Time-dependent localized patterns in a predator-prey model.

Numerical continuation is used to compute solution branches in a two-component reaction-diffusion model of Leslie-Gower type. Two regimes are studied in detail. In the first, the homogeneous state loses stability to supercritical spatially uniform oscillations, followed by a subcritical steady state bifurcation of Turing type.

Origin of jumping oscillons in an excitable reaction-diffusion system.

Oscillons, i.e., immobile spatially localized but temporally oscillating structures, are the subject of intense study since their discovery in Faraday wave experiments.

Localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk.

Axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation posed on a disk with Neumann boundary conditions are studied via numerical continuation and bifurcation analysis. Axisymmetric localized solutions in the form of spots and rings known from earlier studies persist and snake in the usual fashion until they begin to interact with the boundary. Depending on parameters, including the disk radius, these states may or may not connect to the branch of domain-filling target states.

Bending and pinching of three-phase stripes: From secondary instabilities to morphological deformations in organic photovoltaics.

Optimizing the properties of the mosaic nanoscale morphology of bulk heterojunction (BHJ) organic photovoltaics (OPV) is not only challenging technologically but also intriguing from the mechanistic point of view. Among the recent breakthroughs is the identification and utilization of a three-phase (donor-mixed-acceptor) BHJ, where the (intermediate) mixed phase can inhibit mesoscale morphological changes, such as phase separation. Using a mean-field approach, we reveal and distinguish between generic mechanisms that alter, through transverse instabilities, the evolution of stripes: the bending (zigzag mode) and the pinching (cross-roll mode) of the donor-acceptor domains.

Stripes on finite domains: Why the zigzag instability is only a partial story.

Stationary periodic patterns are widespread in natural sciences, ranging from nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical, and biological media and to macro-scale vegetation and cloud patterns. Their formation is usually due to a primary symmetry breaking of a uniform state to stripes, often followed by secondary instabilities to form zigzag and labyrinthine patterns. These secondary instabilities are well studied under idealized conditions of an infinite domain; however, on finite domains, the situation is more subtle since the unstable modes depend also on boundary conditions.

The role of spatial self-organization in the design of agroforestry systems.

The development of sustainable agricultural systems in drylands is currently a crucial issue in the context of mitigating the outcomes of population growth under the conditions of climatic changes. The need to meet the growing demand for food, fodder, and fuel, together with the hazards due to climate change, requires cross-disciplinary studies of ways to increase livelihood while minimizing the impact on the environment. Practices of agroforestry systems, in which herbaceous species are intercropped between rows of woody species plantations, have been shown to mitigate several of the predicaments of climatic changes.

Defectlike structures and localized patterns in the cubic-quintic-septic Swift-Hohenberg equation.

We study numerically the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded one-dimensional domains. Under appropriate conditions stripes with wave number k≈1 bifurcate supercritically from the zero state and form S-shaped branches resulting in bistability between small and large amplitude stripes. Within this bistability range we find stationary heteroclinic connections or fronts between small and large amplitude stripes, and demonstrate that the associated spatially localized defectlike structures either snake or fall on isolas.

From bulk self-assembly to electrical diffuse layer in a continuum approach for ionic liquids: The impact of anion and cation size asymmetry.

Ionic liquids are solvent-free electrolytes, some of which possess an intriguing self-assembly at finite length scale due to Coulombic interactions. Using a continuum framework (based on Onsager's relations), it is shown that bulk nanostructures arise via linear (supercritical) and nonlinear (subcritical) bifurcations (morphological phase transitions), which also directly affect the electrical double layer structure. A Ginzburg-Landau amplitude equation is derived and the bifurcation type is related to model parameters, such as temperature, potential, and interactions.

Desertification by front propagation?

Understanding how desertification takes place in different ecosystems is an important step in attempting to forecast and prevent such transitions. Dryland ecosystems often exhibit patchy vegetation, which has been shown to be an important factor on the possible regime shifts that occur in arid regions in several model studies. In particular, both gradual shifts that occur by front propagation, and abrupt shifts where patches of vegetation vanish at once, are a possibility in dryland ecosystems due to their emergent spatial heterogeneity.

Soliton transport in tubular networks: transmission at vertices in the shrinking limit.

Soliton transport in tubelike networks is studied by solving the nonlinear Schrödinger equation (NLSE) on finite thickness ("fat") graphs. The dependence of the solution and of the reflection at vertices on the graph thickness and on the angle between its bonds is studied and related to a special case considered in our previous work, in the limit when the thickness of the graph goes to zero. It is found that both the wave function and reflection coefficient reproduce the regime of reflectionless vertex transmission studied in our previous work.

Approximating the dynamics of communicating cells in a diffusive medium by ODEs-homogenization with localization.

Bacteria may change their behavior depending on the population density. Here we study a dynamical model in which cells of radius [R] within a diffusive medium communicate with each other via diffusion of a signalling substance produced by the cells. The model consists of an initial boundary value problem for a parabolic PDE describing the exterior concentration [u] of the signalling substance, coupled with [N] ODEs for the masses [ai] of the substance within each cell.

Pattern formation for NO+NH3 on Pt(100): two-dimensional numerical results.

The Lombardo-Fink-Imbihl model of the NO+NH3 reaction on a Pt(100) surface consists of seven coupled ordinary differential equations (ODE) and shows stable relaxation oscillations with sharp transitions in the relevant temperature range. Here we study numerically the effect of coupling of these oscillators by surface diffusion in two dimensions. We find different types of patterns, in particular phase clusters and standing waves.