Stationary Patterns Research Articles

Process Evaluation of Subseasonal North Atlantic Oscillation Prediction in the ECMWF Ensemble Forecast System

AbstractThis study evaluates the prediction skill of the North Atlantic Oscillation (NAO) pattern and its associated energy budget as simulated by the European Center for Medium‐Range Weather Forecasts ensemble forecast system. By classifying NAO events into high‐ and low‐skill cases, we analyzed the stationarity of NAO patterns and the role of baroclinic energy conversion in NAO prediction. In both positive and negative NAO phases, high‐skill cases exhibited more stationary NAO patterns than low‐skill cases. The analysis of processes indicates that high‐skill NAO cases are due to stronger baroclinic maintenance of NAO, with its initial position at the climatological thermal trough, whereas low‐skill NAO cases result from forecast biases in wave propagation from the North Pacific. Specifically, biases in baroclinic energy conversion in the meridional direction from week 2 lead to weak advection of the eddy available potential energy (EAPE), resulting in lower prediction skill.

Self-sustained chaotic movement of electrothermal responsive liquid crystal elastomer pendulum

Self-sustained movement has the ability to absorb energy from the external environment to maintain its own movement. In the present work, a self-sustained chaotic simple pendulum system featuring an electrothermal responsive liquid crystal elastomers fiber and a mass ball is proposed and examined. The power-on state is achieved in the system through the infusion of liquid metal into this fiber. Which enables the fiber to shrink, feeding energy into the system to compensate for the energy dissipation resulting from damping, thus allowing the system to maintain its self-sustained movement. On the basis of the existing dynamic liquid crystal elastomer model, the dynamic governing equations are established, and the movement behavior of the system under electrothermal stimulation is theoretically explored. Numerical findings suggest that the simple pendulum system presents three different movement patterns, namely stationary pattern, self-sustained oscillatory pattern, and self-sustained chaotic pattern. Moreover, five system parameters of elastic coefficient, viscosity coefficient, gravitational acceleration, shrinkage coefficient and electrothermal strength are examined. The present work can contribute to a better understanding of self-sustained chaotic systems driven by electrothermal responsive materials and offer guidance for further exploration of applications such as chaos machine, and mind-brain chaos analysis.

Numerical investigation of the refractive properties of near-horizontal rocky shore platforms and their effects on harmonic and stationary wave patterns

Near-horizontal shore platforms display highly irregular plan shapes, but little is known about the way in which these irregularities influence the significant wave height (Hŝ) on the platforms and the frequency components of the nearshore wavefield. As ocean waves share akin refractive properties to light rays, it can be assumed that, similarly to optical lenses, shore platforms can separate waves according to their frequency depending on their geometry. Thus, we use a non-linear Boussinesq wave model to conduct harmonic and bispectral mode decomposition analyses to study the control of concave and convex platform edges over wind waves (WW: 0.125–0.33 Hz), swell waves (SW: 0.05–0.125 Hz) and infragravity (IG: 0.008–0.05 Hz) waves frequencies. For breaking and non-breaking waves, increasing the platform edge concavity intensified wave divergence and subsequent attenuation of SW and IG across the outer platforms, reducing Hŝ by up to 25 %. Increasing the platform edge convexity intensified focusing and amplification of SW and WW over the outer platforms, increasing Hŝ by up to 18 % and 55 % for breaking and non-breaking waves. In the presence of breaking, IG amplification was affected by wave divergence across the inner platform, a condition determined by a critical convex curvature threshold (K=1.8) balancing wave focusing from refraction and defocusing from breaking. We find that convex curvature can determine the relative dominance of WW, SW and IG across platforms. Alongshore, coherent wave interactions governed IG stationary patterns defined by a node near the platform centreline and two antinodes on either side of concave edges. A node was generated at the platform centreline, and two antinodes were observed on either side of the convex edges for K>1.8, with the opposite pattern observed for K<1.8. Such patterns likely result in alongshore variations in wave-generated currents and erosion shaping rock coasts.

Stationary correlation pattern in highly non-stationary MEG recordings of healthy subjects and its relation to former EEG studies.

