Pattern Selection Mechanisms Research Articles

Pattern selection mechanism from the equilibrium point and limit cycle.

The outbreak of infectious diseases often exhibits periodicity, and this periodic behavior can be mathematically represented as a limit cycle. However, the periodic behavior has rarely been considered in demonstrating the cluster phenomenon of infection induced by diffusion (the instability modes) in the SIR model. We investigate the emergence of Turing instability from a stable equilibrium and a limit cycle to illustrate the dynamical and biological mechanisms of pattern formation. We identify the Hopf bifurcation to demonstrate the existence of a stable limit cycle using First Lyapunov coefficient in our spatiotemporal diffusion-driven SIR model. The competition between different instability modes induces different types of patterns and eventually spot patterns emerge as stable patterns. We investigate the impact of susceptible, infected, and recovered individuals on the type of patterns. Interestingly, these instability modes play a vital role in selecting the pattern formations, which is directly related to the number of observed spot patterns. Subsequently, we explain the dynamical and biological mechanisms of spot patterns to develop an effective epidemic prevention strategy.

Absolute vs Convective Instabilities and Front Propagation in Lipid Membrane Tubes.

We analyze the stability of biological membrane tubes, with and without a base flow of lipids. Membrane dynamics are completely specified by two dimensionless numbers: the well-known Föppl-von Kármán number Γ and the recently introduced Scriven-Love number SL, respectively quantifying the base tension and base flow speed. For unstable tubes, the growth rate of a local perturbation depends only on Γ, whereas SL governs the absolute versus convective nature of the instability. Furthermore, nonlinear simulations of unstable tubes reveal an initially localized disturbance result in propagating fronts, which leave a thin atrophied tube in their wake. Depending on the value of Γ, the thin tube is connected to the unperturbed regions via oscillatory or monotonic shape transitions-reminiscent of recent experimental observations on the retraction and atrophy of axons. We elucidate our findings through a weakly nonlinear analysis, which shows membrane dynamics may be approximated by a model of the class of extended Fisher-Kolmogorov equations. Our study sheds light on the pattern selection mechanism in axonal shapes by recognizing the existence of two Lifshitz points, at which the front dynamics undergo steady-to-oscillatory bifurcations.

Effects of anisotropic diffusion on Turing patterns in heterogeneous environment

Diffusion plays a crucial role in the forming and evolving of Turing patterns. Generally, the diffusion processes in complex systems do not comply to the complete random walk theory, which means that the diffusion is abnormal rather than normal, such as super-diffusion, sub-diffusion and anisotropic diffusion. However, most of previous studies focused on the pattern formation mechanism under the normal diffusion. In this paper, a two-component reaction-diffusion model with anisotropic diffusion is used to study the effect of anisotropic diffusion on Turing patterns in heterogeneous environments. Three different types of anisotropic diffusions are utilized. It is shown that the system gives rise to stripe patterns when the degree of anisotropic diffusion is high. The directions of stripes are determined by the degree of the diffusion coefficient deviating from the bifurcation point. In a low degree of anisotropic diffusion, the pattern type is the same as the counterpart in a low degree of the isotropic diffusion. When the diffusion coefficient grows linearly in the space, different types of patterns compete with each other and survive in different regions under the influence of spatial heterogeneity. When the diffusion coefficient is modulated by a one-dimensional periodic function, both type and wavelength of the pattern are determined by the modulated wavelength and the intrinsic wavelength. The system can exhibit alternating two-scale mixed patterns of different types when the modulated wavelength is larger than the intrinsic wavelength. Note that each of the diffusion coefficients of some special anisotropic media is a tensor, which can be expressed as a matrix in two-dimensional cases. We also study the influence of off-diagonal diffusion coefficient <i>D</i> on Turing pattern. It is found that the Turing pattern induced by off-diagonal diffusion coefficient always selects the oblique stripe pattern. The off-diagonal diffusion coefficient <i>D</i> not only affects the pattern selection mechanism, but also expands the parameter range of Turing space. The critical diffusion coefficient <inline-formula><tex-math id="M3">\begin{document}$ {D_{\text{c}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M3.png"/></alternatives></inline-formula> increases linearly with the diagonal diffusion coefficient <inline-formula><tex-math id="M4">\begin{document}$ {D_u} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M4.png"/></alternatives></inline-formula> increasing. The intrinsic wavelength of the oblique stripe pattern decreases as the off-diagonal diffusion coefficient <i>D</i> increases. It is interesting to note that the critical wavelength corresponding to the critical diffusion coefficient <inline-formula><tex-math id="M6">\begin{document}$ {D_{\text{c}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M6.png"/></alternatives></inline-formula> is independent of the diagonal diffusion coefficient <inline-formula><tex-math id="M7">\begin{document}$ {D_u} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20221294_M7.png"/></alternatives></inline-formula>. These results not only provide a new insight into the formation mechanism of Turing patterns, but also increase the range and complexity of possible patterns.

