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In this work, the influences of geometry and domain size on spatiotemporal pattern formation are investigated to establish the parameter spaces for a cross-diffusive reaction–diffusion model on an annulus. By applying the linear stability theory, we derive the conditions which can give rise to Turing, Hopf and transcritical types of diffusion-driven instabilities. We explore whether the selection of a sufficiently large domain size, together with the appropriate selection of parameters, can give rise to the development of patterns on nonconvex geometries, e.g. annulus. Hence, the key research methodology and outcomes of our studies include a complete analytical exploration of the spatiotemporal dynamics in an activator-depleted reaction–diffusion system; a linear stability analysis to characterize the dual roles of cross-diffusion and domain size of pattern formation on an annulus region; the derivation of the instability conditions through the lower and upper bounds of the domain size; the full classification of the model parameters; and a demonstration of how cross-diffusion relaxes the general conditions for the reaction–diffusion system to exhibit pattern formation. To validate the theoretical findings and predictions, we employ the finite element method to reveal the spatial and spatiotemporal patterns in the dynamics of the cross-diffusive reaction–diffusion system within a two-dimensional annular domain. These observed patterns resemble those found in ring-shaped cross-sectional scans of hypoxic tumors. Specifically, the cross-section of an actively invasive region in a hypoxic tumor can be effectively approximated by an annulus.

We build and investigate a nonstandard model of pattern formation in a system of discrete entities evolving in discrete space and time. We chose a sandpile paradigm to fit our ideas in the frame of current research. In our model sand is hot because a grain can topple against gradient, i.e., the grain can walk to another node even when a number of grains in its current node is less than a number of neighboring nodes. Sand is choosey because behavior of the grains is not determined by any global parameter or any threshold of a number of neighboring grains (called here a grain sensitivity) but depends on the exact number of grains in the neighboring nodes. Namely, we assume that a grain being at a node x goes to one of the eight neighboring nodes, chosen at random, if there is another grain at the node x or if the number of grains in eight neighboring nodes lies in some set of 2^{1,…,8}. These 256 rules of sensitivity are investigated. The classification of the rules if offered, based on the morphology of the patterns generated by each rule. Eight morphological classes are found. Fine structure of every class is investigated and transient phenomena are analyzed. Three kinds of description of class rules by Boolean expressions are offered. Evolution of the classes governed by several one-dimensional parameters is considered.

In the present work, we investigate the ability of recently proposed computational operators to characterize the problem of pattern formation by fluid flow and mass transport in a viscous binary fluid mixture under osmosedimentation. We perform a numerical investigation of these patterns using three computational operators (R² → R), computed from the spatial mass distribution, the streamlines, and the velocity fields. Our main goal is to demonstrate the ability of these computational operators to distinguish different dynamical regimes in the complex patterns arising from the osmosedimentation process.

We model formation and evolution of transverse dune fields. In the model, only the cross section of the dune is simulated. The only physical variable of relevance is the dune height, from which the dune width and velocity are determined, as well as phenomenological rules for interaction between two dunes of different heights. We find that dune fields with no sand on the ground between dunes are unstable, i.e., small dunes leave the higher ones behind. We then introduce a saturation length to simulate transverse dunes on a sand bed and show that this leads to stable dune fields with regular spacing and dune heights. Finally, we show that our model can be used to simulate coastal dune fields if a constant sand influx is considered, where the dune height increases with the distance from the beach, reaching a constant value.

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. And spatial patterns are ubiquitous in nature, which can modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, a predator-prey system with Michaelis-Menten-type functional response and self- and cross-diffusion is investigated. Based on the mathematical analysis, we obtain the condition of the emergence of spatial patterns through diffusion instability, i.e., Turing pattern. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e., stripe-like or spotted or coexistence of both. The obtained results show that the interaction of self-diffusion and cross-diffusion plays an important role on the pattern formation of the predator-prey system.

We present a polarity-driven activator-inhibitor model of budding yeast in a two-dimensional medium wherein impeding metabolites secretion (or growth inhibitors) and growth directionality are determined by the local nutrient level. We found that colony size and morphological features varied with nutrient concentration. A branched-type morphology is associated with high impeding metabolite concentration together with a high fraction of distal budding, while opposite conditions (low impeding metabolite concentration, high fraction of proximal budding) promote Eden-type patterns. Increasing the anisotropy factor (or polarity) produced other spatial patterns akin to the electrical breakdown under varying electric field. Rapid changes in the colony morphology, which we conjecture to be equivalent to a transition from an inactive quiescent state to an active budding state, appeared when nutrients were limited.

Excitable cellular automata with dynamical excitation interval exhibit a wide range of space-time dynamics based on an interplay between propagating excitation patterns which modify excitability of the automaton cells. Such interactions leads to formation of standing domains of excitation, stationary waves and localized excitations. We analyzed morphological and generative diversities of the functions studied and characterized the functions with highest values of the diversities. Amongst other intriguing discoveries we found that upper boundary of excitation interval more significantly affects morphological diversity of configurations generated than lower boundary of the interval does and there is no match between functions which produce configurations of excitation with highest morphological diversity and configurations of interval boundaries with highest morphological diversity. Potential directions of future studies of excitable media with dynamically changing excitability may focus on relations of the automaton model with living excitable media, e.g. neural tissue and muscles, novel materials with memristive properties and networks of conductive polymers.

