Labyrinthine Patterns Research Articles

Spatiotemporal Dynamics of a Reaction Diffusive Predator-Prey Model: A Weak Nonlinear Analysis

In the realm of ecology, species naturally strive to enhance their own survival odds. This study introduces and investigates a predator-prey model incorporating reaction-diffusion through a system of differential equations. We scrutinize how diffusion impacts the model’s stability. By analysing the stability of the model’s uniform equilibrium state, we identify a condition leading to Turing instability. The study delves into how diffusion influences pattern formation within a predator-prey system. Our findings reveal that various spatiotemporal patterns, such as patches, spots, and even chaos, emerge based on species diffusion rates. We derive the amplitude equation by employing the weak nonlinear multiple scales analysis technique and the Taylor series expansion. A novel sinc interpolation approach is introduced. Numerical simulations elucidate the interplay between diffusion and Turing parameters. In a two-dimensional domain, spatial pattern analysis illustrates population density dynamics resulting in isolated groups, spots, stripes, or labyrinthine patterns. Simulation results underscore the method’s effectiveness. The article concludes by discussing the biological implications of these outcomes.

Modification of Turing patterns through the use of time-varying anisotropic diffusion

The Turing mechanism underpinning pattern formation in reaction–diffusion relies on the interplay between diffusion parameters and reaction kinetics. Diffusion is typically assumed isotropic, however anisotropic diffusion is known to arise in both nature and laboratory conditions. We study how the Turing instability and resulting Turing patterns are modified when the underlying diffusion tensor is both anisotropic and time-dependent, modelling, for instance, chemical species reacting and diffusing through time-varying anisotropic media. We show that the set of unstable wavenumber vectors corresponding to Turing modes evolve in a spatially biased manner under this anisotropy, thereby modifying the spatial scale and structure of any resulting Turing patterns differently along each spatial coordinate. We employ this spatial bias to develop control strategies to modify the shape or even structure of Turing patterns over time. We are able to make minor changes to the aspect ratio of Turing patterns, such as morphing circular Schnakenberg spots into elliptical spots, as well as more major changes to the structure of patterns, for instance converting Gierer–Meinhardt spots into stripes or FitzHugh–Nagumo labyrinthine patterns into target patterns. Our results suggest that time-varying anisotropic media may be used as a tool by which to modify and even control Turing patterns.

Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system.

Effect of heterogeneous environmental conditions on labyrinthine vegetation patterns.

Self-organization is a ubiquitous phenomenon in Nature due to the permanent balance between injection and dissipation of energy. The wavelength selection process is the main issue of pattern formation. Stripe, hexagon, square, and labyrinthine patterns are observed in homogeneous conditions. In systems with heterogeneous conditions, a single wavelength is not the rule. Large-scale self-organization of vegetation in arid environments can be affected by heterogeneities, such as interannual precipitation fluctuations, fire occurrences, topographic variations, grazing, soil depth distribution, and soil-moisture islands. Here, we investigate theoretically the emergence and persistence of vegetation labyrinthine patterns in ecosystems under deterministic heterogeneous conditions. Based on a simple local vegetation model with a space-varying parameter, we show evidence of perfect and imperfect labyrinthine patterns, as well as disordered vegetation self-organization. The intensity level and the correlation of the heterogeneities control the regularity of the labyrinthine self-organization. The phase diagram and the transitions of the labyrinthine morphologies are described with the aid of their global spatial features. We also investigate the local spatial structure of labyrinths. Our theoretical findings qualitatively agree with satellite images data of arid ecosystems that show labyrinthinelike textures without a single wavelength.

Truchet-tile structure of a topologically aperiodic metal-organic framework.

When tiles decorated to lower their symmetry are joined together, they can form aperiodic and labyrinthine patterns. Such Truchet tilings offer an efficient mechanism of visual data storage related to that used in barcodes and QR codes. We show that the crystalline metal-organic framework [OZn4][1,3-benzenedicarboxylate]3 (TRUMOF-1) is an atomic-scale realization of a complex three-dimensional Truchet tiling. Its crystal structure consists of a periodically arranged assembly of identical zinc-containing clusters connected uniformly in a well-defined but disordered fashion to give a topologically aperiodic microporous network. We suggest that this unusual structure emerges as a consequence of geometric frustration in the chemical building units from which it is assembled.

