Turing instability for a semi-discrete Gierer-Meinhardt system

In this paper, we investigate the spatial patterns of a Gierer-Meinhardt system where the space is discrete in two dimensions with the periodic boundary condition and time is continuous, in contrast to the continuum models. The conditions of Turing instability are obtained by linear analysis and a series of numerical simulations are performed. In the instability region, we have shown that this system can produce a number of different patterns such as stripes and snowflake pattern, other than ubiquitous fix-spotted patterns. As mentioned, the results are substantiated only by means of snapshots of the apatial grid. However, we also give some analysis by using the time series at three random grids and of the average value of states, that is, the stable state patterns can be observed. On the other hand, the effects of varying parameters on pattern formation are also discussed. Moreover, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. The patterns can form as a consequence of cellular interaction. So patterns arising from a semi-discrete model can present simulations on a geometrically accurate representation in biology. As a result, our work is interesting and important in ecology.     

Turing pattern selection in a reaction-diffusion epidemic model

We present Turing pattern selection in a reaction-diffusion epidemic model under zero-flux boundary conditions.The value of this study is twofold.First,it establishes the amplitude equations for the excited modes,which determines the stability of amplitudes towards uniform and inhomogeneous perturbations.Second,it illustrates all five categories of Turing patterns close to the onset of Turing bifurcation via numerical simulations which indicates that the model dynamics exhibits complex pattern replication:on increasing the control parameter ν,the sequence "H 0 hexagons → H 0-hexagon-stripe mixtures → stripes → H π-hexagon-stripe mixtures → H π hexagons" is observed.This may enrich the pattern dynamics in a diffusive epidemic model.     

Turing instability for a two-dimensional Logistic coupled map lattice

In this Letter, stability analysis is applied to a two-dimensional Logistic coupled map lattice with the periodic boundary conditions. The conditions of Turing instability are obtained, and various patterns can be exhibited by numerical simulations in the Turing instability region. For example, space-time periodic structures, periodic or quasiperiodic traveling wave solutions, stationary wave solutions, spiral waves, and spatiotemporal chaos, etc. have been observed. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The similar patterns have also been seen in relevant references. However, the present Letter owes to pattern formation via diffusion-driven instabilities because the system is stable in the absence of diffusion.     

Phase transition in a three-states reaction-diffusion system

A one-dimensional reaction-diffusion model consisting of two species of particles and vacancies on a ring is introduced. The number of particles in one species is conserved while in the other species it can fluctuate because of creation and annihilation of particles. It has been shown that the model undergoes a continuous phase transition from a phase where the currents of different species of particles are equal to another phase in which they are different. The total density of particles and also their currents in each phase are calculated exactly.     

Superlattice Turing structures in a photosensitive reaction-diffusion system

Families of complex superlattice structures, consisting of combinations of basic hexagonal or square patterns, are found in a photosensitive reaction-diffusion system. The structures are induced by simple illumination patterns whose wavelengths are appropriately related to that of the system's intrinsic Turing pattern. Computer simulations agree with the structures and their stability. The technique offers a general approach to generating superlattices for use in information storage and other applications.     

Turing pattern selection in a reactionben diffusion epidemic modelast

We present Turing pattern selection in a reaction—diffusion epidemic model under zero-flux boundary conditions. The value of this study is twofold. First, it establishes the amplitude equations for the excited modes, which determines the stability of amplitudes towards uniform and inhomogeneous perturbations. Second, it illustrates all five categories of Turing patterns close to the onset of Turing bifurcation via numerical simulations which indicates that the model dynamics exhibits complex pattern replication: on increasing the control parameter v, the sequence “H₀ hexagons → H₀-hexagon-stripe mixtures → stripes → H_π-hexagon-stripe mixtures → H_π hexagons” is observed. This may enrich the pattern dynamics in a diffusive epidemic model.     

