How spatial patterns can lead to less resilient ecosystems

TL;DR

This work questions the widely held assumption that spatial vegetation patterns inherently enhance ecosystem resilience under drought. By deriving a unifying reduced equation that accounts for finite patched regions and anisotropic, non-reciprocal interactions, the study reveals a nonlinear convective instability that can trigger desertification fronts at lower environmental stress than isotropic models predict. The results show that patterning can either delay or hasten tipping depending on parameters such as the pattern-forming strength $\Gamma$, diffusion $d$, and non-reciprocity $\alpha$, with patterned ecosystems sometimes becoming less resilient than homogeneous ones for sufficient non-reciprocity. The findings provide a framework linking boundary effects, anisotropy, and spatiotemporal front dynamics to resilience in drylands and suggest that real-world desertification risk assessments must incorporate directional forcing and finite-domain geometry.

Abstract

Several theoretical models predict that spatial patterning increases ecosystem resilience. However, these predictions rely on simplifying assumptions, such as assuming isotropic and infinitely large ecosystems, and empirical evidence directly linking spatial patterning to enhanced resilience remains scarce. We introduce a unifying framework, encompassing existing models for vegetation pattern formation in water-stressed ecosystems, that relaxes these assumptions. This framework incorporates finite vegetated areas surrounded by desert and anisotropic environmental conditions that lead to non-reciprocal plant interactions. Under these more realistic conditions, we identify a novel desertification mechanism, known as nonlinear convective instability in physics but largely overlooked in ecology. These instabilities form when non-reciprocal interactions destabilize the vegetation-desert interface and can trigger desertification fronts even under stress levels where isotropic models predict stability. Importantly, ecosystems exhibiting periodic vegetation patterns are more susceptible to nonlinear convective instabilities than those with homogeneous vegetation, suggesting that spatial patterning may reduce, rather than enhance, resilience. These findings challenge the prevailing view that self-organized patterning enhances ecosystem resilience and provide a new framework for investigating how spatial dynamics shape the stability and resilience of ecological systems under changing environmental conditions.

Figures (9)

• Figure 1: Schematic summary of model reduction from two representative examples of reaction-diffusion and nonlocal interaction-redistribution models to the reduced equation \ref{['eq1']}. The parameters of the reduced equation encapsulate the ecological feedbacks in each of the original models as described in the table.
• Figure 2: a, b) Bifurcation diagram and front velocity for quasi-homogeneous vegetation ($\Gamma=1$) under varying environmental stress $\eta$ (see \ref{['sec:MatMet']} for details). Brown (dark) and orange (light) symbols/curves denote reciprocal and non-reciprocal plant interactions, respectively. c, d) Same as a,b, but for patterned vegetation ($\Gamma=2.25$). In all panels, $\eta_{(c,\alpha)}^{\text{\tiny{(H,P)}}}$ mark tipping points for quasi-homogeneous (H) and patterned (P) vegetation with reciprocal ($c$) and non-reciprocal ($\alpha$) interactions. e) In the $(\eta,\alpha)$ space, curves $\eta_{\alpha}^{\tiny\text{\tiny{(H,P)}}}$— dashed for patterned and solid for non-patterned vegetation— delineate regime boundaries where different spatial configurations persist (P: patterns, H: quasi-homeogeneous, BS: bare soil). $\alpha_c$ denotes the non-reciprocity threshold beyond which spatial patterning reduces resilience.
• Figure 3: Behavior of the tipping point in the non-reciprocal system $\eta_{\alpha}$ as a function of $\Gamma$ and $\alpha$. Increasing $\Gamma$ triggers the Turing instability and changes the pattern wavelength and amplitude, while increasing $\alpha$ always destabilizes the ecosystem. The color map measures $\eta_{\alpha}$ relative to $\eta_\mathrm{{s.n}}$, and a dashed line indicates the limit $\eta_{\alpha}=\eta_\mathrm{{s.n}}$, separating regions where spatial effects make vegetation more or less resilient compared to the result of non-spatial models.
• Figure 4: Schematic summary of the effect of non-reciprocal plant interactions in homogeneous (a) and patterned (b) ecosystems. In both scenarios, nonlinear convective instabilities anticipate ecosystem collapse relative to the corresponding isotropic tipping point $\eta_{\alpha}<\eta_{c}$. This shift is stronger in patterned vegetation, which can make self-organized ecosystems collapse at stress levels that homogeneous vegetation can withstand.In terms of the vegetation-desert front velocity, we distinguish three regimes. (i) At low environmental stress (green region in a, b; $\eta<0$), the bare-soil state $b=0$ is unstable and vegetation fronts can invade desert regions (leftmost panel in the bottom row of a and b). (ii) At intermediate stress levels (orange region in a, b), the desert state becomes stable but vegetation fronts can still invade deserts. (iii) At high environmental stress (red region in a, b), the front velocity reverts and triggers a desertification front propagation into the vegetated area, exacerbated by nonreciprocal interactions.
• Figure S1: Convective instabilities and ecosystem collapse in the nonlocal (a, c) and water–vegetation (b, d) models. (a,b) Bifurcation diagrams showing homogeneous (squares) and patterned (circles) vegetation. Colors denote the distance between the non-spatial tipping point $\eta_{s.n}$ and the maximum tolerable stress. Teal (brown) indicates higher (lower) resilience. (c,d) Stress level (relative to $\eta_{s.n}$) at which convective instabilities appear versus pattern-forming parameters ($l_c$, $d_0$) and nonreciprocal strengths ($x_{0c}$, $s$). Parameters. Nonlocal: $\chi_f=3.3$, $\chi_c=2$, $D=0.3$, $l_f=1$, $x_{0f}=0$. Water–biomass: $p=1$, $\delta=1$, $\gamma=0.5$, $d_1=0.001$.