Coexistence of large amplitude stationary structures in a model of reaction-diffusion system

Particle dynamics simulations of Turing patterns

The direct simulation Monte Carlo method is used to reproduce Turing patterns at the microscopic level in reaction-diffusion systems. In order to satisfy the basic condition for the development of such a spatial structure, we propose a model involving a solvent, which allows for disparate diffusivities of individual reactive species. One-dimensional structures are simulated in systems of various lengths. Simulation results agree with the macroscopic predictions obtained by integration of the reaction-diffusion equations. Additional effects due to internal fluctuations are observed, such as temporal transitions between structures of different wavelengths in a confined system. For a structure developing behind a propagating wave front, the fluctuations suppress the induction period and accelerate the formation of the Turing pattern. These results support the ability of reaction-diffusion models to robustly reproduce axial segmentation including the formation of early vertebrae or somites in noisy biological environments.

Pattern formation and coexistence domains for a nonlocal population dynamics

In this Rapid Communication we propose a most general equation to study pattern formation for one-species populations and their limit domains in systems of length L. To accomplish this, we include nonlocality in the growth and competition terms, where the integral kernels now depend on characteristic length parameters α and β. Therefore, we derived a parameter space (α,β) where it is possible to analyze a coexistence curve α^{*}=α^{*}(β) that delimits domains for the existence (or absence) of pattern formation in population dynamics systems. We show that this curve is analogous to the coexistence curve in classical thermodynamics and critical phenomena physics. We have successfully compared this model with experimental data for diffusion of Escherichia coli populations.

Oscillons localized inside breathing periodical structures in a two-variable model of a one-dimensional infinite excitable reaction-diffusion system

A two-variable model of a one-dimensional (1D), infinite, excitable, reaction-diffusion system describing oscillons localized inside an expanding breathing periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions.

Stationary periodical structure emitting an infinite number of traveling impulses in a model of a one-dimensional infinite excitable reaction-diffusion system

A two-variable model of a one-dimensional infinite excitable reaction-diffusion system describing an expanding stationary periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions. The chemical scheme consists of mono- and bimolecular reactions.

Turing patterns in three dimensions

We investigate three-dimensional Turing patterns in two-component reaction diffusion systems. The FitzHugh-Nagumo equation, the Brusselator, and the Gray-Scott model are solved numerically in three dimensions. Several interconnected structures of domains as well as lamellar, hexagonal, and spherical domains are obtained as stable motionless equilibrium patterns. The relative stability of these structures is studied analytically based on the reduction approximation. The relation with the microphase-separated structures in block copolymers is also discussed.

Periodic spatiotemporal patterns in a two-dimensional two-variable reaction-diffusion model

Periodic spatiotemporal two-dimensional (2D) asymptotic patterns in an excitable two-variable thermochemical (reaction-diffusion) system are shown. In a one-dimensional system the traveling impulse which reflects from impermeable boundaries is a stable asymptotic solution if the diffusion coefficient of the reactant is greater than the thermal diffusivity of the system. Periodic patterns of two symmetries are presented in the 2D system: the impulse of excitation propagating along the diagonal of a square spatial domain and a structure consisting of curved impulses which propagate in the direction perpendicular to one side of a rectangular domain.

Periodical survival or decay of traveling impulse in a model of a one-dimensional reaction-diffusion system

A two-variable model of a one-dimensional (1D), open, excitable reaction-diffusion system describing space-time evolutions of traveling impulses is investigated. It is shown that depending on the size of the system, the traveling impulse can survive or decay. Continuous increase of the size of the system causes periodical repetitions of surviving and decay of the impulse. The qualitative properties of the model, which allow us to expect the phenomenon, are described. Numerical solutions confirm this expectation. The chemical reaction scheme is realistic and may be a stimulus for seeking the phenomenon in experiments.

Two-dimensional model of a reaction-diffusion system as a typewriter

Pattern formation is a common phenomenon, which appears in biological systems, especially in cell differentiation processes. The proper level for understanding the creation of patterns seems to be a physicochemical description. The most fundamental models should be based on systems, in which only chemical reactions and diffusion transport occur (reaction-diffusion systems). In order to present a richness of patterns, we show here the asymptotic patterns in the form of capital letters obtained in two-dimensional reaction-diffusion systems with zero-flux boundary conditions. All capital letters are obtained in the same model, but initial conditions and sizes of the systems are different for each letter. The chemical model consists of elementary reactions and is realistic. It can be realized experimentally in continuous-flow unstirred reactor with an enzymatic reaction allosterically inhibited by an excess of its reactant and product.