Complex Pattern Formation Research Articles

Spatiotemporal complexity analysis of a discrete space-time cancer growth model with self-diffusion and cross-diffusion

We investigate spatiotemporal pattern formation in cancer growth using discrete time and space variables. We first introduce the coupled map lattices (CMLs) model and provide a dynamical analysis of its fixed points along with stability results. We then offer parameter criteria for flip, Neimark–Sacker, and Turing bifurcations. In the presence of spatial diffusion, we find that stable homogeneous solutions can experience Turing instability under certain conditions. Numerical simulations reveal a variety of spatiotemporal patterns, including patches, spirals, and numerous other regular and irregular patterns. Compared to previous literature, our discrete model captures more complex and richer nonlinear dynamical behaviors, providing new insights into the formation of complex patterns in spatially extended discrete tumor models. These findings demonstrate the model’s ability to capture complex dynamics and offer valuable insights for understanding and treating cancer growth, highlighting its potential applications in biomedical research.

Complex pattern formation governed by a Cahn–Hilliard–Swift–Hohenberg system: Analysis and numerical simulations

This paper investigates a Cahn–Hilliard–Swift–Hohenberg system, focusing on a three-species chemical mixture subject to physical constraints on volume fractions. The resulting system leads to complex patterns involving a separation into phases as typical of the Cahn–Hilliard equation and small scale stripes and dots as seen in the Swift–Hohenberg equation. We introduce singular potentials of logarithmic type to enhance the model’s accuracy in adhering to essential physical constraints. The paper establishes the existence and uniqueness of weak solutions within this extended framework. The insights gained contribute to a deeper understanding of phase separation in complex systems, with potential applications in materials science and related fields. We introduce a stable finite element approximation based on an obstacle formulation. Subsequent numerical simulations demonstrate that the model allows for complex structures as seen in pigment patterns of animals and in porous polymeric materials.

Procedure to reveal the mechanism of pattern formation process by topological data analysis

Topological data analysis (TDA) is a versatile tool that can be used to extract scientific knowledge from complex pattern formation processes. However, the physics correspondence between the features obtained from TDA and pattern dynamics does not agree one-to-one, and the physical interpretation of the TDA features needs to be set appropriately according to the phenomenon to be analyzed. In this study, we propose an analytical procedure to physically interpret pattern dynamics through TDA and machine learning techniques. The proposed procedure was applied to the process of magnetic domain pattern formation to quantify non-trivial domain pattern classifications and reveal the nature of the underlying dynamics. On the basis of these findings, we also propose a candidate reduction model to understand the nature of magnetic domain formation.

Exploring the Element of Form Based on Traditional Chinese Auspicious Patterns

Chinese traditional auspicious patterns, which date back more than 5,000 years, are a vital component of national art and traditional culture. These patterns have special meanings and are often associated with good wishes and blessings. Their development reached its peak in the Ming and Qing Dynasties. This research on auspicious patterns focuses mainly on the creation method and classification, with few forms. It seeks to ascertain the features of the traditional auspicious pattern form in China. This study adopts qualitative analysis, specifically in analyzing patterns by observation. Based on Panofsky's theory of iconography, the study selected six common Chinese traditional auspicious patterns and analyzed them to determine their classification, layout, composition and elements. It can be concluded that their form is characterized by various elements, specifically figurative and complete, complex pattern formations, full layout designs, balanced or symmetrical compositions, and single or fit patterns. Thus, this study summarized the pattern form analysis model or the PFC-Model. By addressing this gap in knowledge, the study enhances understanding of auspicious patterns. At the same time, the pattern analysis model can be used to analyze similar patterns, which makes it easier to analyze other pattern forms and brings the study of traditional Chinese patterns to a new height. In the meantime, this research only studies the external form of auspicious patterns. In this regard, as form and color is the first layer of research mentioned in Panofsky's image theory, it needs to be more comprehensive. In the future, researchers can continue to explore its color characteristics and extend the research into the content of the first layer of the theories. Undoubtedly, researchers can also expand the analysis of auspicious patterns using the theory's second and third layers.

Billiards with Spatial Memory.

