Traveling vegetation-herbivore waves can sustain ecosystems threatened by droughts and population growth

Dryland vegetation can survive water stress by forming spatial patterns but is often subjected to herbivory as an additional stress that puts it at risk of desertification. Understanding the mutual relationships between vegetation patterning and herbivory is crucial for securing food production in drylands, which constitute the majority of rangelands worldwide. Here, we introduce a novel vegetation-herbivore model that captures pattern-forming feedbacks associated with water and herbivory stress and a behavioral aspect of herbivores representing an exploitation strategy.Applying numerical continuation methods, we analyze the bifurcation structure of uniform and patterned vegetation-herbivore solutions, and use direct numerical simulations to study various forms of collective herbivore dynamics. We find that herbivory stress can induce traveling vegetation-herbivore waves and uncover the ecological mechanism that drives their formation. In the traveling-wave state, the herbivore distribution is asymmetric with higher density on one side of each vegetation patch. At low precipitation values their distribution is localized, while at high precipitation the herbivores are spread over the entire landscape. Importantly, their asymmetric distribution results in uneven herbivory stress, strong on one side of each vegetation patch and weak on the opposing side - weaker than the stress exerted in spatially uniform herbivore distribution. Consequently, the formation of traveling waves results in increased sustainability to herbivory stress. We conclude that vegetation-herbivore traveling waves may play an essential role in sustaining herbivore populations under conditions of combined water and herbivory stress, thereby contributing to food security in endangered regions threatened by droughts and population growth.

💡 Research Summary

The authors present a spatially explicit reaction‑diffusion model that couples vegetation biomass (B), soil water content (W) and herbivore biomass (H) to investigate how water limitation and herbivory interact in dryland ecosystems. Vegetation growth follows a water‑dependent logistic term with a root‑shoot enhancement factor, while water dynamics are driven by precipitation, evaporation, shading‑modified transpiration and diffusion. Herbivores consume vegetation with a saturating functional response G(B)=αB/(β+B) and experience natural mortality. Crucially, herbivore movement combines a density‑dependent random walk (D_R(B)) that slows down in the presence of vegetation and a directed movement term (vegetaxis) D_V(B)∇B that drives herbivores up gradients of vegetation biomass, reflecting an exploitation strategy.

Using numerical continuation (MatCont for ODEs, pde2path for PDEs) the authors map the bifurcation structure as precipitation P varies. Three uniform steady states exist: bare soil (BS), vegetation without herbivores (UV), and vegetation with herbivores (UH). At low P, BS is stable; as P exceeds a threshold P_T, a Turing instability creates stationary vegetation patterns. Further increase to P_H allows herbivores to persist, producing the UH state. The presence of herbivores raises soil water because grazed vegetation reduces transpiration.

Pattern analysis reveals two distinct routes to traveling vegetation‑herbivore waves. In the low‑precipitation regime, stationary vegetation patterns lose stability via a Hopf bifurcation, leading to oscillatory dynamics where herbivores concentrate on one side of each vegetation patch. This asymmetry creates a traveling wave that propagates across the landscape. In the high‑precipitation regime, the uniform UH state undergoes a non‑uniform oscillatory instability, generating a wave in which herbivores are spread over the domain but still display a side‑biased density relative to each vegetation patch. The wave speed, amplitude and shape depend sensitively on parameters such as the maximal consumption rate α, satiation biomass β, and the vegetaxis coefficients D_HB and D_HH.

The ecological mechanism uncovered is that the asymmetric herbivore distribution imposes strong grazing pressure on one flank of a vegetation patch while leaving the opposite flank relatively untouched. This uneven stress reduces the average herbivory pressure compared with a spatially uniform herbivore field, thereby enhancing the resilience of the vegetation‑herbivore system. The traveling wave state thus supports higher herbivore populations under combined water and grazing stress, offering a potential buffer against desertification.

Parameter sweeps show that increasing α or decreasing β (i.e., making herbivores more dependent on vegetation) lowers the precipitation threshold for wave emergence, while stronger vegetaxis (higher D_HB) promotes faster wave propagation. The model predicts transitions among stationary patterns, traveling waves, and mixed states, suggesting that real drylands could exhibit similar dynamical regimes depending on climate variability and grazing pressure.

In summary, the study demonstrates that herbivore‑induced traveling vegetation‑herbivore waves are a robust dynamical outcome of coupled water‑vegetation‑herbivore interactions. These waves mitigate both water scarcity and herbivory stress, thereby improving ecosystem sustainability and food security in regions facing drought and growing human populations. The work bridges pattern‑formation theory with ecological realism, offering testable predictions for field observations and management strategies in arid rangelands.