In this study, we analyze magnetoencephalographic (MEG) recordings from 48 clinically healthy subjects obtained from the Human Connectome Project (HCP) while they performed a working memory task and a motor task. Our results reveal a well-developed, stable interrelation pattern that spans the entire scalp and is nearly universal, being almost task- and subject-independent. Additionally, we demonstrate that this pattern closely resembles a stationary correlation pattern (SCP) observed in EEG signals under various physiological and pathological conditions (the distribution of Pearson correlations are centered at about 0.75). Furthermore, we identify the most effective EEG reference for studying the brain's functional network derived from lag-zero cross-correlations. We contextualize these findings within the theory of complex dynamical systems operating near a critical point of a phase transition.

Spatiotemporal Dynamics of an Intraguild Predation Model with Intraspecies Competition

Different species in the environment compete with one another for shared resources and suitable habitats. As a result, it becomes crucial to identify the essential components involved in preserving biodiversity. This paper describes the emergence of a wide range of temporal and spatiotemporal dynamics of an intraguild predation (IGP) model with intraspecies competition and Holling type-II response function between the IG-prey and IG-predator. To explore the dynamics of the temporal model, we study the local stability of all the biologically feasible steady states, global stability of the interior steady state, and Hopf bifurcation. Performing partial rank correlation coefficient (PRCC) sensitivity analysis, we picked more sensitive parameters and plotted the stability regions in two-parameter spaces. The dynamics exhibited by the system can demonstrate some important hallmarks noticed in the environments, such as the existence and nonexistence of oscillatory behavior in the IGP model. The analytical formulations for the stability and bifurcation of the coexistence of the steady-state are not easy to obtain, but we have verified numerically that the limit cycle bifurcates from the coexistence of the steady-state and there eventually emerges a chaotic behavior through period-doubling bifurcations. By plotting the largest Lyapunov exponent (LLE), we have verified that the temporal system experiences chaotic behavior. It is demonstrated that in the case of a diffusive system, diffusion may cause the coexistence of the steady state, which is otherwise stable, to become unstable. Criteria for the occurrence of Turing instability connected to the IGP model are derived. Our numerical illustrations demonstrate that diffusion-driven instability develops different stationary patterns based on the different initial conditions and diffusion parameters. The spatiotemporal patterns are observed in the Turing, Hopf–Turing, and Hopf regions.

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.

Variations in non-local interaction range lead to emergent chase-and-run in heterogeneous populations.

In a chase-and-run dynamic, the interaction between two individuals is such that one moves towards the other (the chaser), while the other moves away (the runner). Examples can be found in both interacting cells and animals. Here, we investigate the behaviours that can emerge at a population level, for a heterogeneous group that contains subpopulations of chasers and runners. We show that a wide variety of patterns can form, from stationary patterns to oscillatory and population-level chase-and-run, where the latter describes a synchronized collective movement of the two populations. We investigate the conditions under which different behaviours arise, specifically focusing on the interaction ranges: the distances over which cells or organisms can sense one another's presence. We find that when the interaction range of the chaser is sufficiently larger than that of the runner-or when the interaction range of the chase is sufficiently larger than that of the run-population-level chase-and-run emerges in a robust manner. We discuss the results in the context of phenomena observed in cellular and ecological systems, with particular attention to the dynamics observed experimentally within populations of neural crest and placode cells.

Interaction of inertia and magnetic force in the liquid metal flow past a magnetic obstacle

In this paper, we present a numerical study of the three-dimensional behavior of a liquid metal flow in an insulating rectangular duct of narrow cross section past a localized magnetic field (i.e., a magnetic obstacle) produced by two parallel square magnets arranged externally on the walls of the duct. A series of simulations are conducted focused mainly on describing the interplay between inertial and magnetic forces in a wide range of interaction parameters (1.8&amp;lt;N&amp;lt;48) by varying the Reynolds number while the Hartmann number is kept fixed (Ha = 75). The analyzed configuration coincides with that studied experimentally by Domínguez et al. [“Experimental and theoretical study of the dynamics of wakes generated by magnetic obstacles,” Magnetohydrodynamics 51(2), 215–224 (2015)] and, as a first step, experimental data from local variables (streamwise velocity component) and global parameters (oscillation frequency and kinetic energy of the wake) are consistently replicated by the numerical model. Furthermore, to complement the flow phenomenology, the transition to different flow structures as the interaction parameter varies is explored. It is found that when the magnetic forces predominate over inertia, stationary vortex patterns with two, four, and six vortices appear while, unlike the hydrodynamic flow past a bluff body, the increase in inertial effects leads to a reduction in the number of vortices and eventually to their disappearance, reaching a state in which the magnetic obstacle becomes imperceptible to the flow. The existence of a critical value of the interaction parameter that maximizes the kinetic energy of the wake is confirmed numerically and corroborated from the experimental data.