Many classes of active matter develop spatial memory by encoding information in space. We present a framework based on mathematical billiards, wherein particles remember their past trajectories. Despite its deterministic rules, such a system is strongly nonergodic and exhibits intermittent statistics and complex pattern formation. We show how these features emerge from the dynamic change of topology. Our work illustrates how the dynamics of a single-body system can dramatically change with spatial memory, laying the groundwork to further explore systems with complex memory kernels.

Bifurcations and pattern formation in a host–parasitoid model with nonlocal effect

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

Robust two-color physically unclonable patterns from controlled aggregation of a single organic luminophore.

This research presents a new approach to create two-color luminescent physically unclonable functions (PUFs) using an organic luminophore with tunable emission colors. These PUFs offer high security and stable performance, significantly enhancing anti-counterfeiting capabilities by exponentially increasing encoding capacity through dual-color integration and complex pattern formation.

Patterns of a General Chemical Model Involving Degn-Harrison Reaction Scheme

This paper reports the pattern formation of a general Degn-Harrison system. We first determine the types and stability of the unique positive equilibrium for the spatially homogeneous system. The equilibrium may be node, focus, or center. Supercritical or subcritical Hopf bifurcation may occur if it is a center. In the sequel, we propose the conditions for the occurrence of Turing instability for the spatially inhomogeneous system. We can theoretically explain that Turing instability exists as the equilibrium transitions from homogeneous stable to inhomogeneous unstable states. Finally, we perform computational experiments to investigate the complex pattern formation of this chemical model. An interesting finding is that if one of the reactants has a high diffusion rate, the pattern formation will be inhibited. Conversely, a low diffusion rate may promote pattern formation.

Rapid bacteria-phage coevolution drives the emergence of multiscale networks.

Interactions between species catalyze the evolution of multiscale ecological networks, including both nested and modular elements that regulate the function of diverse communities. One common assumption is that such complex pattern formation requires spatial isolation or long evolutionary timescales. We show that multiscale network structure can evolve rapidly under simple ecological conditions without spatial structure. In just 21 days of laboratory coevolution, Escherichia coli and bacteriophage Φ21 coevolve and diversify to form elaborate cross-infection networks. By measuring ~10,000 phage-bacteria infections and testing the genetic basis of interactions, we identify the mechanisms that create each component of the multiscale pattern. Our results demonstrate how multiscale networks evolve in parasite-host systems, illustrating Darwin's idea that simple adaptive processes can generate entangled banks of ecological interactions.

Complex pattern dynamics and synchronization in a coupled spatiotemporal plankton system with zooplankton vertical migration

In aquatic environments, zooplankton often exhibits life habit of vertical migration and the phytoplankton-zooplankton predator-prey systems in different water layers are therefore coupled together. To uncover the nonlinear mechanisms of pattern dynamics in such coupled plankton system, a spatiotemporal model is developed in this research with the consideration of plankton horizontal diffusion and zooplankton vertical migration. Equation of dispersion curve for the coupled system is derived, and an extension of regular Turing instability is disclosed by determining the number of peaks on the dispersion curve. We find a mechanism named as “projection of characteristics” that local or global characteristics projecting between different subsystems causes the self-organization of complex patterns with multi-scale structures. The combination of various bifurcations with Turing instability controls the system stability and can lead to more complex pattern formation. Notably, when flip bifurcation takes place, the coupled system generates an interesting result that the phytoplankton populations in some water layers become extinct but the zooplankton populations in these layers can still be persistent due to the effect of population compensation. It is also found that the increase of zooplankton migration rate can result to spatiotemporal synchronization of pattern evolution, and that particularly, the pattern dynamics in just a part of subsystems achieve local synchronization. The methods and results presented herein may lead to a better understanding on the emergence of complex plankton patchiness in nature.