Horizontal patterns from finite speed directional quenching

In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface; in the case presented here the interface moves with speed $c$ in such a way that the bistable region grows, which is the most physically relevant scenario. According to previous results in the literature (Phase separation patterns from directional quenching, Journal of Nonlinear Science, 2017, by Monteiro, R. and Scheel, A.), several patterns exist when $c \gtrsim0$, and here we investigate their persistence for finite $c>0$, clarifying the pattern selection mechanism related to the speed $c$ of the quenching front.

Dynamical pattern selection of growing cellular mosaic in fish retina.

A Markovian lattice model for photoreceptor cells is introduced to describe the growth of mosaic patterns on fish retina. The radial stripe pattern observed in wild-type zebrafish is shown to be selected naturally during retina growth, against the geometrically equivalent circular stripe pattern. The mechanism of such dynamical pattern selection is clarified on the basis of both numerical simulations and theoretical analyses, which find that the successive emergence of local defects plays a critical role in the realization of the wild-type pattern.

Pattern selection in ternary mushy layers

We consider finite-amplitude convection in a mushy layer during the primary solidification of a ternary alloy. Previous linear stability theories applied to ternary alloy primary-solidification models have identified an exceptional class of direct convective instability when all the individual stratifying agencies (one thermal and two solutal) were statically stabilizing. A reduced model, in which the effects of latent heat, solute rejection and background solidification are neglected, contains the essential interactions that admit qualitatively the same instability. We examine pattern selection for steady convection in this model. We find that roll, square or hexagonal convection patterns can be nonlinearly stable, depending on the relative importance of a number of physical effects, namely the solutal diffusion rates, the liquidus slopes and the background thermal and solutal density stratifications. The results for a special case are found to isolate a purely double-diffusive phase-change mechanism of pattern selection. Subcritical behaviour is identified inside the domain of individual static stability. A physical system is proposed that may be a promising one in which to experimentally identify these novel instabilities.

Vortex nucleation in a dissipative variant of the nonlinear Schrödinger equation under rotation

In the present work, we motivate and explore the dynamics of a dissipative variant of the nonlinear Schrödinger equation under the impact of external rotation. As in the well established Hamiltonian case, the rotation gives rise to the formation of vortices. We show, however, that the most unstable mode leading to this instability scales with an appropriate power of the chemical potential μ of the system, increasing proportionally to μ2/3. The precise form of the relevant formula, obtained through our asymptotic analysis, provides the most unstable mode as a function of the atomic density and the trap strength. We show how these unstable modes typically nucleate a large number of vortices in the periphery of the atomic cloud. However, through a pattern selection mechanism, prompted by symmetry-breaking, only few isolated vortices are pulled in sequentially from the periphery towards the bulk of the cloud resulting in highly symmetric stable vortex configurations with far fewer vortices than the original unstable mode. These results may be of relevance to the experimentally tractable realm of finite temperature atomic condensates.

Self-organization of waves and pulse trains by molecular motors in cellular protrusions.

Actin-based cellular protrusions are an ubiquitous feature of cells, performing a variety of critical functions ranging from cell-cell communication to cell motility. The formation and maintenance of these protrusions relies on the transport of proteins via myosin motors, to the protrusion tip. While tip-directed motion leads to accumulation of motors (and their molecular cargo) at the protrusion tip, it is observed that motors also form rearward moving, periodic and isolated aggregates. The origins and mechanisms of these aggregates, and whether they are important for the recycling of motors, remain open puzzles. Motivated by novel myosin-XV experiments, a mass conserving reaction-diffusion-advection model is proposed. The model incorporates a non-linear cooperative interaction between motors, which converts them between an active and an inactive state. Specifically, the type of aggregate formed (traveling waves or pulse-trains) is linked to the kinetics of motors at the protrusion tip which is introduced by a boundary condition. These pattern selection mechanisms are found not only to qualitatively agree with empirical observations but open new vistas to the transport phenomena by molecular motors in general.