Theory of Turing pattern formation and its experimental realization in the CIMA reaction system in the presence of materials lowering the diffusivity of activators

In 1952, Alan Turing accomplished a pioneering theoretical study to show that the coupling of nonlinear chemical reactions and diffusion leads to the instability of spatially homogeneous states. The activator and inhibitor are synthesized as intermediates of the reaction system in the Turing model. Turing found that spatially periodic stationary concentration patterns are spontaneously generated when the diffusion coefficient of the activator is lower than that of the inhibitor. The first experimental realization of the Turing pattern was achieved in 1990 in a chlorite–iodide–malonic acid (CIMA) reaction system. Iodide and chlorite anions act as the activator and inhibitor of this reaction system, respectively. Although there is no significant difference in the diffusion coefficient of iodide and chlorite anions, the Turing pattern was generated because starch was added to the gel reactor to enhance the color tone. This formed a complex with iodide to inhibit its diffusion to satisfy the condition for the Turing instability. Several examples were found after this finding. We focused on the high affinity of quaternary alkyl ammonium cations to iodide. The CIMA reaction was performed in an open gel reactor by adding a quaternary alkyl ammonium cationic surfactant. In addition, the polymer gel consists of the quaternary alkyl ammonium group as the side chain was utilized for the open gel reactor. The micelles of the surfactants and the polymer gels trapped iodide in their vicinity as a counter anion to lower the effective diffusivity to satisfy the condition for the Turing instability.

Classifying ambulation patterns in institutional settings

Observational studies of older adults show pacing, lapping, and stationary ambulation patterns can be associated with an increased risks for falls or an early sign of an acute or other health event in long-term care. The aim of this study is to use classical machine learning algorithms to automate the process of recognizing these patterns with the goal of assisting health care staff in monitoring the health and well-being of their residents. This study utilized data from six residents whose movements were tracked with a real-time locating system while performing everyday activities of daily living for up to 1.9 years. No residents exhibited lapping patterns over the course of the study. Machine learning statistical techniques recognized stationary and pacing with accuracy≥70%, with indirect and direct patterns having an accuracy of around 50% due to environmental constraints. Study findings suggest automated methods may be used with real-time locating data to recognize ambulation patterns that have been associated with poor health in this population. Study findings may be utilized by health care staff to tailor resident care plans and develop timely interventions that may affect falls and provide for the earlier recognition of acute events in this population.

Bifurcation Analysis and Spatiotemporal Dynamics in a Diffusive Predator–Prey System Incorporating a Holling Type II Functional Response

This study aims to investigate a diffusive predator–prey system incorporating additional food for predators, prey refuge, fear effect, and its carry-over effects. For the temporal model, the well-posedness and persistence of the system have been discussed. We investigated the existence and the stability behavior of the various equilibria. Furthermore, we explored the bifurcations of codimension-1 including transcritical, saddle-node, and Hopf, concerning the crucial parameters. The system also presents codimension-2 bifurcations such as Bogdanov–Takens and cusp bifurcation along with the global homoclinic bifurcation. We observed the bubbling phenomena, which illustrate the fluctuations in the amplitudes of the periodic oscillations. For the spatiotemporal system, we established the non-negativity and boundedness of the solutions. We derived the conditions for the diffusion-driven instabilities in a confined region with Neumann boundary conditions. Extensive numerical simulations have been conducted to depict the various stationary patterns in Turing space. It is observed that incorporating cross-diffusion divides the bi-parametric plane into various sub-regions and dynamic patterns are analyzed in these different regions. The intricate spatiotemporal dynamics exhibited by prey–predator interactions are crucial for unraveling the intricacies within ecological systems.

Transverse instability of traveling wave created by longwave Marangoni convection in the liquid layer covered by insoluble surfactant

We consider the Marangoni convection in a liquid layer heated from below. The liquid interface is covered by insoluble surfactant that plays an active role in the pattern formation together with inhomogeneity of temperature along the interface and surface deformability. In the vicinity of the onset of Marangoni convection, besides different kinds of stationary patterns (hexagons, rolls, squares), the appearance of wave patterns is possible. We analyze the transverse instability of the single traveling wave (TW) using generalizations of the complex Ginzburg–Landau equation (CGLE).