Electrodeposition of Alkali Metals in Nonaqueous Electrolytes: Classical and Nonclassical Phenomena

The electrodeposition of metals in aqueous electrolytes is an important technology for corrosion protection, surface beautification, complex pattern formation, and many other applications. In most cases, a smooth surface finish is desired and achieved, thanks to the precision understanding of the nucleation and growth mechanisms. However, the electrodeposition of alkali metals in nonaqueous electrolytes, which is critical for anode-free alkali-ion batteries to achieve the highest possible energy density, is still poorly understood. During battery recharge, alkali ions stored in the positive electrode of the battery would move toward the negative electrode to deposit on the current collector. However, due to the very low intrinsic redox potentials of alkali metals (Li, Na, and K for battery applications), even nonaqueous electrolytes would spontaneously react with the metal deposits to form a solid-electrolyte interphase (SEI) layer composed of organic and inorganic reduction products that allow ions to pass through but not electrons. The SEI layer, which is heterogeneous in nature, not only undermines the uniformity of incoming ion flux but also impedes the surface diffusion of adatoms, leading to undesired localized nucleation and growths that pose short-circuiting risks. In this presentation, we will introduce our latest results on the operando characterization and mathematical modeling of electrodeposition of alkali metals in various nonaqueous electrolytes, yet against a porous separator. Our analyses in three different dynamic regimes will facilitate the precision understanding of the nonclassical electrodeposition behaviors and reveal the remaining challenge that the electrodeposition community can contribute toward achieving revolutionary stable metal electrodes.

Morphological Diagram of Dynamic-Interfacial-Release-Induced Surface Instability.

In this study, a morphological diagram was constructed for quantitatively predicting various modes of surface instabilities caused by the dynamic interfacial release of strain in initially flat bilayer composites comprising an elastomer and a capping layer. Theory, experiment, and simulation were combined to produce the diagram, which enables systematic generation of the following instability patterns: wrinkle, fold, period-double, delamination, and coexisting patterns. The pattern that forms is most strongly affected by three experimental parameters: the elastic modulus of the elastomer, the elastic modulus of the capping layer, and the thickness of the capping layer. The morphological diagram offers understanding of the formation of complex patterns and development of their applications. Critically, the wrinkle alignment can be well controlled by changing the direction of the interfacial release to enable the creation of centimeter-sized and highly ordered lamellar wrinkled patterns with a single orientation on a soft elastomer without the need for costly high-vacuum instruments or complex fabrication processes. The method and diagram have great potential for broad use in many practical applications ranging from flexible electronic devices to smart windows.

Probabilistic cellular automata modelling of intercellular interactions in airways: complex pattern formation in patients with chronic obstructive pulmonary disease

Chronic obstructive pulmonary disease (COPD) is a highly prevalent lung disease characterized by chronic inflammation and tissue remodeling possibly induced by unusual interactions between fibrocytes and CD8+ T lymphocytes in the peribronchial area. To investigate this phenomenon, we developed a probabilistic cellular automata type model where the two types of cells follow simple local interaction rules taking into account cell death, proliferation, migration and infiltration. We conducted a rigorous mathematical analysis using multiscale experimental data obtained in control and disease conditions to estimate the model’s parameters accurately. The simulation of the model is straightforward to implement, and two distinct patterns emerged that we can analyse quantitatively. In particular, we show that the change in fibrocyte density in the COPD condition is mainly the consequence of their infiltration into the lung during exacerbations, suggesting possible explanations for experimental observations in normal and COPD tissue. Our integrated approach that combines a probabilistic cellular automata model and experimental findings will provide further insights into COPD in future studies.

Mechanisms underlying the formation of complex color patterns on Nigella orientalis (Ranunculaceae) petals.

Complex color patterns on petals are widespread in flowering plants, yet the mechanisms underlying their formation remain largely unclear. Here, by conducting detailed morphological, anatomical, biochemical, optical, transcriptomic, and functional studies, we investigated the cellular bases, chromogenic substances, reflectance spectra, developmental processes, and underlying mechanisms of complex color pattern formation on Nigella orientalis petals. We found that the complexity of the N. orientalis petals in color pattern is reflected at multiple levels, with the amount and arrangement of different pigmented cells being the key. We also found that biosynthesis of the chromogenic substances of different colors is sequential, so that one color/pattern is superimposed on another. Expression and functional studies further revealed that a pair of R2R3-MYB genes function cooperatively to specify the formation of the eyebrow-like horizontal stripe and the Mohawk haircut-like splatters. Specifically, while NiorMYB113-1 functions to draw a large splatter region, NiorMYB113-2 functions to suppress the production of anthocyanins from the region where a gap will form, thereby forming the highly specialized pattern. Our results provide a detailed portrait for the spatiotemporal dynamics of the coloration of N. orientalis petals and help better understand the mechanisms underlying complex color pattern formation in plants.