Dynamic transitions and pattern formations for a Cahn–Hilliard model with long-range repulsive interactions

The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on the Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing the long-range repulsive interaction. First, we show that as soon as the linear problem loses stability, the system always undergoes a dynamic transition to one of the three types, forming different patterns/structures. The types of transition are then dictated by a nondimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. The derived explicit form of this parameter offers precise information for the phase diagrams. Second, we obtain a novel and explicit pattern selection mechanism associated with the competition between the long-range repulsive interaction and the short-range attractive interactions. In particular, the hexagonal pattern is unique to the long-range interaction, and is associated with a novel two-dimensional reduced transition equations on the center manifold generated by the unstable modes, consisting of (degenerate) quadratic terms and non-degenerate cubic terms. Finally, explicit information on the metastability and basin of attraction of different disordered/ordered states and patterns are derived as well.

Instabilities and pattern evolution in a vertically heated annulus

The convection in an annular container with heated bottom, cooled top and insulated side walls are studied by both linear instability analysis and direct numerical simulation. The onset of convection is investigated by linear stability analysis and corresponding pattern selection mechanisms are discussed. The nonlinear evolution of different flow patterns and the convective heat transfer are simulated. The transition to oscillatory flow is also given by stability analysis where the base flow is a steady three dimensional flow. The stability predictions are in good agreement with the numerical simulations, including both the growth rate and the dimensionless frequency.

Pattern formation in Rayleigh–Bénard convection

The main objective of this article is to study the three-dimensional Rayleigh-Benard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an (m-1)-dimensional homological sphere where m is the multiplicity of the first critical eigenvalue. When m=1, the structure of this attractor is trivial. When m=2, it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where m=2. We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable.

Magnetically induced spreading and pattern selection in thin ferrofluid drops

We report an experimental study of a fingering pattern formation which occurs during the spreading of an immiscible thin ferrofluid drop subjected to a radial magnetic field. Our results indicate that this ferrohydrodynamic system works as a magnetic analog of conventional spin coating, where centrifugal driving is replaced with a magnetic body force induced by the radial applied field. In this context, a magnetically tunable pattern selection mechanism is proposed in which the shape and number of the arising fingered structures can be properly controlled.

Mechanism of nonlinear flow pattern selection in moderately non-Boussinesq mixed convection

Nonlinear (non-Boussinesq) variations in fluid's density, viscosity, and thermal conductivity caused by a large temperature gradient in a flow domain lead to a wide variety of instability phenomena in mixed convection channel flow of a simple gas such as air. It is known that in strongly nonisothermal flows, the instabilities and the resulting flow patterns are caused by competing buoyancy and shear effects [see S. A. Suslov and S. Paolucci, J. Fluid Mech. 302, 91 (1995)]. However, as is the case in the Boussinesq limit of small temperature gradients, in moderately non-Boussinesq regimes, only a shear instability mechanism is active. Yet in contrast to Boussinesq flows, multiple instability modes are still detected. By reducing the system of full governing Navier-Stokes equations to a dynamical system of coupled Landau-type disturbance amplitude equations we compute a comprehensive parametric map of various shear-driven instabilities observed in a representative moderately non-Boussinesq regime. Subsequently, we analyze nonlinear interaction of unstable modes and reveal physical reasons for their appearance.

Crossover from a square to a hexagonal pattern in faraday surface waves

We report on surface wave pattern formation in a Faraday experiment operated at a very shallow filling level, where modes with a subharmonic and harmonic time dependence interact. Associated with this distinct temporal behavior are different pattern selection mechanisms, favoring squares or hexagons, respectively. In a series of bifurcations running through a pair of superlattices the surface wave pattern transforms between the two incompatible symmetries. The close analogy to two- and three-dimensional crystallography is pointed out.

Dynamics and Selection of Giant Spirals in Rayleigh-Bénard Convection

For Rayleigh-Benard convection of a fluid with Prandtl number \sigma \approx 1, we report experimental and theoretical results on a pattern selection mechanism for cell-filling, giant, rotating spirals. We show that the pattern selection in a certain limit can be explained quantitatively by a phase-diffusion mechanism. This mechanism for pattern selection is very different from that for spirals in excitable media.