A new methodology was introduced to investigate the pattern formation in time series systems due to their viscoelastic behavior. Four stochastic processes, uniform distribution, normal distribution, Nasdaq-100 stock market index, and a melody were studied within this context. The time series data were converted into vectorial forms in a scattering diagram. The sequential vectors can be split into its in-line (or conservative) and out-of-line (or dissipative) components. Thus, one can define the storage and loss modulus for conservative, and dissipative components, respectively. Instead of using the geometric Brownian equation which involves Wiener noise term, the changes were taken into consideration at every step by introducing “lethargy” concept and the deviation from it. Thus, the mathematics is somehow simplified, and the dynamical behavior of time series systems can be elucidated at every step of change. The viscoelastic behavior of time series systems reveals patterns of the viscoelastic parameters such as storage and loss modulus, and also of thermodynamic work-like and heat-like properties. Besides, there occur some minima and maxima in the distribution of the angles between the sequential vectors in the scattering diagram. The same is true for the change of entropy of the system.

We discuss the effects of spatial interference between two infectious hotspots as a function of the mobility of individuals (wind speed) between the two and their relative degree of infectivity. As long as the upstream hotspot is less contagious than the downstream one, increasing the wind speed leads to a monotonic decrease of the infection peak in the downstream hotspot. Once the upstream hotspot becomes about between twice and five times more infectious than the downstream one, an optimal wind speed emerges, whereby a local minimum peak intensity is attained in the downstream hotspot, along with a local maximum beyond which the beneficial effect of the wind is restored. Since this nonmonotonic trend is reminiscent of the equation of state of nonideal fluids, we dub the above phenomena “epidemic condensation”. When the relative infectivity of the upstream hotspot exceeds about a factor five, the beneficial effect of the wind above the optimal speed is completely lost: any wind speed above the optimal one leads to a higher infection peak. It is also found that spatial correlation between the two hotspots decay much more slowly than their inverse distance. It is hoped that the above findings may offer a qualitative clue for optimal confinement policies between different cities and urban agglomerates.

Spatial fluctuations in dissipative systems, such as rapid granular flows, behave very differently from those in elastic fluids. Fluctuations in the flow field drive the linear and nonlinear instability in the density field (clustering), while vortex structures appear and grow through the mechanism of noise reduction. The dynamics of the flow field on the largest space and time scales is described by diffusion equations with different diffusivities for the transverse and longitudinal flow fields. The results are obtained from analytic and simulation methods.

The problem of morphogenesis and Turing instability are revisited from the point of view of dimensionality effects. First the linear analysis of a generic Turing model is elaborated to the case of multiple stationary states, which may lead the system to bistability. The difference between two- and three-dimensional pattern formation with respect to pattern selection and robustness is discussed. Preliminary results concerning the transition between quasi-two-dimensional and three-dimensional structures are presented and their relation to experimental results are addressed.

The complex arrangements of atoms near grain boundaries are difficult to understand theoretically. We propose a phenomenological (Ginzburg–Landau-like) description of crystalline phases based on symmetries and some fairly general stability arguments. This method allows a very detailed description of defects at the lattice scale with virtually no tunning parameters, unlike the usual phase-field methods. The model equations are directly inspired from those used in a very different physical context, namely, the formation of periodic patterns in systems out-of-equilibrium (e.g. Rayleigh–Bénard convection, Turing patterns). We apply the formalism to the study of symmetric tilt boundaries. Our results are in quantitative agreement with those predicted by a recent crystallographic theory of grain boundaries based on a geometrical quasicrystal-like construction. These results suggest that frustration and competition effects near a defect in crystalline arrangements have some universal features, of interest in solids or other periodic phases.

We present a family of methods, analytical and numerical, which can describe behaviour in (non) equilibrium ensembles, both classical and quantum, especially in the complex systems, where the standard approaches cannot be applied. We demonstrate the creation of nontrivial (meta) stable states (patterns), localized, chaotic, entangled or decoherent, from basic localized modes in various collective models arising from the quantum hierarchy of Wigner-von Neumann-Moyal-Lindblad equations, which are the result of "wignerization" procedure of classical BBGKY hierarchy. We present the explicit description of internal quantum dynamics by means of exact analytical/numerical computations.