Development of ornamentation on ventifacts: An examination of flow and saltation kinematic mechanisms

AbstractVentifacts are rocks that have been shaped through abrasion by wind‐transported sediment. Their surfaces may be ornamented with characteristic microscale features that are replicated and overlain to form complex patterns. Compared to a large body of work on the transport of unconsolidated sediment in which the atmospheric boundary‐layer flow and sand cloud respond to perturbations in the developing topography, no experimental work to this date has addressed morphodynamic feedback at the particle scale in aeolian systems dominated by abrasion. This article reports on a case study in which laser Doppler anemometry was used in wind tunnel experiments to measure air flow and sand transport over a highly ornamented, tabular ventifact from the Taylor Valley, Antarctica. Periodicity characterizing the static microtopography of the ventifact was observed to be strongly imprinted on the kinetics of both the airflow and particle cloud when sampled along a transect of the ventifact aligned with flow in the wind tunnel matching the dominant sand transporting wind direction for the ventifact in situ. The vertical velocity component was especially well synchronized, with changes in elevation on the order of 2–4 mm. The horizontal component of particle velocity demonstrated an offset in the downwind direction associated with the large forward momentum of the saltating sand particles. However, submillimeter relief on the fixed ventifact surface was not sufficient to initiate kinetic feedback, as observed along a transect aligned normal to a series of elongated flute structures and the dominant sand transporting wind direction. The particle‐scale experimental results confirm that (i) abrasion of this ventifact principally occurred under a unimodal wind and sand transport regime of low‐moderate intensity, and (ii) relief arising from weathering, as governed by the ventifact's lithology, likely initiated development of the present‐day ornamentation.

Complex pattern formations induced by the presence of cross-diffusion in a generalized predator–prey model incorporating the Holling type functional response and generalization of habitat complexity effect

The interaction between predator and their prey is one of the most complex processes in ecosystems due to its structure and nature. Many extrinsic and intrinsic factors can affect the population dynamics. In this manuscript, we have focused our study on the process of Turing-pattern formation in a generalized predator–prey system with cross-diffusion and habitat complexity effects. By applying the linear stability theory, sufficient conditions for the occurrence of Hopf bifurcation and Turing-driven instability are successfully established. Moreover, the amplitude equations are derived near the critical value of Turing instability according to the standard multiple-scale analysis. From the mathematical view, our investigation reveals that the whole patterns generated by the system are governed by two control parameters, which are the cross-diffusion parameter δ21 and the predator mortality rate m. Complex but interesting dynamical pattern formations are specified theoretically and numerically such as spot pattern, stripe pattern, and spot-strip pattern in terms of varying the two control parameters. To support the theoretical results, a series of numerical tests are carried out, especially, to illustrate the diverse patterns near the codimension- Turing–Hopf bifurcation point (δ21T,m(0)H) and the significant impact of δ21 and m on the distribution of the two species.

A morphoelastic stability framework for post-critical pattern formation in growing thin biomaterials

Morphological instabilities play a key role in the evolution of form and function in growing biomaterials. Spatially varying differential growth, in particular, leads to residual stresses that for soft materials or slender geometries are favourably relieved by out-of-plane buckling or wrinkling. While the onset of instability in growing biomaterials has been studied extensively, the post-critical regime remains poorly understood. To this end, this paper presents a robust computational modelling framework for morphoelastic instabilities and associated post-critical pattern formation. The seven-parameter shell element—commonly used for elasto-plastic analysis of engineering materials—is implemented alongside the multiplicative decomposition of the deformation gradient tensor into a growth and an elastic part. The governing nonlinear equations are solved using a generalised path-following/numerical continuation solver that facilitates comprehensive exploration of the stability landscape through pinpointing of critical points and branch switching at bifurcations. The utility and power of the computational framework is demonstrated by unveiling complex pattern formation phenomena that arise as a result of sequentially occurring morphological instabilities. In particular, we highlight the central role that exponential edge growth plays in fractal wrinkling patterns, the blooming of doubly-curved flower petals, and wavy daffodil trumpets. In addition, the ability of the solver to track critical points through parameter space enables efficient sensitivity studies into the role of material parameters on pattern formation. The presented computational framework is thus a versatile tool for modelling the evolution of form and function in growing biological systems and the design of biotechnology applications.