Pattern selection and the route to turbulence in incompressible polar active fluids

Active fluids, such as suspensions of microswimmers, are well known to self-organize into complex spatio-temporal flow patterns. An intriguing example is mesoscale turbulence, a state of dynamic vortex structures exhibiting a characteristic length scale. Here, we employ a minimal model for the effective microswimmer velocity field to explore how the turbulent state develops from regular, stationary vortex patterns when the strength of activity resp. related parameters such as nonlinear advection or polar alignment strength—is increased. First, we demonstrate analytically that the system, without any spatial constraints, develops a stationary square vortex lattice in the absence of nonlinear advection. Subsequently, we perform an extended stability analysis of this nonuniform ‘ground state’ and uncover a linear instability, which follows from the mutual excitement and simultaneous growth of multiple perturbative modes. This extended analysis is based on linearization around an approximation of the analytical vortex lattice solution and allows us to calculate a critical advection or alignment strength, above which the square vortex lattice becomes unstable. Above these critical values, the vortex lattice develops into mesoscale turbulence in numerical simulations. Utilizing the numerical approach, we uncover an extended region of hysteresis where both patterns are possible depending on the initial condition. Here, we find that turbulence persists below the instability of the vortex lattice. We further determine the stability of square vortex patterns as a function of their wavenumber and represent the results analogous to the well-known Busse balloons known from classical pattern-forming systems such as Rayleigh–Bénard convection experiments and corresponding models such as the Swift–Hohenberg equation. Here, the region of stable periodic patterns shrinks and eventually disappears with increasing activity parameters. Our results show that the strength of activity plays a similar role for active turbulence as the Reynolds number does in driven flow exhibiting inertial turbulence.

Emergence of non-trivial solutions from trivial solutions in reaction–diffusion equations for pattern formation

Reaction–diffusion equations serve as fundamental tools in describing pattern formation in biology. In these models, nonuniform steady states often represent stationary spatial patterns. Notably, these steady states are not unique, and unveiling them mathematically presents challenges. In this paper, we introduce a framework based on bifurcation theory to address pattern formation problems, specifically examining whether nonuniform steady states can bifurcate from trivial ones. Furthermore, we employ linear stability analysis to investigate the stability of the trivial steady-state solutions. We apply the method to two classic reaction–diffusion models: the Schnakenberg model and the Gray–Scott model. For both models, our approach effectively reveals many nonuniform steady states and assesses the stability of the trivial solution. Numerical computations are also presented to validate the solution structures for these models.

A two-timescale model of plankton–oxygen dynamics predicts formation of oxygen minimum zones and global anoxia

Decline of the dissolved oxygen in the ocean is a growing concern, as it may eventually lead to global anoxia, an elevated mortality of marine fauna and even a mass extinction. Deoxygenation of the ocean often results in the formation of oxygen minimum zones (OMZ): large domains where the abundance of oxygen is much lower than that in the surrounding ocean environment. Factors and processes resulting in the OMZ formation remain controversial. We consider a conceptual model of coupled plankton–oxygen dynamics that, apart from the plankton growth and the oxygen production by phytoplankton, also accounts for the difference in the timescales for phyto- and zooplankton (making it a “slow-fast system”) and for the implicit effect of upper trophic levels resulting in density dependent (nonlinear) zooplankton mortality. The model is investigated using a combination of analytical techniques and numerical simulations. The slow-fast system is decomposed into its slow and fast subsystems. The critical manifold of the slow-fast system and its stability is then studied by analyzing the bifurcation structure of the fast subsystem. We obtain the canard cycles of the slow-fast system for a range of parameter values. However, the system does not allow for persistent relaxation oscillations; instead, the blowup of the canard cycle results in plankton extinction and oxygen depletion. For the spatially explicit model, the earlier works in this direction did not take into account the density dependent mortality rate of the zooplankton, and thus could exhibit Turing pattern. However, the inclusion of the density dependent mortality into the system can lead to stationary Turing patterns. The dynamics of the system is then studied near the Turing bifurcation threshold. We further consider the effect of the self-movement of the zooplankton along with the turbulent mixing. We show that an initial non-uniform perturbation can lead to the formation of an OMZ, which then grows in size and spreads over space. For a sufficiently large timescale separation, the spread of the OMZ can result in global anoxia.