Wave Pattern Selection in an Excitable System

Wave pattern selection between periodic spirals and periodic circular pacemaker waves in excitable media is investigated with numerical simulations in a FitzHugh-Nagumo reaction-diffusion model. When spirals have oscillatory tails circular waves are more likely dominant over spirals, whereas it is opposite when spirals have monotonic tails. In both cases a coherent state is achieved in an asymptotic limit with a single period and a velocity. The observed wave pattern selection mechanism seems to underlie the competition between spirals and circular waves observed in signaling activity of Dictyostelium discoideum amoebae population.

Cell number selection in electrohydrodynamic convection in liquid crystals

Results of an experimental investigation into a pattern selection mechanism in electrohydrodynamic flow in a nematic liquid crystal device are presented. The apparatus consists of a small aspect ratio cell so that only a few spatial modes are activated. The initial bifurcation to cellular flow is found to be disconnected, and the ensuing cell number selection process is in agreement with the mechanism proposed by Benjamin for Navier–Stokes flows. The results point to a greater generality of this mechanism, since the flow we consider here is of a complex fluid on a microscopic length scale.

Pattern selection mechanism in mesophase carbon fibres

Carbonaceous mesophases are spun into high performance carbon fibres using the melt spinning process. The spinning process produces a wide range of different fibre textures whose origins are not well understood. The cross section fibre texture is a strong function of temperature, such that at lower temperatures the selected pattern is known as radial, while at higher temperatures it is known as onion. More intriguing and unexpected is the random pattern that is found in the intermediate temperature range. This paper uses theory and simulation to reproduce the fibre textures, and to identify the selection mechanisms that explain the origin of the observed patterns.

Thin film forced velocity cells and cellular dendrites—I. Experiments

A new, compreheensive set of experimental systematics for the alloys succinonitrile-salol (S-Sa), succinonitrile-acetone (S-A), and pivalic acid-ethanol(P-E) is summarized. The contribution deals synthetically with transients involving the Mullins and Serkerka instability together with Ostwald ripening, pattern selection mechanisms, cell root instabilities, steady state wavelength and amplitude systematics spanning near threshold cellular and dendritic forms, anisotropic effects, facetting and hysteresis and systematics of the cell to dendrite transition. The observations reveal that there are two main stable or nearly stable branches of non-planar cellular solutions, both significantly involving solid state diffusion and excluding single-valued local equilibrium spatial structures. The normal single-valued, non-local equilibrium branch which we designated as dynamical has a scale decreasing with increasing velocity. The other triple-valued, near-marginal branch which we designate as quasi-equilibrium has a scale increasing with increasing velocity.

Thin film forced velocity cells and cellular dendrites—II. Analysis of data

Based upon new accurate diffusion data this paper reanalyses a comprehensive set of experimental systematics for the alloys succinonitrile-salol (S-Sa), succinonitrile-acetone (S-A), and pivalic acid-ethanol (P-E). Using these experiments we have tested Mullins and Sekerka and Zener-Hillert based theories which have generated a variety of scaling formulae. The contribution deals quantitatively in most cases with wavenumber degeneracy, non-degenerate local equilibrium periodic shapes, pattern selection mechanisms and their phase field or dissipation optimal representations, cell root instabilities, steady state wavelength and amplitude systematics spanning near threshold cellular and dendritic forms, anisotropic effects including crystal mode selection, facetting and hysteresis, and prediction of systematics of the cell to dendrite transition. There are at least six characteristics lengths which bear significantly upon the theoretical problem as evidenced by the current complex experimental record. We undertake for the first time the explanation of the two main stable or nearly stable branches of non-planar cellular solutions and their mutual transitions, both significantly involving solid state diffusion and excluding single-valued local equilibrium spatial structures. The normal single-valued, non-local equilibrium dynamical branch has a scale decreasing with increasing velocity. The triple-valued, near-marginal quasi-equilibrium branch has a scale increasing with increasing velocity and can be comprehended supermarginally as an Ostwald-ripened Mullins and Sekerka instability. The latter branch extends submarginally but it is not necessarily the subcritical branch predicted by bifurcation theory.