Pattern dynamics and Turing instability induced by self-super-cross-diffusive predator-prey model via amplitude equations

<abstract><p>Incorporating self-diffusion and super-cross diffusion factors into the modeling approach enhances efficiency and realism by having a substantial impact on the scenario of pattern formation. Accordingly, this work analyzes self and super-cross diffusion for a predator-prey model. First, the stability of equilibrium points is explored. Utilizing stability analysis of local equilibrium points, we stabilize the properties that guarantee the emergence of the Turing instability. Weakly nonlinear analysis is used to get the amplitude equations at the Turing bifurcation point (WNA). The stability analysis of the amplitude equations establishes the conditions for the formation of small spots, hexagons, huge spots, squares, labyrinthine, and stripe patterns. Analytical findings have been validated using numerical simulations. Extensive data that may be used analytically and numerically to assess the effect of self-super-cross diffusion on a variety of predator-prey systems.</p></abstract>

Injected 3T-3D-FLAIR-MRI labyrinthine patterns match with the severity and tonotopic alteration in sudden sensorineural hearing loss.

The aim of the study was to assess a correlation between MRI labyrinthine changes detected with IV-gadolinium optimized high-resolution 3D-FLAIR sequences 4h after injection (OPT4-3DFLAIR) and the type of SSNHL, in terms of frequency alteration and severity. This was a prospective monocentric study achieved from July 2019 to December 2020. The inclusion criterion was acute hearing loss of at least 30dB over three contiguous frequencies occurring within a 72-h period, documented by a pure-tone audiometry (PTA). The primary endpoint was the visual assessment of hyperintensity in labyrinthine structures on OPT4-3DFLAIR performed on 3T MRI. Thirty-six affected ears were included (20 men, 15 women; mean age: 54.5 ± 16.3years) with 69.4% full-spectrum hearing loss. The median hearing loss, expressed as median and interquartile range [IQR] was 91dB [74-120], with 47.2% of concomitant acute vestibular syndrome. Pathological signal was found in 26 out of 36 ears (72.2%). Basal turn enhancement was found in all abnormal MRIs, with 73.1% of apical turn enhancement and 50% of vestibular enhancement. Seventeen on 19 cases (89.5%) with apical involvement on MRI had low-frequency hearing loss. Vestibular involvement on MRI was significantly associated with a wider frequency range of hearing loss (p = 0.0002) and the severity of SSNHL (84.5 [71.7-92.5] dB versus 120 [85.8-120] dB, p = 0.0158). This report shows that in pathological MRI in SSNHL, a pathologic cochlear base signal is always detected, a cochlear apical turn enhancement matches with low-tone impairment, and a pathological signal within the posterior labyrinth is associated with an impairment of all frequencies and the severity of SSNHL.

Localized states with nontrivial symmetries: Localized labyrinthine patterns.

The formation of self-organized patterns and localized states are ubiquitous in Nature. Localized states containing trivial symmetries such as stripes, hexagons, or squares have been profusely studied. Disordered patterns with nontrivial symmetries such as labyrinthine patterns are observed in different physical contexts. Here we report stable localized disordered patterns in spatially extended dissipative systems. These two- and three-dimensional localized structures consist of an isolated labyrinth embedded in a homogeneous steady state. Their partial bifurcation diagram allows us to explain this phenomenon as a manifestation of a pinning-depinning transition. We illustrate our findings on the Swift-Hohenberg-type of equationsand other well-established models for plant ecology, nonlinear optics, and reaction-diffusion systems.

The angiosperm pollen Volkheimerites labyrinthus gen. et sp. nov. from the earliest Paleogene (Danian) of Patagonia, Argentina

Volkheimerites labyrinthus gen. et sp. nov., an angiosperm pollen type, is fully described and illustrated. The specimens were retrieved from lower Paleocene strata of the Salamanca Formation (San Jorge Basin) located in Chubut Province, Patagonia, Argentina. Medium to large size (25.5–55 μm) and a characteristic striate sculpture with labyrinthine pattern are distinguishing features of the grains in this new genus. Volkheimerites labyrinthus shares the presence of a coarse tectum supported by columellae with the fossil genera Dichastopollenites, Trisectoris and Periretisyncolpites, and some Winteraceae and Annonaceae species, but differs in having striate-labyrinthine as opposed to reticulate ornamentation. Our material is also distinguishable by the occurrence of solitary grains (monads) instead of the more frequent presence of tetrads in the Winteraceae family or grains separated into two hemispheres as in Dichastopollenites. The large thin area of the exine appears to be morphologically analogous to the proximal thinning of the exine in the monads of tetrads as occurs in some Annonaceae species. The detailed description and illustration of this new angiosperm pollen genus expands our understanding of Paleocene plant diversity in Patagonia.

