Desertification by front propagation?

Characterizing Oscillatory and Excitability Regimes in a Protein-Free Lipid Membrane

Observations of electric potential oscillations in artificial lipid bilayers near the order-disorder transition indicate the existence of a stable limit cycle and, therefore, the possibility of producing excitable signals close to the bifurcation. We present a theoretical investigation of membrane oscillatory and excitability regimes induced by an increase in ion permeability at the order-disorder transition. The model accounts for the coupled effects of state-dependent permeability, membrane charge density, and hydrogen ion adsorption. A bifurcation diagram shows a transition between fixed-point and limit cycle solutions, enabling both oscillatory and excitability responses at different values of the acid association parameter. Oscillations are identified in terms of the membrane state, electric potential difference, and ion concentration near the membrane. The emerging voltage and time scales agree with measurements. Excitability is demonstrated by applying an external electric current stimulus, and the emerging signals display a threshold response and the appearance of repetitive signals upon using a long-lasting stimulus. The approach highlights the important role of the order-disorder transition, enabling membrane excitability in the absence of specialized proteins.

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.

Spatial dynamics of a vegetation model with uptake-diffusion feedback in an arid environment

Vegetation patterns with a variety of structures is amazing phenomena in arid or semi-arid areas, which can identify the evolution law of vegetation and are typical signals of ecosystem functions. Many achievements have been made in this respect, yet the mechanisms of uptake-diffusion feedback on the pattern structures of vegetation is not fully understood. To well reveal the influences of parameters perturbation on the pattern formation of vegetation, we give a comprehensive analysis on a vegetation-water model in the forms of reaction-diffusion equation which is posed by Zelnik et al. (Proc Natl Acad Sci 112:12,327-12,331, 2015). We obtain the exact parameters range for stationary patterns and show the dynamical behaviors near the bifurcation point based on nonlinear analysis. It is found that the model has the properties of spot, labyrinth and gap patterns. Moreover, water diffusion rate prohibits the growth of vegetation while shading parameter promotes the increase of vegetation biomass. Our results show that gradual transitions from uniform state to gap pattern can occur for suitable value of parameters which may induce the emergence of desertification.

Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models

In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involved on the inertial times, reveals some intriguing consequences. To show in detail the richness of such a scenario, we present, as an illustrative example, the pattern dynamics occurring in the hyperbolic generalization of the extended Klausmeier model. This is a simple two-species model used to describe the migration of vegetation stripes along the hillslope of semiarid environments. By means of a thorough comparison between analytical predictions and numerical simulations, we show that inertia, apart from enlarging the region of the parameter plane where wave instability occurs, may also modulate the key features of the coherent structures, solution of the CCGL equation. In particular, it is proven that inertial effects play a role, not only during transient regime from the spatially-homogeneous steady state toward the patterned state, but also in altering the amplitude, the wavelength, the angular frequency, and even the stability of the phase-winding solutions.

Unified framework for localized patterns in reaction-diffusion systems; the Gray-Scott and Gierer-Meinhardt cases

A recent study of canonical activator-inhibitor Schnakenberg-like models posed on an infinite line is extended to include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model to the Gray-Scott model. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localized pattern parameter regime as the homotopy parameter varies. Several distinct codimension-two bifurcations are discovered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other pattern-formation systems arising as models of natural phenomena. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Agricultural land use and the sustainability of social-ecological systems

Agricultural land expansion and intensification, driven by human consumption of agricultural goods, are among the major threats to environmental degradation and biodiversity conservation. Land degradation can ultimately hamper agricultural production through a decrease in ecosystem services. Thus, designing viable land use policies is a key sustainability challenge. We develop a model describing the coupled dynamics of human demography and landscape composition, while imposing a trade-off between agricultural expansion and in-tensification. We model land use strategies spanning from low-intensity agriculture and high land conversion rates per person to high-intensity agriculture and low land conversion rates per person; and explore their consequences on the long-term dynamics of the coupled human-land system. We seek to characterise the strategies' viability in the long run; and understand the mechanisms that potentially lead to large-scale land degradation and population collapse due to resource scarcity. We show that the viability of land use strategies strongly depends on the land's intrinsic recovery rate. We also find that socio-ecological collapses occur when agricultural intensification is not accompanied by a sufficient decrease in land conversion. Based on these findings we stress the dangers of uninformed land use planning and the importance of precautionary behaviour for land use management and land use policy design.