When repelling particles are confined in a quasi-one-dimensional trap by a transverse potential, a configurational phase transition happens. All particles are aligned along the trap axis at large confinement, but below a critical transverse confinement they adopt a staggered row configuration (zigzag phase). This zigzag transition is a subcritical pitchfork bifurcation in extended systems and in systems with cyclic boundary conditions in the longitudinal direction. Among many evidences, phase coexistence is exhibited by localized nonlinear patterns made of a zigzag phase embedded in otherwise aligned particles. We give the normal form at the bifurcation and we show that these patterns can be described as solitary wave envelopes that we call bubbles. They are stable in a large temperature range and can diffuse as quasi-particles, with a diffusion coefficient that may be deduced from the normal form. The potential energy of a bubble is found to be lower than that of the homogeneous bifurcated phase, which explains their stability. We observe also metastable states, that are pairs of solitary wave envelopes which spontaneously evolve toward a stable single bubble. We evidence a strong effect of the discreteness of the underlying particles system and introduce the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive so that the bubbles come closer and eventually merge as a single bubble. In contrast, the bubbles interaction is found to be repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: it is attractive for NF-systems, repulsive for F-systems, and decreases exponentially with the bubbles distance.

Based on a class of chaotic system composed of hidden attractors, in which the equilibrium points are described by a circular function, complete synchronization between two identical systems, pattern formation and synchronization of network is investigated, respectively. A statistical factor of synchronization is defined and calculated by using the mean field theory, the dependence of synchronization on bifurcation parameters discussed in numerical way. By setting a chain network, which local kinetic is described by hidden attractors, synchronization approach is investigated. It is found that the synchronization and pattern formation are dependent on the coupling intensity and also the selection of coupling variables. In the end, open problems are proposed for readers’ extensive guidance and investigation.

Pattern estimation and selection in media can give important clues to understand the collective response to external stimulus by detecting the observable variables. Both reaction–diffusion systems (RDs) and neuronal networks can be treated as multi-agent systems from molecular level, intrinsic cooperation, competition. An external stimulus or attack can cause collapse of spatial order and distribution, while appropriate noise can enhance the consensus in the spatiotemporal systems. Pattern formation and synchronization stability can bridge isolated oscillators and the network by coupling these nodes with appropriate connection types. As a result, the dynamical behaviors can be detected and discussed by developing different spatial patterns and realizing network synchronization. Indeed, the collective response of network and multi-agent system depends on the local kinetics of nodes and cells. It is better to know the standard bifurcation analysis and stability control schemes before dealing with network problems. In this review, dynamics discussion and synchronization control on low-dimensional systems, pattern formation and synchronization stability on network, wave stability in RDs and neuronal network are summarized. Finally, possible guidance is presented when some physical effects such as polarization field and electromagnetic induction are considered.

In this paper, we explore the emergence of patterns in a fractional cross-diffusion model with Beddington–DeAngelis type functional response. First, we explore the stability of the equilibrium points with or without fractional cross-diffusion. Instability of equilibria can be induced by cross-diffusion. We perform the linear stability analysis to obtain the constraints for the Turing instability. It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns. For the dynamics of pattern, the weakly nonlinear multi-scaling analysis has been performed to obtain the amplitude equations. Finally, we ensure the existence of Turing patterns such as squares, spots and stripes by using the stability analysis of the amplitude equations. Moreover, with the assistance of numerical simulations, we verify the theoretical results.

The epidemic often spreads along social networks and shows the effect of memorability on the outbreak. But the dynamic mechanism remains to be illustrated in the fractional-order epidemic system with a network. In this paper, Turing instability induced by the network and the memorability of the epidemic are investigated in a fractional-order epidemic model. A method is proposed to analyze the stability of the fractional-order model with a network through the Laplace transform. Meanwhile, the conditions of Turing instability and Hopf bifurcation are obtained to discuss the role of fractional order in the pattern selection and the Hopf bifurcation point. These results prove the fractional-order epidemic model may describe dynamical behavior more accurately than the integer epidemic model, which provides the bridge between Turing instability and the outbreak of infectious diseases. Also, the early warning area is discussed, which can be treated as a controlled area to avoid the spread of infectious diseases. Finally, the numerical simulation of the fractional-order system verifies the academic results is qualitatively consistent with the instances of COVID-19.

We review investigations on defects in systems described by real scalar fields in (D, 1) space-time dimensions. We first work in one spatial dimension, with models described by one and two real scalar fields, and in higher dimensions. We show that when the potential assumes specific form, there are models which support stable global defects for D arbitrary. We also show how to find first-order differential equations that solve the equations of motion, and how to solve models in D dimensions via soluble problems in D = 1. We illustrate the procedure examining specific models and showing how they may be used in applications in different contexts in condensed matter physics, and in other areas.

Nanomanipulation as a new emerging area enables precise manipulation, interaction and control at the nanoscale. Currently, the modeling schemes are based on continuum mechanics approaches. A main consideration in the nanomanipulation process is the fact that surface attraction forces are greater than gravitational forces at the nanoscale. In other words, surface area properties dominate volume properties. Especially at the nanoscale (i.e. the manipulation of fine nanoparticles with size of about 5 nm) the physical phenomena have not been completely understood. Along this line of reasoning, the aim of this paper is to conduct an atomistic investigation of physical interaction analysis of particle–substrate system for manipulation and positioning purposes. In the present paper, 2D molecular dynamics have been conducted to simulate metallic nanoparticle behavior during the pushing process. Dependency of the aforementioned behavior on size, matter, temperature, etc. is investigated.