DYNAMICS ANALYSIS OF THREE-SPECIES REACTION-DIFFUSION SYSTEM VIA THE MULTIPLE SCALE PERTURBATION METHOD

In this paper, the general analysis of spatiotemporal dynamics of three-species reaction-diffusion system induced by Turing bifurcation is given. Firstly, by employing the Routh-Hurwitz criterion the conditions for Turing bifurcation in three-species reaction-diffusion equations are obtained. Secondly, through the tool of the multiple scale perturbation method the amplitude equations of Turing patterns are also given. Finally, we take a three-species predator-prey model as an example to illustrate the application of these general theoretical results, and meanwhile carry out many numerical simulations to depict spots pattern, stripes pattern and labyrinthine pattern and demonstrate the validity of these theories.

Yolk Sac Tumor of the Ovary: A Report of 150 Cases and Review of the Literature.

One hundred fifty yolk sac tumors (YSTs) of the ovary in patients from 1 to 61 (mean: 21.5) years of age are described; 75% of the patients were in the second and third decades and only 1 was above 50 years of age. The clinical manifestations were typically related to a fast-growing adnexal mass; endocrine manifestations (hirsutism) were present in only 2 cases. The tumors were all unilateral and 70% were ≥15 cm; an associated dermoid cyst was present in 20 cases. The tumors were solid and cystic in 57% of the cases, 25% were multicystic, and 18% uniformly solid. The solid tissue was typically tan to pink or yellow and often friable with hemorrhage and necrosis; smaller solid neoplasms were sometime uniformly yellow. The most common histologic pattern was reticular composed of an irregular meshwork of spaces that was conspicuous in 68% of the neoplasms but present to at least a minor degree in all of them. That appearance almost always merged with small to large cysts that were prominent in about 40% of tumors. In 25% of the tumors, cysts sometimes associated with a cellular stroma (the polyvesicular pattern), were present but conspicuous in only half these cases. One third of the tumors had a labyrinthine pattern, 22% glands, and 6% a festoon pattern. Papillae with a central blood vessel (Schiller-Duval bodies) were seen in one-third of the tumors but were numerous in only 5% of them. Nonspecific appearing papillae were seen in 10% of the tumors. A solid growth of cells with pale cytoplasm was seen in one-third of the tumors but was conspicuous in only half of that subset. The solid appearance was typically reminiscent of that of dysgerminoma, but lacked the septa and lymphocytic infiltrate of that neoplasm. Nine tumors had a component of cells with scant cytoplasm resulting in a blastema-like appearance and 3 had cells with abundant clear cytoplasm. Cords and clusters of cells were common but did not dominate the microscopic appearance. The stroma typically had a nonspecific collagenous to edematous appearance. Stromal luteinization was seen in 12 tumors; in 5 this was likely due to the patient being pregnant. Two tumors had minor foci of cells that resembled hepatocytes. Hyaline bodies were seen in most of the tumors and were often conspicuous. The neoplastic cells typically had modest amounts of lightly staining cytoplasm and only mild nuclear pleomorphism. Cells lining cysts were often flattened sometimes resulting in a deceptively innocuous appearance. Many of the tumors (mostly consultation cases), caused diagnostic difficulty; tumors in the differential diagnosis included clear cell carcinoma, embryonal carcinoma, Sertoli-Leydig cell tumor, and juvenile granulosa cell tumor. The patient age and marked elevation of the serum alpha-fetoprotein level (if measured) is helpful in many of these considerations. The overtly malignant gross appearance of most YSTs contrasts with certain other tumors in the differential and the association of some YSTs with dermoid cyst and many clear cell carcinomas with endometriosis may be helpful. The vast majority of ovarian YSTs are dominated microscopically by merging of reticular and cystic patterns which, although focally mimicked by other neoplasms, are in general characteristic, and distinctive features of other neoplasms are absent. Immunohistochemistry, particularly for alpha-fetoprotein and glypican 3, and lack of staining for various markers of other neoplasms is helpful but overlap exists and these results must be considered in the context of the overall clinical, gross, and microscopic features. YSTs dominated by hepatoid and glandular features are rare and their categorization as YSTs should be done cautiously if thoroughly sampled tumors show no evidence of classic features of YST emphasized herein and first elaborated by the Danish investigator Gunnar Teilum whose seminal observations have stood the test of time.