From Historical Narratives to Circular Economy: De-Complexifying the "Desertification" Debate

Assuming the importance of a “socioeconomic mosaic” influencing soil and land degradation at the landscape scale, spatial contexts should be considered in the analysis of desertification risk as a base for the design of appropriate counteracting strategies. A holistic approach grounded on a multi-scale qualitative and quantitative assessment is required to identify optimal development strategies regulating the socioeconomic dimensions of land degradation. In the last few decades, the operational thinking at the base of a comprehensive, holistic theory of land degradation evolved toward many different conceptual steps. Moving from empirical, qualitative and unstructured frameworks to a more structured, rational and articulated thinking, such theoretical approaches have been usually oriented toward complex and non-linear dynamics benefiting from progressive and refined approximations. Based on these premises, eleven disciplinary approaches were identified and commented extensively on in the present study, and were classified along a gradient of increasing complexity, from more qualitative and de-structured frameworks to more articulated, non-linear thinking aimed at interpreting the intrinsic fragmentation and heterogeneity of environmental and socioeconomic processes underlying land degradation. Identifying, reviewing and classifying such approaches demonstrated that the evolution of global thinking in land degradation was intimately non-linear, developing narrative and deductive approaches together with inferential, experimentally oriented visions. Focusing specifically on advanced economies in the world, our review contributes to systematize multiple—sometimes entropic—interpretations of desertification processes into a more organized framework, giving value to methodological interplays and specific interpretations of the latent processes underlying land degradation.

Formation of localized states in dryland vegetation: Bifurcation structure and stability

We study theoretically the emergence of localized states of vegetation close to the onset of desertification. These states are formed through the locking of vegetation fronts, connecting a uniform vegetation state with a bare soil state, which occurs nearby the Maxwell point of the system. To study these structures we consider a universal model of vegetation dynamics in drylands, which has been obtained as the normal form for different vegetation models. Close to the Maxwell point localized gaps and spots of vegetation exist and undergo collapsed snaking. The presence of gaps strongly suggest that the ecosystem may undergo a recovering process. In contrast, the presence of spots may indicate that the ecosystem is close to desertification.

Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion

The origin, stability, and bifurcation structure of different types of bright localized structures described by the Lugiato-Lefever equation are studied. This mean field model describes the nonlinear dynamics of light circulating in fiber cavities and microresonators. In the case of anomalous group velocity dispersion and low values of the intracavity phase detuning these bright states are organized in a homoclinic snaking bifurcation structure. We describe how this bifurcation structure is destroyed when the detuning is increased across a critical value, and determine how a bifurcation structure known as foliated snaking emerges.

Implications of tristability in pattern-forming ecosystems

Many ecosystems show both self-organized spatial patterns and multistability of possible states. The combination of these two phenomena in different forms has a significant impact on the behavior of ecosystems in changing environments. One notable case is connected to tristability of two distinct uniform states together with patterned states, which has recently been found in model studies of dryland ecosystems. Using a simple model, we determine the extent of tristability in parameter space, explore its effects on the system dynamics, and consider its implications for state transitions or regime shifts. We analyze the bifurcation structure of model solutions that describe uniform states, periodic patterns, and hybrid states between the former two. We map out the parameter space where these states exist, and note how the different states interact with each other. We further focus on two special implications with ecological significance, breakdown of the snaking range and complex fronts. We find that the organization of the hybrid states within a homoclinic snaking structure breaks down as it meets a Maxwell point where simple fronts are stationary. We also discover a new series of complex fronts between the uniform states, each with its own velocity. We conclude with a brief discussion of the significance of these findings for the dynamics of regime shifts and their potential control.

Nonlocal grazing in patterned ecosystems

Many ecosystems exhibit gapped, labyrinthine, striped or spotted patterns. Important examples are vegetation patterns in drylands: these patterns are viewed as precursors of a catastrophic transition to a degraded state. A possible source of degradation is overgrazing, but many current spatially extended models include grazing in a local linear way. In this article nonlocal grazing responses are derived, taking into account (1) how many consumers there are (demographic response) (2) where they are (aggregative response) and (3) how much they forage (functional response). Different assumptions lead to different grazing responses, the type of grazing has a large influence on how ecosystems adapt to changing environmental conditions. In dryland simulations the different types of grazing are shown to alter the desertification process driven by decreasing rainfall. A sufficiently strong aggregative response leads to the suppression of vegetation patterns, nuancing their role as generic early warning signals.