Zig-zag structures in silver dichromate precipitate.

Precipitation patterns are commonly concentric rings forming in a Petri dish or parallel bands appearing in a test tube (Liesegang phenomenon). The rings frequently consist of a number of convex segments that are separated from each other by spaces devoid of precipitate resulting in small gaps (dislocations). Along these gaps, the so-called zig-zag structures can form, which connect one side of a gap with its opposite side. We observe that the occurrence of zig-zags requires a minimum thickness of the reactive layer (≥ 0.8 mm). This fact together with microscopic evidence indicates their three-dimensional character. One finds that at the very beginning of the precipitation reaction a curling process starts in the corresponding contour lines. These observations suggest structures of a helicoid with the axis perpendicular to the plane of the reaction-diffusion front to pass through the layer. Zig-zags are not parallel to the reaction plane, i.e., they are not formed periodically, but evolve continuously as a rotating spiral wave. Thus, their topology is closely related to helices in a test tube.

Mathematical modeling of spatiotemporal patterns formed at a traveling reaction front.

In some chemical systems, the reaction proceeds in the form of a propagating wave. An example is the propagation of a combustion wave. At the front of such a wave, different oscillatory regimes and the appearance of spatiotemporal structures can be observed. We propose a qualitative mechanism for the formation of patterns at the front of the reaction. It is assumed that the reason is the interaction of two subsystems, one corresponding to the propagating front and the other describing the emerging patterns. The appropriate mathematical model contains two blocks: for the travelling front, we use a model of the Fisher-Kolmogorov-Petrovsky-Piskunov type, while patterns at the front are described by the FitzHugh-Nagumo type model. Earlier, we applied this approach to explain the occurrence of autowaves-target waves and spirals-at the front of the reaction. In the present paper, we demonstrate in numerical simulations that this approach also works effectively to explain stationary relative to the front patterns, the so-called Turing or cellular structures, that are observed experimentally, in particular, at the front of a combustion wave. We also investigate the dependence of these patterns on the thickness of the front and its speed, as well as on the degree of diffusion instability achieved within the front layer.

Widening the criteria for emergence of Turing patterns.

The classical concept for emergence of Turing patterns in reaction-diffusion systems requires that a system should be composed of complementary subsystems, one of which is unstable and diffuses sufficiently slowly while the other one is stable and diffuses sufficiently rapidly. In this work, the phenomena of emergence of Turing patterns are studied and do not fit into this concept, yielding the following results. (1) The criteria are derived, under which a reaction-diffusion system with immobile species should spontaneously produce Turing patterns under any diffusion coefficients of its mobile species. It is shown for such systems that under certain sets of types of interactions between their species, Turing patterns should be produced under any parameter values, at least provided that the corresponding spatially non-distributed system is stable. (2) It is demonstrated that in a reaction-diffusion system, which contains more than two species and is stable in absence of diffusion, the presence of a sufficiently slowly diffusing unstable subsystem is already sufficient for diffusion instability (i.e., Turing or wave instability), while its complementary subsystem can also be unstable. (3) It is shown that the presence of an immobile unstable subsystem, which leads to destabilization of waves within an infinite range of wavenumbers, in a spatially discrete case can result in the generation of large-scale stationary or oscillatory patterns. (4) It is demonstrated that under the presence of subcritical Turing and supercritical wave bifurcations, the interaction of two diffusion instabilities can result in the spontaneous formation of Turing structures outside the region of Turing instability.

Investigation of the mechanism of emergence of autowave structures at the reaction front.

A qualitative mechanism for autowave pattern formation at the reaction front, observed in certain chemical systems including combustion, is suggested. It is assumed that patterns are formed as a result of interaction of two subsystems, one of which is responsible for the reaction front propagation while the other determines the formation of waves at the front. A corresponding phenomenological model is constructed in which reaction front propagation is described by a submodel of the Fisher-Kolmogorov-Petrovskii-Piskunov type and waves on the front are described by a submodel of the FitzHugh-Nagumo type. In the three-dimensional numerical analysis, it is demonstrated that the model is able to qualitatively explain the emergence of wave patterns of both spiral and target types, which are experimentally observed at the reaction front. The dependence of these patterns on the velocity and thickness of the front is examined.

Pattern formation in a reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation.

We investigate numerically the behavior of a two-component reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation in response to local rigid perturbation. In a large region of parameters, the initial perturbation evolves into a localized structure. In a part of that region, closer to the bifurcation line, this structure turns out to be unstable and covers all the available space over the course of time in a process of self-completion. Depending on the parameter values in two-dimensional (2D) space, this process happens either through generation and evolution of new peaks on oscillatory tails of the initial pattern, or through the elongation, deformation, and rupture of initial structure, leading to space-filling nonbranching snakelike patterns. Transient regimes are also possible. Comparison of these results with 1D simulations shows that the prebifurcation region of parameters where the self-completion process is observed is much larger in the 2D case.

Spatial patterns formed by chemotactic bacteria Escherichia coli.

In certain experimental conditions, bacteria form complex spatial-temporal patterns. A striking example of such kind was reported by Budrene and Berg (1991), who observed a wide variety of different colony structures ranging from arrays of spots to radially oriented stripes or arrangements of more complex elongated spots, formed by Escherichia coli. We discuss the relevant mechanisms of intercellular regulation in bacterial colony which may cause pattern formation, and formulate the corresponding mathematical model. In numerical experiments a variety of patterns, observed in real systems, is reproduced. The dynamics of their formation is investigated.