ARPEC: A novel staggered perforated permeable caisson breakwater for wave absorption and harbour flushing

Perforated multichamber caisson breakwaters are widely used for deep water harbour protection from waves, causing a partial dissipation of wave energy, hence reducing wave reflection. The innovative patented geometry presented in this paper, the ARPEC, is an Anti Reflection PErmeable Caisson breakwater that allows a significant wave energy dissipation and a hydraulic connection between the sea side and the port side by means of a labyrinthine pattern of offset openings in all external and internal walls. The hydraulic response of this innovative structure is evaluated by means of both 2D physical and numerical models. It is shown that, when compared with traditional impermeable perforated caissons, the marginal increase of wave transmission (CT=HT/HI in the approximate range of 0.1–0.2, varying with wave period) is compensated by a slight reduction of wave reflection (CR=HR/HI in the approximate range of 0.3–0.6, varying with wave period) and by an enhanced water circulation in the sheltered basin, which is beneficial for flushing of microtidal harbours.

Vortex ordering and dynamics on Santa Fe artificial ice pinning arrays

We numerically examine the ordering, pinning, and flow of superconducting vortices interacting with a Santa Fe artificial ice pinning array. We find that as a function of magnetic field and pinning density, a wide variety of vortex states occur, including ice rule obeying states and labyrinthine patterns. In contrast to square pinning arrays, we find no sharp peaks in the critical current due to the inherent frustration effect imposed by the Santa Fe ice geometry; however, there are some smoothed peaks when the number of vortices matches the number of pinning sites. For some fillings, the Santa Fe array exhibits stronger pinning than the square array due to the suppression of one-dimensional flow channels when the vortex motion in the Santa Fe lattice occurs through the formation of both longitudinal and transverse flow channels.

Inkjet-Printed Quantum Dot Fluorescent Security Labels with Triple-Level Optical Encryption.

Optical security labels play a significant role in protecting both our wealth and health. However, simultaneously meeting the requirements including low-cost fabrication, easy detection, and high-level security is still challenging for security labels. Here, we design an unclonable anti-counterfeiting system with triple-level security by using the inkjet printing technique, which can be authenticated by naked eyes, a portable microscope, and a fluorescence microscope. These labels are achieved by printing microscale quantum dot (QD) ink droplets on premodified substrates with random-distributed glass microspheres. Due to the unique capillary action induced by the glass microspheres, QDs in the ink droplets are deposited around the microspheres, forming microscale multicircular patterns. Multiple pinning of QDs at the three-phase contact lines appears during the evaporation of the droplet, resulting in the formation of a nanoscale labyrinthine pattern around the microspheres. The nanoscale labyrinth pattern and the microscale multicircular microsphere array, together with the printed macroscopic image, constitute a triple-level progressive anti-counterfeiting system. Moreover, the system is compatible with an artificial intelligence-based identification strategy that allows rapid identification and verification of the unclonable security labels.

Permanent encoding of nano‐ to macro‐scale hierarchies of order from evaporative magnetic fluids

AbstractMagnetic field‐directed assemblies of magnetic nanoparticles (MNPs) in ferrofluids exhibit complex interconvertible metastable patterns and structures. Formally, ferrofluid patterns are unstable – they disappear when the magnetic field is removed. The present study shows that ferrofluid patterns can be “trapped” as kinetically stable structures that encode a surprising degree of morphological detail over nanometer to millimeter length scales. An external magnetic field is used to direct assembly of oleic acid‐decorated magnetite (Fe3O4) nanoparticles to make spike and labyrinthine patterns in volatile host solvents of heptane, octane and nonane. Solvent evaporation coupled with increases in sample magnetization drive pattern formation and its permanent recording. Use of a crosslinking siloxane polymer host yields remarkably different material responses. From the trapped states in both fluid systems, previously unreported hierarchies of order emerge in nanocomposite spike structures that also exhibit orientational and magnetic anisotropy. The possibility of designing hierarchical matter from initially uncorrelated MNPs is demonstrated by a directed solid‐state transformation of the magnetic nanocomposite; the spikes template memory of their origin onto the transformation products.