Unraveling biochemical spatial patterns: Machine learning approaches to the inverse problem of stationary Turing patterns

The diffusion-driven Turing instability is a potential mechanism for spatial pattern formation in numerous biological and chemical systems. However, engineering these patterns and demonstrating that they are produced by this mechanism is challenging. To address this, we aim to solve the inverse problem in artificial and experimental Turing patterns. This task is challenging since patterns are often corrupted by noise and slight changes in initial conditions can lead to different patterns. We used both least squares to explore the problem and physics-informed neural networks to build a noise-robust method. We elucidate the functionality of our network in scenarios mimicking biological noise levels and showcase its application using an experimentally obtained chemical pattern. The findings reveal the significant promise of machine learning in steering the creation of synthetic patterns in bioengineering, thereby advancing our grasp of morphological intricacies within biological systems while acknowledging existing limitations.

Intersection control based on Fine-grained Phases for connected and automated vehicles

Traditional traffic signal control is one of the most widely used urban traffic control methods, due to its high efficiency, robustness, and simplicity. However, in the era of connected and automated vehicles (CAVs), it is prospective to enhance traffic control precision and achieve vehicle coordination in a more efficient manner. Focusing on an isolated intersection, this study proposes an innovative concept of Fine-grained Phase (FP) for CAV-enabled traffic control, which realizes more refined right-of-way allocation than traditional signal phases while preserving the simplicity and scalability. The development of FP jointly optimizes traffic control strategy and intersection layout design to fully utilize time-space resources, and multiple forms of FP could be generated in response to various traffic demands; meanwhile, vehicle coordination is achieved in an offline–online manner to balance solution quality and efficiency. Under the FP-based control, each lane is assigned with periodical entry time points using FP design results, then vehicles are supposed to catch up with the appropriate time points to fulfill their trips. Theoretical analyses are conducted under stationary vehicle arrival patterns, where the average delay and the intersection capacity associated with different forms of FP can be analytically acquired. For the implementation of FP-based control, we construct a general offline–online framework, consisting of an offline model that optimizes the geometry of the conflict zone at intersections and the control strategy of vehicles simultaneously, which is formulated as a mixed-integer linear programming problem, and an online FP adoption mechanism that can adapt to varying demand patterns. Under the framework, limited online computation is required, facilitating its practical real-time implementation. Simulation studies under a variety of demand settings show that FP-based control is advantageous over benchmark methods in reducing vehicle delay and also improving intersection capacity in most cases; even for cases with extremely imbalanced demands, FP-based control can admit comparable throughput, while the average delay is much smaller. In addition, results of computation time reveal that FP-based control could realize real-time implementation with negligible computation burden.

Stability and bifurcation diagram for a shadow Gierer–Meinhardt system in one spatial dimension

We are concerned with a Neumann problem of a shadow system of the Gierer–Meinhardt model in an interval I=(0,1) . A stationary problem is studied, and we consider the diffusion coefficient ɛ > 0 as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each n⩾1 , two branches of n-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small τ > 0, and all n-mode solutions, n⩾2 , are unstable for all τ > 0, where τ > 0 is a time constant. The system is known for having stationary spiky patterns with large amplitude for small ɛ > 0. Then, asymptotic expansions of maximum and minimum values of a stationary solution as ɛ → 0 are also obtained.

Spatiotemporal and Trade-Off Dynamics in Prey–Predator Model with Domed Functional Response and Fear Effect

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion

In this study, we have investigated the temporal and spatio-temporal behavior of a prey–predator model with weak Allee effect in prey and the quality of being cannibalistic in a specialist predator. The parameters responsible for the Allee effect and cannibalism impact both the existence and stability of coexistence steady states of the temporal system. The temporal system exhibits various kinds of local bifurcations such as saddle–node, Hopf, Generalized Hopf (Bautin), Bogdanov–Takens, and global bifurcation like homoclinic, saddle–node bifurcation of limit cycles. For the model with self-diffusion, we establish the non-negativity and prior bounds of the solution. Subsequently, we derive the theoretical conditions in which self-diffusion leads to the destabilization of the interior equilibrium. Additionally, we explore the conditions under which cross-diffusion induces the Turing-instability where self-diffusion fails to do so. Further, we present different kinds of stationary and dynamic patterns on varying the values of diffusion coefficients to depict the spatio-temporal model’s rich dynamics. It has been found that the addition of self and cross-diffusion in a prey–predator model with the Allee effect in prey and cannibalistic predator play essential roles in comprehending the pattern formation of a distributed population model. It is expected that the comprehensive mathematical analysis and extensive numerical simulations used in investigating the global dynamics of the proposed model can facilitate researchers in studying the temporal and spatial aspects of prey–predator models in more significant detail.