Turing Patterns in a Reaction-Diffusion System

We have further investigated Turing patterns in a reaction-diffusion system by theoretical analysis and numerical simulations. Simple Turing patterns and complex superlattice structures are observed. We find that the shape and type of Turing patterns depend on dynamical parameters and external periodic forcing, and is independent of effective diffusivity rate σ in the Lengyel-Epstein model. Our numerical results provide additional insight into understanding the mechanism of development of Turing patterns and predicting new pattern formations.     

Note on a universal quantum Turing machine

In this Letter, we construct a novel model of universal quantum Turing machine (QTM) by means of a property of Riemann zeta function, which is free from the specific time for an input data and efficiently simulates each step of a given QTM.     

Dynamics of interface in three-dimensional anisotropic bistable reaction-diffusion system

This paper presents a theoretical investigation of dynamics of interface (wave front) in three-dimensional (3D) reaction-diffusion (RD) system for bistable media with anisotropy constructed by means of anisotropic surface tension. An equation of motion for the wave front is derived to carry out stability analysis of transverse perturbations, which discloses mechanism of pattern formation such as labyrinthine in 3D bistable media. Particularly, the effects of anisotropy on wave propagation are studied. It was found that, sufficiently strong anisotropy can induce dynamical instabilities and lead to breakup of the wave front. With the fast-inhibitor limit, the bistable system can further be described by a variational dynamics so that the boundary integral method is adopted to study the dynamics of wave fronts.     

Turing instability in reaction-diffusion systems with nonlinear diffusion

The Turing instability is studied in two-component reaction-diffusion systems with nonlinear diffusion terms, and the regions in parametric space where Turing patterns can form are determined. The boundaries between super- and subcritical bifurcations are found. Calculations are performed for one-dimensional brusselator and oregonator models.     

Diffusion in a bistable potential: A systematic WKB treatment

We study the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, in the limit of constant and small diffusion coefficient. A systematic WKB analysis of the lowest eigenmodes of the equivalent Schrödinger-like equation yields the following results: the final approach to equilibrium is governed by the Kramers high-viscosity rate, which is shown to be exact in this limit; for intermediate times, we show that Suzuki's scaling statement does give the correct behavior for the transition between the one-peak and the two-peak structure forP. However, the intermediate time domain also contains a second half, whereP enters the diffusive equilibrium regions, characterized by a time scale of the same order as Suzuki's time.     

A WKB treatment of diffusion in a multidimensional bistable potential

We extend to the case of a finite set of stochastic variables whose distributionP obeys a nonlinear Fokker-Planck equation our previous treatment of diffusion in a bistable potentialU, in the limit of small, constant diffusion coefficient. This is done with the help of an extended WKB approximation due to Gervais and Sakita. The treatment is valid if there exists a well-defined most probable path connecting the minima ofU, and if the valley ofU along that path has a slowly varying width, and weak curvature and twisting. We find that: (i) the final approach to equilibrium is governed by Eyring's generalization of the Kramers high-viscosity rate, which we rederive; (ii) for intermediate times, if the initial distribution is concentrated in the region of instability (close vicinity of the saddle point ofU),P has, along the most probable path, the behavior described by Suzuki's scaling statement for a one-dimensional system. In a second part of this time domain,P enters the diffusive regions around the minima ofU and relaxes toward local longitudinal equilibrium on a time comparable with Suzuki's time scale. The time for relaxation toward transverse local equilibrium may, depending on the initial conditions, compete with these longitudinal times.We dedicate this work to our colleague, Yuri Orlov.     

Traveling wave solutions of a reaction-diffusion model for bacterial growth

In this paper, we consider a reaction-diffusion model for the bacterial growth. Mathematical analysis on the traveling wave solutions of the model is performed. This includes traveling wave analysis and numerical simulations of wave front propagation for a special case. Specifically, we show that such solutions exist only for wave speeds greater than some minimum speed giving wave with a sharp front. The minimum speed is estimated and the wave profile is calculated and compared with different numerical methods.     

