far-from-equilibrium conditions. For the first class, results of LSA reveal that inertia has a twofold role at onset: it acts as a destabilising mechanism, thereby enlarging the parameter region in which uphill migrating vegetation bands can emerge, and it reduces the pattern migration speed. Its role also manifests itself close to onset, as proved by the Stuart-Landau equation for the pattern amplitude deduced via multiple-scale WNA. Indeed, it is shown that inertial effects may reverse the dynamical regime, from supercritical to subcritical, thus leading to hysteresis. For the second class of solutions, the travelling vegetation pulses are first captured via numerical simulations and then investigated via Geometric Singular Perturbation Theory (GSPT). In far-from-equilibrium conditions, inertia is shown to increase pulse speed while preserving the intrinsic multiscale structure of the solution, in full agreement with the numerical findings. Overall, the proposed combined analytical-numerical investigations have depicted several ecological scenarios as a function of the distance from the instability threshold, elucidating that inertia does not exclusively act as a time lag.

Motivation & Objective

• Investigate how inertial effects influence the formation and evolution of vegetation patterns on sloped semi-arid terrains.
• Characterize pattern types across regimes from close-to onset to far-from-equilibrium.
• Incorporate autotoxic effects and assess their interaction with inertia and rainfall.
• Develop and apply a mix of analytical and numerical techniques to map pattern regimes and transitions.

Proposed method

• Formulate a hyperbolic (inertia-including) extension of the Klausmeier model with autotoxicity (dimensionless form in Eq. 2–4).
• Perform linear stability analysis around desert and vegetated steady states to identify onset of wave instabilities (wave bifurcation locus).
• Apply multiple-scale weakly nonlinear analysis to derive a Stuart–Landau-type amplitude equation near onset.
• Use Geometric Singular Perturbation Theory to construct and analyze large-amplitude travelling pulses far from onset.
• Validate theoretical findings with numerical simulations (e.g., COMSOL) to explore pattern formation and pulse dynamics.
• Solve reduced/layer systems (Equations 16–17) to determine migration speed and critical rainfall thresholds at onset.

Figure 1: Subdivision of the $\left(\mathcal{B},\mathcal{A}\right)$ -plane into three different zones according to the location of the existence threshold $\mathcal{A}=\mathcal{A}_{ex}$ and the wave bifurcation locus $\mathcal{A}=\mathcal{A}_{c}$ obtained via numerical integration of System ( 16 )-(

Experimental results

Research questions

• RQ1 How does vegetation inertia modify the onset of wave instability and the associated migration speed of patterns?
• RQ2 What parameter regimes yield migrating bands near onset versus travelling pulses far from onset?
• RQ3 Does inertia promote subcritical bifurcations and hysteresis in pattern formation?
• RQ4 How do autotoxicity and rainfall interact with inertia to shape pattern morphology and dynamics?

Key findings

• Inertia enlarges the pattern-forming region and can slow migration speed near onset, acting as a destabilising mechanism.
• Inertia can induce a transition from supercritical to subcritical bifurcation, leading to hysteresis in pattern formation.
• Far from onset, inertia increases travelling pulse speed and preserves a multiscale structure of pulses.
• Two main pattern classes emerge: small-amplitude migrating bands near onset and large-amplitude travelling pulses far from equilibrium.
• Increasing inertia expands the pattern-forming region, especially at higher rainfall.
• Numerical and analytical results for migration speed at onset show consistent agreement, with inertia modulating speed and sensitivity to autotoxicity.

Figure 2: Vegetation pattern dynamics observed into the wave instability region under worsening environmental conditions for increasing aridity (panels a-d) and increasing plant mortality (panels e-h). Panels (a) and (h) show the spatio-temporal evolution of the patterned solution obtained by sweepi