Localised pattern formation in a model for dryland vegetation

Role of herbivory in shaping the dryland vegetation ecosystem: Linking spiral vegetation patterns and nonlinear, nonlocal grazing

Self-organized vegetation patterns are an amazing aspect of dryland ecosystems; in addition to being visually appealing, patterns control how these water-deprived systems react to escalating environmental stress. Although there is a wide variety of vegetation patterns, little is known about the mechanisms behind spiral patterns. The well-known models that explain other vegetation patterns such stripes, rings, and fairy circles cannot account for these spirals. Here we have adopted a modeling approach in which the interplay between herbivore grazing and vegetation is found to be the reason why spirals form. To comprehend the nonlinear dependence of grazing on the availability vegetation, we have introduced a grazing term that gets saturated when forage is abundant. To account for the impact of the spatial nonhomogeneity in vegetation layout, it is thought that grazing is dependent on mean vegetation density rather than density at a single site. Results show how the system dynamics is changed fundamentally depending on the different types of grazing response. Incorporation of nonlocality into the herbivore grazing leads to spiral-shaped vegetation patterns only in natural grazing scenarios; however, no patterning is seen in human controlled herbivory. Overall, our research points to the nonlocal, nonlinear grazing behavior of herbivores as one of the major driving forces for the development of spiral patterns.

Transitions between dissipative localized structures in the simplified Gilad-Meron model for dryland plant ecology

Spatially extended patterns and multistability of possible different states are common in many ecosystems, and their combination has an important impact on their dynamical behaviors. One potential combination involves tristability between a patterned state and two different uniform states. Using a simplified version of the Gilad-Meron model for dryland ecosystems, we study the organization, in bifurcation terms, of the localized structures arising in tristable regimes. These states are generally related to the concept of wave front locking and appear in the form of spots and gaps of vegetation. We find that the coexistence of localized spots and gaps, within tristable configurations, yields the appearance of hybrid states. We also study the emergence of spatiotemporal localized states consisting of a portion of a periodic pattern embedded in a uniform Hopf-like oscillatory background in a subcritical Turing-Hopf dynamical regime.

Effect of nonlocal grazing on dry-land vegetation dynamics

Dry-land ecosystems have become a matter of grave concern, due to the growing threat of land degradation and bioproductivity loss. Self-organized vegetation patterns are a remarkable characteristic of these ecosystems; apart from being visually captivating, patterns modulate the system response to increasing environmental stress. Empirical studies hinted that herbivory is one the key regulatory mechanisms behind pattern formation and overall ecosystem functioning. However, most of the mathematical models have taken a mean-field strategy to grazing; foraging has been considered to be independent of spatial distribution of vegetation. To this end, an extended version of the celebrated plant-water model due to Klausmeier has been taken as the base here. To encompass the effect of heterogeneous vegetation distribution on foraging intensity and subsequent impact on entire ecosystem, grazing is considered here to depend on spatially weighted average vegetation density instead of density at a particular point. Moreover, varying influence of vegetation at any location over gazing elsewhere is incorporated by choosing a suitable averaging function. A comprehensive analysis demonstrates that inclusion of spatial nonlocality alters the understanding of system dynamics significantly. The grazing ecosystem is found to be more resilient to increasing aridity than it was anticipated to be in earlier studies on nonlocal grazing. The system response to rising environmental pressure is also observed to vary depending on the grazer. Obtained results also suggest the possibility of multistability due to the history dependence of the system response. Overall, this work indicates that the spatial heterogeneity in grazing intensity has a decisive role to play in the functioning of water-limited ecosystems.

Linking spatial self-organization to community assembly and biodiversity

Temporal shifts to drier climates impose environmental stresses on plant communities that may result in community reassembly and threatened ecosystem services, but also may trigger self-organization in spatial patterns of biota and resources, which act to relax these stresses. The complex relationships between these counteracting processes - community reassembly and spatial self-organization - have hardly been studied. Using a spatio-temporal model of dryland plant communities and a trait-based approach, we study the response of such communities to increasing water-deficit stress. We first show that spatial patterning acts to reverse shifts from fast-growing species to stress-tolerant species, as well as to reverse functional-diversity loss. We then show that spatial self-organization buffers the impact of further stress on community structure. Finally, we identify multistability ranges of uniform and patterned community states and use them to propose forms of non-uniform ecosystem management that integrate the need for provisioning ecosystem services with the need to preserve community structure.

Continuum Modeling of Discrete Plant Communities: Why Does It Work and Why Is It Advantageous?

Understanding ecosystem response to drier climates calls for modeling the dynamics of dryland plant populations, which are crucial determinants of ecosystem function, as they constitute the basal level of whole food webs. Two modeling approaches are widely used in population dynamics, individual (agent)-based models and continuum partial-differential-equation (PDE) models. The latter are advantageous in lending themselves to powerful methodologies of mathematical analysis, but the question of whether they are suitable to describe small discrete plant populations, as is often found in dryland ecosystems, has remained largely unaddressed. In this paper, we first draw attention to two aspects of plants that distinguish them from most other organisms—high phenotypic plasticity and dispersal of stress-tolerant seeds—and argue in favor of PDE modeling, where the state variables that describe population sizes are not discrete number densities, but rather continuous biomass densities. We then discuss a few examples that demonstrate the utility of PDE models in providing deep insights into landscape-scale behaviors, such as the onset of pattern forming instabilities, multiplicity of stable ecosystem states, regular and irregular, and the possible roles of front instabilities in reversing desertification. We briefly mention a few additional examples, and conclude by outlining the nature of the information we should and should not expect to gain from PDE model studies.

Assessing the robustness of spatial pattern sequences in a dryland vegetation model

A particular sequence of patterns, 'gaps→labyrinth→spots', occurs with decreasing precipitation in previously reported numerical simulations of partial differential equation dryland vegetation models. These observations have led to the suggestion that this sequence of patterns can serve as an early indicator of desertification in some ecosystems. Because parameter values in the vegetation models can take on a range of plausible values, it is important to investigate whether the pattern sequence prediction is robust to variation. For a particular model, we find that a quantity calculated via bifurcation-theoretic analysis appears to serve as a proxy for the pattern sequences that occur in numerical simulations across a range of parameter values. We find in further analysis that the quantity takes on values consistent with the standard sequence in an ecologically relevant limit of the model parameter values. This suggests that the standard sequence is a robust prediction of the model, and we conclude by proposing a methodology for assessing the robustness of the standard sequence in other models and formulations.