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Extinction and persistence criteria in non-local Klausmeier model of vegetation dynamics on flat landscapes

Maciej Tadej, Ricardo Martinez-Garcia, Michael Hecht

TL;DR

This work analyzes vegetation dynamics on bounded flat landscapes using a non-local Klausmeier model with dispersal kernel $J$ and finite-domain boundary conditions. By combining semigroup techniques, elliptic regularity, and non-local fixed-point methods, it proves well-posedness, non-negativity, and invariant-region properties, and establishes a rigorous framework for stationary states. A key contribution is the identification of a critical patch size tied to the principal eigenvalue $\beta_1$ of $-\mathcal{L}$ and a biomass persistence threshold $\|V\|_\infty \sim B/A$, with non-local dispersal (especially fat-tailed kernels) enhancing persistence in fragmented habitats. Numerical experiments corroborate that non-local kernels can sustain vegetation in smaller patches than local diffusion predicts and reveal boundary-sharp stationary profiles, while bifurcation analyses illustrate extinction and persistence regimes across parameter regimes. Overall, the results deepen understanding of how non-local dispersal shapes extinction/persistence and pattern formation in water-limited ecosystems with realistic boundaries.

Abstract

This paper investigates the dynamics of vegetation patterns in water-limited ecosystems using a generalized Klausmeier model that incorporates non-local plant dispersal within a finite habitat. We establish the well-posedness of the system and provide a rigorous analysis of the conditions required for vegetation survival. Our results identify a critical patch size governed by the trade-off between local growth and boundary losses; habitats smaller than this threshold lead to inevitable extinction. Furthermore, we derive a critical maximal biomass density below which the population collapses to a desert state, regardless of the domain size. We determine stability criteria for stationary solutions and describe the emergence of stable, non-trivial biomass distributions. Numerical experiments comparing sub-Gaussian and super-Gaussian kernels confirm that non-local dispersal mechanisms, particularly those with fat tails, enhance ecosystem resilience by allowing vegetation to persist in smaller, fragmented habitats than predicted by classical local diffusion models.

Extinction and persistence criteria in non-local Klausmeier model of vegetation dynamics on flat landscapes

TL;DR

This work analyzes vegetation dynamics on bounded flat landscapes using a non-local Klausmeier model with dispersal kernel and finite-domain boundary conditions. By combining semigroup techniques, elliptic regularity, and non-local fixed-point methods, it proves well-posedness, non-negativity, and invariant-region properties, and establishes a rigorous framework for stationary states. A key contribution is the identification of a critical patch size tied to the principal eigenvalue of and a biomass persistence threshold , with non-local dispersal (especially fat-tailed kernels) enhancing persistence in fragmented habitats. Numerical experiments corroborate that non-local kernels can sustain vegetation in smaller patches than local diffusion predicts and reveal boundary-sharp stationary profiles, while bifurcation analyses illustrate extinction and persistence regimes across parameter regimes. Overall, the results deepen understanding of how non-local dispersal shapes extinction/persistence and pattern formation in water-limited ecosystems with realistic boundaries.

Abstract

This paper investigates the dynamics of vegetation patterns in water-limited ecosystems using a generalized Klausmeier model that incorporates non-local plant dispersal within a finite habitat. We establish the well-posedness of the system and provide a rigorous analysis of the conditions required for vegetation survival. Our results identify a critical patch size governed by the trade-off between local growth and boundary losses; habitats smaller than this threshold lead to inevitable extinction. Furthermore, we derive a critical maximal biomass density below which the population collapses to a desert state, regardless of the domain size. We determine stability criteria for stationary solutions and describe the emergence of stable, non-trivial biomass distributions. Numerical experiments comparing sub-Gaussian and super-Gaussian kernels confirm that non-local dispersal mechanisms, particularly those with fat tails, enhance ecosystem resilience by allowing vegetation to persist in smaller, fragmented habitats than predicted by classical local diffusion models.
Paper Structure (14 sections, 13 theorems, 83 equations, 4 figures)

This paper contains 14 sections, 13 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The non-local Klausmeier model
  3. Structure
  4. Existence of solutions
  5. Local-in-time solutions
  6. Non-negative and bounded solutions
  7. Stationary solutions
  8. Existence of non-constant water profiles
  9. Existence of non-constant vegetation profiles
  10. Stability analysis
  11. Extinction of vegetation
  12. Survival of vegetation
  13. Numerical experiments
  14. Final conclusions and future work

Key Result

Lemma 2.1

Let $\mathcal{L}: C_{0, \Omega}(\mathbb{R}^n) \to C_{0, \Omega}(\mathbb{R}^n)$ be the non-local dispersal operator from equation eq:DispersalOperatorDefinition_1, $B>0$. Consider the linear operator $\mathcal{A}: C_{0, \Omega}(\mathbb{R}^n) \to C_{0, \Omega}(\mathbb{R}^n)$ given as We denote with $\mathcal{T}(t) : C_{0, \Omega} (\mathbb{R}^n) \to C_{0, \Omega} (\mathbb{R}^n)$, $t \geq 0$, the s

Figures (4)

  • Figure 1: In the left panel, we show the spatially uniform steady-state solution $v_{*,3}$ (red dashed line), the average stationary biomass in the non-local models with thin and fat tails (cyan and orange, respectively) and in the local model (blue) as a function of the patch half-width $L$. We also present the corresponding, numerically estimated critical patch sizes (dashed vertical lines). In the right panel, we present a gallery of stationary biomass density profiles (orange, cyan and blue dots) for different patch half-widths $L$, corresponding to the results shown in the left panel.
  • Figure 2: Comparison of bifurcation diagrams for the non-local models using a super-Gaussian (thin tails) and a sub-Gaussian (fat tails) dispersal kernel, both shown against the standard local model. Solid orange and green lines represent the maximum and average biomass for the non-local models, while dashed cyan and blue lines show the same for the local model. We mark the critical biomass threshold (red line, $\sim B/A$) and the critical rainfall for the kinetic system (purple line, $A=2B$). Here $d_v = 2.0, d_w = 0.1, B = 0.45$.
  • Figure 3: Comparison of bifurcation diagrams for the non-local models using a super-Gaussian (thin tails) and a sub-Gaussian (fat tails) dispersal kernel, both shown against the standard local model. Solid orange and green lines represent the maximum and average biomass for the non-local models, while dashed cyan and blue lines show the same for the local model. We mark the critical biomass threshold (red line, $\sim B/A$) and the critical rainfall for the kinetic system (purple line, $A=2B$). Here $d_v = 2.0, d_w = 80.0, B = 0.45$.
  • Figure 4: A gallery of stationary biomass densities for non-local and local models (green and blue dashed lines respectively). We selected several points from the upper branch of the rightmost bifurcation diagram in Figure \ref{['fig:bifurcation_diagram_fast']}. Here $d_v =2.0, d_w = 80.0, B = 0.45, L=25$.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 17 more