Reversing desertification as a spatial resonance problem

An important environmental application of pattern control by periodic spatial forcing is the restoration of vegetation patterns in water-limited ecosystems that went through desertification. Vegetation restoration is often based on periodic landscape modulations that intercept overland water flow and form favorable conditions for vegetation growth. Viewing this method as a spatial resonance problem, we show that plain realizations of this method, assuming a complete vegetation response to the imposed modulation pattern, suffer from poor resilience to rainfall variability. By contrast, less intuitive realizations, based on the inherent spatial modes of vegetation growth and involving partial vegetation implantation, can be highly resilient and equally productive. We derive these results using two complementary models, a realistic vegetation model, and a simple pattern formation model that lends itself to mathematical analysis and highlights the universal aspects of the behaviors found with the vegetation model. We focus on reversing desertification as an outstanding environmental problem, but the main conclusions hold for any spatially forced system near the onset of a finite-wave-number instability that is subjected to noisy conditions.

Multiplicative jump processes and applications to leaching of salt and contaminants in the soil

We consider simple systems driven multiplicatively by white shot noise, which appear in the modeling of the dynamics of soil nutrients and contaminants. The dynamics of these systems is analyzed in two ways: solving a hierarchy of linear ordinary differential equations for the moments, which gives a time scale of convergence of the stationary probability density function; and characterizing the crossing properties, such as the mean first-passage time and the mean frequency of level crossing. These results are readily applicable to the study of geophysical systems, such as the problem of accumulation of salt in the root zone, i.e., soil salinization.

Spatial forcing of pattern-forming systems that lack inversion symmetry

The entrainment of periodic patterns to spatially periodic parametric forcing is studied. Using a weak nonlinear analysis of a simple pattern formation model we study the resonant responses of one-dimensional systems that lack inversion symmetry. Focusing on the first three n:1 resonances, in which the system adjusts its wavenumber to one nth of the forcing wavenumber, we delineate commonalities and differences among the resonances. Surprisingly, we find that all resonances show multiplicity of stable phase states, including the 1:1 resonance. The phase states in the 2:1 and 3:1 resonances, however, differ from those in the 1:1 resonance in remaining symmetric even when the inversion symmetry is broken. This is because of the existence of a discrete translation symmetry in the forced system. As a consequence, the 2:1 and 3:1 resonances show stationary phase fronts and patterns, whereas phase fronts within the 1:1 resonance are propagating and phase patterns are transients. In addition, we find substantial differences between the 2:1 resonance and the other two resonances. While the pattern forming instability in the 2:1 resonance is supercritical, in the 1:1 and 3:1 resonances it is subcritical, and while the inversion asymmetry extends the ranges of resonant solutions in the 1:1 and 3:1 resonances, it has no effect on the 2:1 resonance range. We conclude by discussing a few open questions.

Competing resonances in spatially forced pattern-forming systems

Spatial periodic forcing can entrain a pattern-forming system in the same way as temporal periodic forcing can entrain an oscillator. The forcing can lock the pattern's wave number to a fraction of the forcing wave number within tonguelike domains in the forcing parameter plane, it can increase the pattern's amplitude, and it can also create patterns below their onset. We derive these results using a multiple-scale analysis of a spatially forced Swift-Hohenberg equation in one spatial dimension. In two spatial dimensions the one-dimensional forcing can induce a symmetry-breaking instability that leads to two-dimensional (2D) patterns, rectangular or oblique. These patterns resonate with the forcing by locking their wave-vector component in the forcing direction to half the forcing wave number. The range of this type of 2:1 resonance overlaps with the 1:1 resonance tongue of stripe patterns. Using a multiple-scale analysis in the overlap region we show that the 2D patterns can destabilize the 1:1 resonant stripes even at exact resonance. This result sheds new light on the use of spatial periodic forcing for controlling patterns.

Spatial periodic forcing can displace patterns it is intended to control

Spatial periodic forcing of pattern-forming systems is an important, but lightly studied, method of controlling patterns. It can be used to control the amplitude and wave number of one-dimensional periodic patterns, to stabilize unstable patterns, and to induce them below instability onset. We show that, although in one spatial dimension the forcing acts to reinforce the patterns, in two dimensions it acts to destabilize or displace them by inducing two-dimensional rectangular and oblique patterns.

Dual-mode spiral vortices

We show that spiral vortices in oscillatory systems can lose stability to secondary modes to form dual-mode spiral vortices. The secondary modes grow at the vortex core where the oscillation amplitude vanishes but are nonlinearly damped by the oscillatory mode away from the core. Gradients of the oscillation phase, induced by the hosted secondary mode, can lead to additional hosting events that culminate in periodic core oscillations or in a novel form of spatiotemporal chaos. The results of this study apply to physical, chemical, and biological systems that go through cusp-Hopf, fold-Hopf, and Hopf-Turing bifurcations.