Ehud Meron

Pattern formation in excitable media

Abstract Excitable media are extended nonequilibrium systems having uniform rest states that are linearly stable but susceptible to finite perturbations. Depending on the forms of these perturbations, a variety of wave patterns can be triggered; solitary waves, target like patterns, and spiral waves are a few examples. These media are naturally encountered in biological systems and are also generated by a class of chemical systems, the most familiar of which is the Belousov-Zhabotinsky reaction. The recent progress that has been made in understanding patterns in one and two space dimensions is reviewed. Special attention is given to theoretical aspects, but experiments and numerical simulations are described as well. On the theoretical side two basic approaches are described, singular perturbation theories and kinematical theories. The two approaches have different ranges of validity and address different questions, but also have an overlap range that allows for comparison. The availability of large fast computers made extensive explorations of parameter spaces possible. New regimes of dynamical behavior found in that way are described. Finally, an account is given of the significant experimental progress that has been made recently by exploiting spectrophotometric and digital imaging techniques and by using new reactor designs.

Vegetation patterns along a rainfall gradient

A continuum model for vegetation patterns in water limited systems is presented. The model involves two variables, the vegetation biomass density and the soil water density, and takes into account positive feedback relations between the two. The model predicts transitions from bare-soil at low precipitation to homogeneous vegetation at high precipitation through intermediate states of spot, stripe and gap patterns. It also predicts the appearance of ring-like shapes as transient forms toward asymptotic stripes. All these patterns have been identified in observations made on two types of perennial grasses in the Northern Negev. Another prediction of the model is the existence of wide precipitation ranges where different stable states coexist, e.g. a bare soil state and a spot pattern, a spot pattern and a stripe pattern, and so on. This result suggests the interpretation of desertification followed by recovery as an hysteresis loop and sheds light on the irreversibility of desertification. 2003 Elsevier Ltd. All rights reserved.

Woody Species as Landscape Modulators and Their Effect on Biodiversity Patterns

ABSTRACT Ecological research on organism-environment interactions has developed asymmetrically. Modulation of organisms by the environment has received much attention, while theoretical studies on the environmental impact of organisms have until recently been limited. We propose a theoretical framework for studying the environmental impacts of woody plants in order to understand their effects on biodiversity. We adopt pattern formation theory to discuss how woody plants organize ecological systems on the patch and landscape levels through patch formation, and how organism patchiness creates resource patchiness that affects biodiversity. We suggest an integrative model that links organisms as landscape modulators through resource distribution and species filtering from larger to smaller spatial scales. Our “biodiversity cycling hypothesis” states that in organism-modulated landscapes, disturbance enables the coexistence of different developmental stages of vegetation patches, thereby increasing biodiversity. This hypothesis emphasizes that species and landscape diversity vary with the development, renewal, maturation, and decay of biotically induced patches.

Patterns of propagating pulses

The complex dynamics that arise in certain nonlinear partial differential equations in time and in one space dimension are studied. In the general case considered, the equation admits a solitary wave in the form of a pulse tailing off exponentially, fore and aft, with possibly oscillatory character. Complicated solutions are described by a superposition of many such solitary structures in interaction. The description is asymptotic in terms of a parameter that becomes exponentially small as the ratio of typical pulse separation to pulse width becomes large. The outcome is a set of dynamical equations for the motion of the individual pulses with nearest neighbor interactions. This system of ordinary differential equations (ODEs) admits a wide range of patterns, both regular and chaotic. The stability theory of such patterns is sketched and the continuum limit of the lattice-dynamical equations of the pulses is given.

Periodic versus scale-free patterns in dryland vegetation

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.

Complex patterns in reaction-diffusion systems: A tale of two front instabilities.

Two front instabilities in a reaction-diffusion system are shown to lead to the formation of complex patterns. The first is an instability to transverse modulations that drives the formation of labyrinthine patterns. The second is a nonequilibrium Ising-Bloch (NIB) bifurcation that renders a stationary planar front unstable and gives rise to a pair of counterpropagating fronts. Near the NIB bifurcation the relation of the front velocity to curvature is highly nonlinear and transitions between counterpropagating fronts become feasible. Nonuniformly curved fronts may undergo local front transitions that nucleate spiral-vortex pairs. These nucleation events provide the ingredient needed to initiate spot splitting and spiral turbulence. Similar spatiotemporal processes have been observed recently in the ferrocyanide-iodate-sulfite reaction.

Localized structures in dryland vegetation: Forms and functions

Vegetation patches in drylands are localized structures of biomass and water. We study these structures using a mathematical modeling approach that captures biomass-water feedbacks. Biomass-water structures are found to differ in their spatial forms and ecological functions, depending on species type, soil conditions, precipitation range, and other environmental factors. Asymptotic spot structures can destabilize to form ring structures, expanding in the radial direction, or crescent structures, migrating uphill. Stable spot structures can differ in their soil-water distributions, forming water-enriched patches or water-deprived patches. The various biomass-water structures are expected to function differently in the context of a plant community, forming landscapes of varying species diversity.

Gradual regime shifts in spatially extended ecosystems

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.

Patterned vegetation and rainfall intermittency

We study a mathematical model for the dynamics of patterned dryland vegetation in the presence of rainfall intermittency, adopting a spatially explicit approach. We find that most results found for constant precipitation carry over to the case of intermittent rainfall, with a few important novelties. For intermittent precipitation, the functional forms of the water uptake and consequently of the vegetation growth rate play an important role. Nonlinear, concave-up forms of water uptake as a function of soil moisture lead to a beneficial effect of rainfall intermittency, with a stronger effect when vegetation feedbacks are absent. The results obtained with the explicit-space model employed here are in keeping with those provided by simpler, implicit-space approaches, and provide a more complete view of vegetation dynamics in arid ecosystems.

Modelling the survival of bacteria in drylands: the advantage of being dormant

We introduce a simple mathematical model for the description of ‘dormancy’, a survival strategy used by some bacterial populations that are intermittently exposed to external stress. We focus on the case of the cyanobacterial crust in drylands, exposed to severe water shortage, and compare the fate of ideal populations that are, respectively, capable or incapable of becoming dormant. The results of the simple model introduced here indicate that under a constant, even though low, supply of water the dormant strategy does not provide any benefit and it can, instead, decrease the chances of survival of the population. The situation is reversed for highly intermittent external stress, due to the presence of prolonged periods of dry conditions intermingled with short periods of intense precipitation. In this case, dormancy allows for the survival of the population during the dry periods. In contrast, bacteria that are incapable of turning into a dormant state cannot overcome the difficult times. The model also rationalizes why dormant bacteria, such as those composing the cyanobacterial crust in the desert, are extremely sensitive to other disturbances, such as trampling cattle.