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Analysis and Patterns of Nonlocal Klausmeier Model

Md Shah Alam

TL;DR

This work introduces a nonlocal Klausmeier-type model where biomass dispersal is governed by a symmetric convolution operator $\Gamma_{\phi}$ while water diffusion remains local. Using a semigroup framework and a duality argument, the authors establish global existence, uniqueness, and uniform $L^{\infty}$-bounds for positive classical solutions in $\mathbb{R}^N$. Numerical simulations compare local and nonlocal versions, showing that larger kernel radii lead to more coherent and morphologically rich vegetation patterns, highlighting the impact of nonlocal interactions on pattern formation. Overall, the study provides a rigorous analytical foundation for nonlocal reaction–diffusion systems in ecology and demonstrates the significant influence of nonlocal dispersal on vegetation patterning with potential applicability to broader nonlocal ecological models.

Abstract

This work studies a nonlocal extension of the Klausmeier vegetation model in $\mathbb{R}^N$ $(N \ge 1)$ that incorporates both local and nonlocal diffusion. The biomass dynamics are driven by a nonlocal convolution operator, representing anomalous and faster dispersal than the standard Laplacian acting on the water component. Using semigroup theory combined with a duality argument, we establish global well-posedness and uniform boundedness of classical solutions. Numerical simulations based on the Finite Difference Method with Forward Euler integration illustrate the qualitative effects of nonlocal diffusion and kernel size on vegetation patterns. The results demonstrate that nonlocal interactions significantly influence the spatial organization of vegetation, producing richer and more coherent structures than those arising in the classical local model.

Analysis and Patterns of Nonlocal Klausmeier Model

TL;DR

This work introduces a nonlocal Klausmeier-type model where biomass dispersal is governed by a symmetric convolution operator while water diffusion remains local. Using a semigroup framework and a duality argument, the authors establish global existence, uniqueness, and uniform -bounds for positive classical solutions in . Numerical simulations compare local and nonlocal versions, showing that larger kernel radii lead to more coherent and morphologically rich vegetation patterns, highlighting the impact of nonlocal interactions on pattern formation. Overall, the study provides a rigorous analytical foundation for nonlocal reaction–diffusion systems in ecology and demonstrates the significant influence of nonlocal dispersal on vegetation patterning with potential applicability to broader nonlocal ecological models.

Abstract

This work studies a nonlocal extension of the Klausmeier vegetation model in that incorporates both local and nonlocal diffusion. The biomass dynamics are driven by a nonlocal convolution operator, representing anomalous and faster dispersal than the standard Laplacian acting on the water component. Using semigroup theory combined with a duality argument, we establish global well-posedness and uniform boundedness of classical solutions. Numerical simulations based on the Finite Difference Method with Forward Euler integration illustrate the qualitative effects of nonlocal diffusion and kernel size on vegetation patterns. The results demonstrate that nonlocal interactions significantly influence the spatial organization of vegetation, producing richer and more coherent structures than those arising in the classical local model.

Paper Structure

This paper contains 6 sections, 3 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction and Main Results
  2. Introduction
  3. Statement of the Main Results
  4. Proof of Main Theorems
  5. Patterns
  6. Conclusion

Key Result

Theorem 1

Assume that (item:D, item:f1, item:f2, (eq:1.3) and (eq:1.4) hold and let $n_0 \in C(\overline{\Omega}, \mathbb{R}_+)$ and $w_0 \in C^2(\overline{\Omega}, \mathbb{R}_+)$ satisfy $\dfrac{\partial w_0}{\partial \eta} = 0$ on $\partial \Omega$. Then there exists a unique global component-wise nonnegati

Figures (4)

  • Figure 1: Patterns of water, $n(x,t)$ and biomass, $w(x,t)$ in local Klausmeier model.
  • Figure 2: Patterns of water, $n(x,t)$ and biomass, $w(x,t)$ in nonlocal Klausmeier model.
  • Figure 3: Patterns of water, $n(x,t)$ and biomass, $w(x,t)$ in nonlocal Klausmeier model for $\sigma=5.0$.
  • Figure 4: Patterns of water, $n(x,t)$ and biomass, $w(x,t)$ in nonlocal Klausmeier model for $\sigma=5.0$.

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 2.1
  • Lemma 2.2