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.

Significance: Combining model and empirical data analyses, we show that transitions between alternative stable states (regime shifts) in spatially extended ecosystems are not necessarily abrupt; cascades of local shifts between a multitude of stable states, composed of patterned and uniform domains, can result in global regime shifts that proceed gradually. In the Namibian fairy circle ecosystem (barren circular gaps in grasslands), such local shifts appear as fairy circle birth or death processes. This mechanism of regime shifts has never been demonstrated in a specific natural context. In addition, the results reported here further support the view of fairy circles as a self-organization phenomenon by providing a new type of evidence based on dynamical processes.

Many mathematical models have been proposed to explain the emergence of vegetation patterns in arid and semiarid environments, but only a few of them take into account the heterogeneity in the system properties. Here we present a rigorous study of the effects of heterogeneous soil‐water diffusivity on vegetation patterns, using two mathematical models. The two models differ in the pattern‐forming feedback that they capture; one model captures the infiltration contrast between vegetated and bare‐soil domains, whereas the other model captures the increased growth rate of denser vegetation due to an enhanced ability to extract water from the soil. In both models, the most significant effect of the heterogeneity on the soil‐water diffusivity is the increased durability of patterned vegetation to a reduced precipitation rate. An additional effect is that the heterogeneity makes the desertification process, namely, the transition from a spotted vegetation pattern to a bare‐soil state, more gradual than in the homogeneous system. Our findings suggest that the heterogeneity cannot be neglected in the study of critical transitions in heterogeneous ecosystems and, particularly, in the study of the desertification process due to climate changes or anthropogenic disturbances.

We use the context of dryland vegetation to study a general problem of complex pattern-forming systems: multiple pattern-forming instabilities that are driven by distinct mechanisms but share the same spectral properties. We find that the co-occurrence of two Turing instabilities when the driving mechanisms counteract each other in some region of the parameter space results in the growth of a single mode rather than two interacting modes. The interplay between the two mechanisms compensates for the simpler dynamics of a single mode by inducing a wider variety of patterns, which implies higher biodiversity in dryland ecosystems.

The dynamics of active particles is of interest at many levels and is the focus of theoretical and experimental research. There have been many attempts to describe the dynamics of particles affected by random active forces in terms of an effective temperature. This kind of description is tempting due to the similarities (or lack thereof) to systems in or near thermal equilibrium. However, the generality and validity of the effective temperature is not yet fully understood. Here we study the dynamics of trapped particles subjected to both thermal and active forces. The particles are not overdamped. Expressions for the effective temperature due to the potential and kinetic energies are derived, and they differ from each other. A third possible effective temperature can be derived from the escape time of the particle from the trap, using a Kramers-like expression for the mean escape time. We find that over a large fraction of the parameter space, the potential energy effective temperature is in agreement with the escape temperature, while the kinetic effective temperature only agrees with the former two in the overdamped limit. Moreover, we show that the specific implementation of the random active force, and not only its first two moments and the two point autocorrelation function, affects the escape-time distribution.

The underlying physics governing the diffusion of a tracer particle in a viscoelastic material is a topic of some dispute. The long-term memory in the mechanical response of such materials should induce diffusive motion with a memory kernel, such as fractional Brownian motion (fBM). This is the reason that microrheology is able to provide the shear modulus of polymer networks. Surprisingly, the diffusion of a tracer particle in a network of a purified protein, actin, was found to conform to the continuous time random walk type (CTRW). We set out to resolve this discrepancy by studying the tracer particle diffusion using two different tracer particle sizes, in actin networks of different mesh sizes. We find that the ratio of tracer particle size to the characteristic length scale of a bio-polymer network plays a crucial role in determining the type of diffusion it performs. We find that the diffusion of the tracer particles has features of fBm when the particle is large compared to the mesh size, of normal diffusion when the particle is much smaller than the mesh size, and of the CTRW in between these two limits. Based on our findings, we propose and verify numerically a new model for the motion of the tracer in all regimes. Our model suggests that diffusion in actin networks consists of fBm of the tracer particle coupled with caging events with power-law distributed escape times.