Physical realization of complex dynamical pattern formation in magnetic active feedback rings

We report the clean experimental realization of cubic–quintic complex Ginzburg–Landau (CQCGL) physics in a single driven, damped system. Four numerically predicted categories of complex dynamical behavior and pattern formation are identified for bright and dark solitary waves propagating around an active magnetic thin film-based feedback ring: (1) periodic breathing; (2) complex recurrence; (3) spontaneous spatial shifting; and (4) intermittency. These nontransient, long lifetime behaviors are observed in self-generated spin wave envelopes circulating within a dispersive, nonlinear yttrium iron garnet waveguide. The waveguide is operated in a ring geometry in which the net losses are directly compensated for via linear amplification on each round trip (of the order of 100 ns). These behaviors exhibit periods ranging from tens to thousands of round trip times (of the order of μs) and are stable for 1000s of periods (of the order of ms). We present ten observations of these dynamical behaviors which span the experimentally accessible ranges of attractive cubic nonlinearity, dispersion, and external field strength that support the self-generation of backward volume spin waves in a four-wave-mixing dominant regime. Three-wave splitting is not explicitly forbidden and is treated as an additional source of nonlinear losses. All observed behaviors are robust over wide parameter regimes, making them promising for technological applications. We present ten experimental observations which span all categories of dynamical behavior previously theoretically predicted to be observable. This represents a complete experimental verification of the CQCGL equation as a model for the study of fundamental, complex nonlinear dynamics for driven, damped waves evolving in nonlinear, dispersive systems. The reported dynamical pattern formation of self-generated dark solitary waves in attractive nonlinearity without external sources or potentials, however, is entirely novel and is presented for both the periodic breather and complex recurrence behaviors.

Superelongation of Liquid Metal.

The ability to control interfacial tension electrochemically is uniquely available for liquid metals (LMs), in particular gallium‐based LM alloys. This imparts them with excellent locomotion and deformation capabilities and enables diverse applications. However, electrochemical oxidation of LM is a highly dynamic process, which often induces Marangoni instabilities that make it almost impossible to elongate LM and manipulate its morphology directly and precisely on a 2D plane without the assistance of other patterning methods. To overcome these limitations, this study investigates the use of an LM–iron (Fe) particle mixture that is capable of suppressing instabilities during the electrochemical oxidation process, thereby allowing for superelongation of the LM core of the mixture to form a thin wire that is tens of times of its original length. More importantly, the elongated LM core can be manipulated freely on a 2D plane to form complex patterns. Eliminating Marangoni instabilities also allows for the effective spreading and filling of the LM–Fe mixture into molds with complex structures and small features. Harnessing these excellent abilities, a channel‐less patterning method for fabricating elastomeric wearable sensors is demonstrated to detect motions. This study shows the potential for developing functional and flexible structures of LM with superior performance.

Exploring complex pattern formation with convolutional neural networks

Many nonequilibrium systems, such as biochemical reactions and socioeconomic interactions, can be described by reaction–diffusion equations that demonstrate a wide variety of complex spatiotemporal patterns. The diversity of the morphology of these patterns makes it difficult to classify them quantitatively, and they are often described visually. Hence, searching through a large parameter space for patterns is a tedious manual task. We discuss how convolutional neural networks can be used to scan the parameter space, investigate existing patterns in more detail, and aid in finding new groups of patterns. As an example, we consider the Gray–Scott model for which training data are easy to obtain. Due to the popularity of machine learning in many scientific fields, well maintained open source toolkits are available that make it easy to implement the methods that we discuss in advanced undergraduate and graduate computational physics projects.