Vegetation pattern formation induced by local growth outpacing susceptibility to non-local competition

TL;DR

This work develops a general, single-component integro-differential framework for vegetation dynamics with local growth $g(u)$, non-local competition via $s(u) \int \phi(x'-x)\, c(u)\, dx'$ and diffusion $D\Delta u$, unifying non-local competition models and enabling systematic analysis of Turing instabilities. Linear stability analysis yields a dispersion relation $\omega(k)=s(\bar{u})\left(\left.\frac{d}{du}\left(\frac{g(u)}{s(u)}\right)\right|_{u=\bar{u}}-c'(\bar{u})\hat{\phi}(k)\right)-Dk^2$, with two ecologically meaningful routes to pattern formation: (i) stronger competitive pressure in the gaps between high-biomass patches when $\inf_{k>0} \hat{\phi}(k)<(1/c'(\bar{u}))\left.\frac{d}{du}\left(\frac{g(u)}{s(u)}\right)\right|_{u=\bar{u}}$, and (ii) growth outpacing susceptibility when $\inf_{k>0} \hat{\phi}(k)=0$ and $0<\left.\frac{d}{du}\left(\frac{g(u)}{s(u)}\right)\right|_{u=\bar{u}}<c'(\bar{u})$. Through two benchmark models—the non-local Fisher-KPP and the Growth Outpacing Susceptibility (GOS) model—the study demonstrates that both mechanisms can produce a supercritical Turing bifurcation, yielding stable, robust vegetation patterns with similar qualitative features across kernels. Numerical simulations reveal that the spatial phase of competitive pressure tracks the sign of $\hat{\phi}(k)$ and that patterns persist far from onset, sometimes exhibiting exclusion zones and multi-stability (Busse balloon). The framework provides ecological interpretations of pattern formation, clarifies when non-local competition suffices for self-organization, and offers a foundation for extending to higher dimensions and multi-component plant communities, with potential implications for ecosystem resilience and management.

Abstract

In this work, we present and analyze a general framework for vegetation dynamics in arid and semi-arid ecosystems in which non-local interactions are purely competitive. The generality of the formulation enables a systematic search for ecological mechanisms that may lead to self-organized patterns. We identify two distinct mechanisms generating Turing instabilities across a broad class of models. The first mechanism arises from intensified competition in the areas between vegetated patches due to the cumulative pressure from their surroundings, and is well-documented in the literature. The second mechanism is novel and occurs when local growth outpaces competitive susceptibility near the uniform equilibrium. The analytical findings are complemented by numerical simulations of two benchmark models, both exhibiting a supercritical Turing bifurcation that leads to the formation of stable and robust vegetation patterns.

Figures (9)

• Figure 1: Plots of the top-hat, parabolic, cosine and triangular kernel (top row) and their Fourier transforms (bottom row).
• Figure 2: $D_\text{max}$ and $k_c$ for the non-local Fisher-KPP model (see (\ref{['model: non-local Fisher KPP']})) with a top-hat kernel for $a = 10$ and $\ell = 5$.
• Figure 3: The dispersion relation $\omega(k)$ for the non-local Fisher-KPP model with a top-hat kernel ($\ell = 1$), local growth rate $a = 0.3$ and diffusion coefficient $D = 0.0035628$. A Turing bifurcation occurs when $\omega(k)$ crosses zero, either as $D$ is decreased (left panel) or as $a$ is increased (right panel).
• Figure 4: The crossing of $\frac{g(u)}{s(u)}$ with $c_\alpha(\bar{u})$ gives rise to a Turing instability for sufficiently small $D$, whereas the crossing of $\frac{g(u)}{s(u)}$ with $c_\beta(\bar{u})$ does not.
• Figure 5: Bifurcation diagrams for the non-local Fisher-KPP model (left) and the GOS model (right), both with a top-hat kernel. The bifurcation parameter is the local growth rate $a$. At $a=0$, a positive and stable vegetated steady state emerges and at $a=a_c$ this equilibrium becomes Turing unstable.