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How inertia affects autotoxicity-mediated vegetation dynamics: from close-to to far-from-equilibrium patterns

Giancarlo Consolo, Carmela Currò, Gabriele Grifò, Annalisa Iuorio, Giovanna Valenti, Frits Veerman

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.

How inertia affects autotoxicity-mediated vegetation dynamics: from close-to to far-from-equilibrium patterns

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.
Paper Structure (15 sections, 5 theorems, 78 equations, 13 figures)

This paper contains 15 sections, 5 theorems, 78 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Pattern-forming region
  4. Vegetation patterns close to onset
  5. Characterisation of the wave bifurcation locus
  6. Evolution of pattern amplitude
  7. Travelling vegetation pulses far from onset
  8. Existence of travelling pulses
  9. Critical manifolds and reduced dynamics
  10. Layer problem
  11. Singular orbit
  12. Persistence
  13. Numerical continuation
  14. Discussion
  15. Acknowledgements.

Key Result

Theorem 1

For $0 < \varepsilon \ll 1$ sufficiently small and $0 < \tau < \tau_{\rm max} \leq \tau_{\rm upper}$, there exists a unique $\theta(\tau)>0$ such that there exists a travelling wave solution $\left( U, V, S, J \right) (x,t) = \left( U, V, S, J \right) (x-\mathcal{C} t)$ of System eq:modadim with wav

Figures (13)

  • 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 \ref{['eq:kc']}-\ref{['eq:Bc']}. Parameter set: $\tau=1$, $\mathcal{H}=0.05$, $\mathcal{V} = 182.5$ and $\mathcal{D} = 4.5$. Note that a different choice of parameter values does not affect qualitatively the abovementioned subdivision of the parameter plane.
  • 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 sweeping the rainfall or the plant loss, respectively, starting from the critical value $(\mathcal{B}_c,\mathcal{A}_c)$. Panels (b,e), (c,f) and (d,g) depict the spatial profiles of such solutions very close to the threshold, in the middle of the pattern-forming region and very far from the instability threshold, respectively. Parameter set: $\tau=1$, $\mathcal{H}=0.05$, $\mathcal{V} = 182.5$ and $\mathcal{D} = 4.5$ (same as in Fig. \ref{['firstfigure']}).
  • Figure 3: Wave bifurcation locus in the $\left(\mathcal{B},\mathcal{A}\right)$-plane for two different values of the toxicity strength: (a) $\mathcal{H}=0$, (b) $\mathcal{H}=0.05$. The loci are computed for different values of inertial time: $\tau\leq0.1$ (solid blue line), $\tau=1$ (dash-dotted red line) and $\tau=3$ (dashed green line). The solid black line denotes the existence locus $\mathcal{A}=\mathcal{A}_{ex}$.
  • Figure 4: Vegetation patterns obtained by integrating numerically the governing System \ref{['model_compact']}-\ref{['vectors_model']} by using the parameter set $(\mathcal{B},\mathcal{A})$ corresponding to points $\mathrm{P}_1$ (first column), $\mathrm{P}_2$ (second column) and $\mathrm{P}_3$ (last column) depicted in Fig. \ref{['figure1']}. The panels represent spatio-temporal evolutions attained for different values of autotoxicity strength and inertial time: $\mathcal{H}=0$ and $\tau=0.1$ (first row), $\mathcal{H}=0$ and $\tau=3$ (second row), $\mathcal{H}=0.05$ and $\tau=0.1$ (third row), and $\mathcal{H}=0.05$ and $\tau=3$ (last row).
  • Figure 5: Migration speed $\mathcal{C}$ in the $\left(\mathcal{H},\mathcal{A}\right)$-plane at onset of wave instability obtained by solving System \ref{['eq:kc']}-\ref{['eq:Bc']} for (a) $\tau=0.1$ and (b) $\tau=3$.
  • ...and 8 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • ...and 3 more