Ecosystem regime shifts are regarded as abrupt global transitions from one stable state to an alternative stable state, induced by slow environmental changes or by global disturbances. Spatially extended ecosystems, however, can also respond to local disturbances by the formation of small domains of the alternative state. Such a response can lead to gradual regime shifts involving front propagation and the coalescence of alternative-state domains. When one of the states is spatially patterned, a multitude of intermediate stable states appears, giving rise to step-like gradual shifts with extended pauses at these states. Using a minimal model, we study gradual state transitions and show that they precede abrupt transitions. We propose indicators to probe gradual regime shifts, and suggest that a combination of abrupt-shift indicators and gradual-shift indicators might be needed to unambiguously identify regime shifts. Our results are particularly relevant to desertification in drylands where transitions to bare soil take place from spotted vegetation, and the degradation process appears to involve step-like events of local vegetation mortality caused by repeated droughts.

Two major forms of vegetation patterns have been observed in drylands: nearly periodic patterns with characteristic length scales, and amorphous, scale-free patterns with wide patch-size distributions. The emergence of scale-free patterns has been attributed to global competition over a limiting resource, but the physical and ecological origin of this phenomenon is not understood. Using a spatially explicit mathematical model for vegetation dynamics in water-limited systems, we unravel a general mechanism for global competition: fast spatial distribution of the water resource relative to processes that exploit or absorb it. We study two possible realizations of this mechanism and identify physical and ecological conditions for scale-free patterns. We conclude by discussing the implications of this study for interpreting signals of imminent desertification.

Spatial patterns are ubiquitous in animate matter. Besides their intricate structure and beauty they generally play functional roles. The capacity of living systems to remain functional in changing environments is a question of utmost importance, but its intimate relationship to pattern formation is largely unexplored. Here, we address this relationship using dryland vegetation as a case study. Following a brief introduction to pattern-formation theory, we describe a mathematical model that captures several mechanisms of vegetation pattern formation and discuss ecological contexts that showcase different mechanisms. Using this model, we unravel the different vegetation patterns that keep dryland ecosystems viable along the rainfall gradient, identify multistability ranges where fronts separating domains of alternative stable states exist, and highlight the roles of front dynamics in mitigating or reversing desertification. The utility of satellite images in testing model predictions is discussed. An outlook on outstanding open problems concludes this paper.

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.