Chaotic Turing pattern formation in spatiotemporal systems

The problem of Turing pattern formation has attracted much attention in nonlinear science as well as physics, chemistry and biology. So far spatially ordered Turing patterns have been observed in stationary and oscillatory media only. In this paper we find that spatially ordered Turing patterns exist in chaotic extended systems. And chaotic Turing patterns are strikingly rich and surprisingly beautiful with their space structures. These findings are in sharp contrast with the intuition of pseudo-randomness of chaos. The richness and beauty of the chaotic Turing patterns are attributed to a large variety of symmetry properties realized by various types of self-organizations of partial chaos synchronizations.     

Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system

We study a one-dimensional reaction-diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic (A + B → 2B) and a cubic (A + 2B → 3B) autocatalysis. The parameters of this system are in the ratio D = D_B/D_A of the diffusion constants of the reactant A and the autocatalyst B, and the relative activity k of the cubic reaction. First, for all values of D > 0 and k ≥ 0, we prove the existence of a family of propagating fronts (or travelling waves) describing the advance of the reaction. In particular, in the quadratic case k = 0, we recover the results of Billingham and Needham [Phil. Trans. R. Soc. London A 334 (1991) 1–24]. Then, if D is close to 1 and k is sufficiently small, we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted Sobolev spaces. This extends part of the results that are known for the scalar equation to which our system reduces when D = 1.     

Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers

A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include "twinkling eyes," where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spot-like or stripe-like Turing structures. A new approach to generating the short-wave instability is proposed.     

A class of solvable reaction-diffusion processes on a Cayley tree

Considering the most general one-species reaction-diffusion processes on a Cayley tree, it has been shown that there exist two integrable models. In the first model, the reactions are the various creation processes, i.e. °°→•°, °°→•• and °•→••, and in the second model, only the diffusion process •°→°• exists. For the first model, the probabilities P_l(m;t), of finding m particles on the lth shell of the Cayley tree, have been found exactly, and for the second model, the functions P_l(1;t) have been calculated. It has been shown that these are the only integrable models if one restricts consideration to the L+1-shell probabilities P(m₀,m₁,…,m_L;t).     

Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes

Spatial resonances leading to superlattice hexagonal patterns, known as "black-eyes," and superposition patterns combining stripes and/or spots are studied in a reaction-diffusion model of two interacting Turing modes with different wavelengths. A three-phase oscillatory interlacing hexagonal lattice pattern is also found, and its appearance is attributed to resonance between a Turing mode and its subharmonic.     

Selection and coexistence by reaction-diffusion dynamics in fitness landscapes

The phenomena of selection and coexistence are modelled as chemical pattern formation by competition for a given resource in heterogeneous spatial fields (fitness landscapes). The reactions form a closed reproductive cycle so that total concentration of species and their building-blocks is kept constant in time. Eigen's ‘constant overall organization' conditions can be view as a particular case of proposed conditions.

Species whose fitness values exceed the replication threshold govern the system dynamics. In the limit of vanishing diffusion of species and full diffusional mixing of resource, all individuals share a common niche and only the fittest survives during an evolution. In the opposite limit of negligible diffusivity of resource, there is no competition between individuals and all of them coexist. At the intermediate values of resource diffusion, a number of long-lived spatial niches are formed, which lead to long-term coexistence of locally fittest individuals.     

Phase equation with nonlinear excitation for nonlocally coupled oscillators

Some reaction-diffusion systems feature nonlocal interaction and, near the point of Hopf bifurcation, can be represented as a system of nonlocally coupled oscillators. Phase of oscillations satisfies an evolution pde which takes different forms depending on the values of parameters. In the simplest case the equation is effectively a diffusion equation which is excitation-free. However, more complex forms are possible such as the Nikolaevskii equation and the Kuramoto–Sivashinsky equation incorporating linear excitation. We analyse a situation when the phase equation is based on nonlinear excitation. We derive conditions on the values of the parameters leading to the situation and show that the values satisfying the conditions exist.