Dynamics and spatial organization of plant communities in water-limited systems

A mathematical model for plant communities in water-limited systems is introduced and applied to a mixed woody-herbaceous community. Two feedbacks between biomass and water are found to be of crucial importance for understanding woody-herbaceous interactions: water uptake by plants' roots and increased water infiltration at vegetation patches. The former acts to increase interspecific competition while the latter favors facilitation. The net interspecific interaction is determined by the relative strength of the two feedbacks. The model is used to highlight new mechanisms of plant-interaction change by studying factors that tilt the balance between the two feedbacks. Factors addressed in this study include environmental stresses and patch dynamics of the woody species. The model is further used to study mechanisms of species-diversity change by taking into consideration tradeoffs in species traits and conditions giving rise to irregular patch patterns.

A mathematical model of plants as ecosystem engineers

Understanding the structure and dynamics of plant communities in water-limited systems often calls for the identification of ecosystem engineers--key species that modify the landscape, redistribute resources and facilitate the growth of other species. Shrubs are excellent examples; they self-organize to form patterns of mesic patches which provide habitats for herbaceous species. In this paper we present a mathematical model for studying ecosystem engineering by woody plant species in drylands. The model captures various feedbacks between biomass and water including water uptake by plants' roots and increased water infiltration at vegetation patches. Both the uptake and the infiltration feedbacks act as mechanisms for vegetation pattern formation, but have opposite effects on the water resource; the former depletes the soil-water content under a vegetation patch, whereas the latter acts to increase it. Varying the relative strength of the two feedbacks we find a trade-off between the engineering capacity of a plant species and its resilience to disturbances. We further identify two basic soil-water distributions associated with engineering at the single patch level, hump-shaped and ring-shaped, and discuss the niches they form for herbaceous species. Finally, we study how pattern transitions at the landscape level feedback to the single patch level by affecting engineering strength.

Ecosystem engineers: from pattern formation to habitat creation

Habitat and species richness in drylands are affected by the dynamics of a few key species, termed "ecosystem engineers." These species modulate the landscape and redistribute the water resources so as to allow the introduction of other species. A mathematical model is developed for a pair of ecosystem engineers commonly found in drylands: plants forming vegetation patterns and cyanobacteria forming soil crusts. The model highlights conditions for habitat creation and for high habitat richness, and suggests a novel mechanism for species loss events as a result of environmental changes.

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.

Diversity of vegetation patterns and desertification

A new model for vegetation patterns is introduced. The model reproduces a wide range of patterns observed in water-limited regions, including drifting bands, spots, and labyrinths. It predicts transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation, through intermediate states of spot, stripe, and hole patterns. It also predicts wide precipitation ranges where different stable states coexist. Using these predictions we propose a novel explanation of desertification phenomena and a new approach to classifying aridity.

Four-phase patterns in forced oscillatory systems

We investigate pattern formation in self-oscillating systems forced by an external periodic perturbation. Experimental observations and numerical studies of reaction-diffusion systems and an analysis of an amplitude equation are presented. The oscillations in each of these systems entrain to rational multiples of the perturbation frequency for certain values of the forcing frequency and amplitude. We focus on the subharmonic resonant case where the system locks at one-fourth the driving frequency, and four-phase rotating spiral patterns are observed at low forcing amplitudes. The spiral patterns are studied using an amplitude equation for periodically forced oscillating systems. The analysis predicts a bifurcation (with increasing forcing) from rotating four-phase spirals to standing two-phase patterns. This bifurcation is also found in periodically forced reaction-diffusion equations, the FitzHugh-Nagumo and Brusselator models, even far from the onset of oscillations where the amplitude equation analysis is not strictly valid. In a Belousov-Zhabotinsky chemical system periodically forced with light we also observe four-phase rotating spiral wave patterns. However, we have not observed the transition to standing two-phase patterns, possibly because with increasing light intensity the reaction kinetics become excitable rather than oscillatory.

Front propagation and pattern formation in anisotropic bistable media

The effects of diffusion anisotropy on pattern formation in bistable media are studied using a FitzHugh-Nagumo reaction-diffusion model. A relation between the normal velocity of a front and its curvature is derived and used to identify distinct spatiotemporal patterns induced by the diffusion anisotropy. In a wide parameter range anisotropy is found to have an ordering effect: initial patterns evolve into stationary or breathing periodic stripes parallel to one of the principal axes. In a different parameter range, anisotropy is found to induce spatiotemporal chaos confined to one space dimension, a state we term "stratified chaos."

Phase dynamics of nearly stationary patterns in activator-inhibitor systems

The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model are studied using a phase dynamics approach. A Cross-Newell phase equation describing slow and weak modulations of periodic stationary solutions is derived. The derivation applies to the bistable, excitable, and Turing unstable regimes. In the bistable case stability thresholds are obtained for the Eckhaus and zigzag instabilities and for the transition to traveling waves. Neutral stability curves demonstrate the destabilization of stationary planar patterns at low wave numbers to zigzag and traveling modes. Numerical solutions of the model system support the theoretical findings.

Multiphase patterns in periodically forced oscillatory systems

Periodic forcing of an oscillatory system produces frequency locking bands within which the system frequency is rationally related to the forcing frequency. We study extended oscillatory systems that respond to uniform periodic forcing at one quarter of the forcing frequency (the 4:1 resonance). These systems possess four coexisting stable states, corresponding to uniform oscillations with successive phase shifts of pi/2. Using an amplitude equation approach near a Hopf bifurcation to uniform oscillations, we study front solutions connecting different phase states. These solutions divide into two groups: pi fronts separating states with a phase shift of pi and pi/2 fronts separating states with a phase shift of pi/2. We find a type of front instability where a stationary pi front "decomposes" into a pair of traveling pi/2 fronts as the forcing strength is decreased. The instability is degenerate for an amplitude equation with cubic nonlinearities. At the instability point a continuous family of pair solutions exists, consisting of pi/2 fronts separated by distances ranging from zero to infinity. Quintic nonlinearities lift the degeneracy at the instability point but do not change the basic nature of the instability. We conjecture the existence of similar instabilities in higher 2n:1 resonances (n=3,4, em leader) where stationary pi fronts decompose into n traveling pi/n fronts. The instabilities designate transitions from stationary two-phase patterns to traveling 2n-phase patterns. As an example, we demonstrate with a numerical solution the collapse of a four-phase spiral wave into a stationary two-phase pattern as the forcing strength within the 4:1 resonance is increased.

Impulse patterning and relaxational propagation in excitable media

Wavetrains of impulses in homogeneous excitable media relax during propagation toward constant-speed patterns. Here we present a study of this relaxation process. Starting with the basic reaction-diffusion or cable equations, we derive kinematics for the trajectories of widely spaced impulses in the form of ordinary differential equations for the set of times at which impulses arrive at a given point in space. Stability criteria derived from these equations allow us to determine the possible asymptotic forms of propagating trains. When the recovery after excitation is monotonic, only one stable train exists for a given propagation speed. In the case of an oscillatory recovery, however, many stable trains are possible. This essential difference between monotonic and oscillatory recoveries manifests itself in qualitatively distinct relaxational behaviors.