Labyrinthine patterns transitions

Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1180903 Millennium Institute for Research in Optics (MIRO) ANID by Beca Doctorado Nacional 2020-21201376

A positive and energy stable numerical scheme for the Poisson–Nernst–Planck–Cahn–Hilliard equations with steric interactions

In this work, we consider numerical methods for the Poisson–Nernst–Planck–Cahn–Hilliard (PNPCH) equations with steric interactions, which correspond to a non-constant mobility H−1 gradient flow of a free-energy functional that consists of electrostatic free energies, steric interaction energies of short range, entropic contribution of ions, and concentration gradient energies. We propose a novel energy stable numerical scheme that respects mass conservation and positivity at the discrete level. Existence and uniqueness of the solution to the proposed nonlinear scheme are established by showing that the solution is a unique minimizer of a convex functional over a closed, convex domain. The positivity of numerical solutions is further theoretically justified by the singularity of the entropy terms, which prevents the minimizer from approaching zero concentrations. A further numerical analysis proves discrete free-energy dissipation. Extensive numerical tests are performed to validate that the numerical scheme is first-order accurate in time and second-order accurate in space, and is capable of preserving the desired properties, such as mass conservation, positivity, and free energy dissipation, at the discrete level. Moreover, the PNPCH equations and the proposed scheme are applied to study charge dynamics and self-assembled nanopatterns in highly concentrated electrolytes that are widely used in electrochemical energy devices. Numerical results demonstrate that the PNPCH equations and our numerical scheme are able to capture nanostructures, such as lamellar patterns and labyrinthine patterns in electric double layers and the bulk, and multiple time relaxation with multiple time scales. The multiple time relaxation dynamics with metastability take long time to reach an equilibrium, highlighting the need for robust, energy stable numerical schemes that allow large time stepping. In addition, we numerically characterize the interplay between cross steric interactions of short range and the concentration gradient regularization, and their impact on the development of nanostructures in the equilibrium state.

A comparison of computational models for wrinkling of pressurized shell-core systems

Four nonlinear computational models for the surface wrinkling of curved shell-core systems under external pressure are presented. Three of the considered finite element models neglect the displacements tangential to the shell surface. Two of the models are static formulations and the other two are derived in the dynamic framework. For the latter, the energy-decaying time-stepping algorithm is applied, which is suitable for numerically stiff problems, such as shell-core systems, characterized by stiff membrane and soft wrinkling deformation modes. In all cases the core is modeled by elastic springs. As a comparative problem we choose the surface wrinkling of pressurized shell-core spheres. In particular, five systems with different material and geometric properties are computed, which have different wrinkle patterns. A good agreement is found between the results of the models as well as with the relevant references, which provide numerical and experimental results. However, it has been observed that our reduced-order models are blind to the prediction of the secondary transformation – from the dimple-like pattern to the labyrinthine pattern. Another conclusion is that a non-tailored (i.e. standard) shell finite element on an elastic foundation combined with the energy-decaying scheme, provides excellent predictions of the surface wrinkle patterns.

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. Using two prototypical models, the Swift-Hohenberg equation and the forced complex Ginzburg-Landau equation, we consider finite size domains with no flux boundary conditions transversal to the stripes and reveal a distinct mixed-mode instability that lies in between the classical zigzag and the Eckhaus lines. This explains the stability of stripes in the mildly zigzag unstable regime and, after crossing the mixed-mode line, the evolution of zigzag stripes in the bulk of the domain and the formation of defects near the boundaries. The results are of particular importance for problems with large timescale separation, such as bulk-heterojunction deformations in organic photovoltaic and vegetation in semi-arid regions, where early temporal transients may play an important role.

Ferrofluid Microdroplet Splitting for Population-Based Microfluidics and Interfacial Tensiometry.

Ferrofluids exhibit a unique combination of liquid properties and strong magnetic response, which leads to a rich variety of interesting functional properties. Here, the magnetic‐field‐induced splitting of ferrofluid droplets immersed in an immiscible liquid is presented, and related fascinating dynamics and applications are discussed. A magnetic field created by a permanent magnet induces instability on a mother droplet, which divides into two daughter droplets in less than 0.1 s. During the splitting process, the droplet undergoes a Plateau–Rayleigh‐like instability, which is investigated using high‐speed imaging. The dynamics of the resulting satellite droplet formation is shown to depend on the roughness of the supporting surface. Further increasing the field results in additional splitting events and self‐assembly of microdroplet populations, which can be magnetically actuated. The effects of magnetization and interfacial tension are systematically investigated by varying magnetic nanoparticles and surfactant concentrations, and a variety of outcomes from labyrinthine patterns to discrete droplets are observed. As the splitting process depends on interfacial tension, the droplet splitting can be used as a measure for interfacial tension as low as 0.1 mN m−1. Finally, a population‐based digital microfluidics concept based on the self‐assembled microdroplets is presented.