Stationary Patterns Research Articles

Spatial dynamics of a vegetation model with uptake-diffusion feedback in an arid environment.

Vegetation patterns with a variety of structures is amazing phenomena in arid or semi-arid areas, which can identify the evolution law of vegetation and are typical signals of ecosystem functions. Many achievements have been made in this respect, yet the mechanisms of uptake-diffusion feedback on the pattern structures of vegetation is not fully understood. To well reveal the influences of parameters perturbation on the pattern formation of vegetation, we give a comprehensive analysis on a vegetation-water model in the forms of reaction-diffusion equation which is posed by Zelnik et al.(Proc Natl Acad Sci 112:12,327-12,331, 2015). We obtain the exact parameters range for stationary patterns and show the dynamical behaviors near the bifurcation point based on nonlinear analysis. It is found that the model has the properties of spot, labyrinth and gap patterns. Moreover, water diffusion rate prohibits the growth of vegetation while shading parameter promotes the increase of vegetation biomass. Our results show that gradual transitions from uniform state to gap pattern can occur for suitable value of parameters which may induce the emergence of desertification.

Supercritical and subcritical rotating convection in a horizontally periodic box with no-slip walls at the top and bottom

The study of instabilities in the convection of rotating fluids is one of the classical topics of research. However, in spite of more than five decades of research, the instabilities and related transition scenarios near the onset of rotating convection of low Prandtl number fluids are not well understood. Here, we investigate the transition scenario in rotating Rayleigh–Bénard convection with no-slip boundary conditions by performing 3D direct numerical simulations (DNS) and low-dimensional modeling. The governing parameters, namely, the Taylor number (Ta), Rayleigh number (Ra), and Prandtl number (Pr), are varied in the ranges 0<Ta≤8×103, 0<Ra<1×104, and 0<Pr≤0.35, where convection appears as a stationary cellular pattern. In DNS, for Pr<0.31, the supercritical or subcritical onset of convection appears, according as Ta>Tac(Pr) or Ta<Tac(Pr), where Tac(Pr) is a Pr dependent threshold of Ta. On the other hand, only supercritical onset of convection is observed for Pr≥0.31. At the subcritical onset, both finite amplitude stationary and time dependent solutions are manifested. The origin of these solutions are explained using a low dimensional model. DNS show that as Ra is increased beyond the onset of convection, the system becomes time dependent and depending on Pr, standing and traveling wave solutions are observed. For very small Pr (≤0.045), interestingly, finite amplitude time dependent solutions are manifested at the onset for higher Ta.

Self-Training Strategy Based on Finite Element Method for Adaptive Bioluminescence Tomography Reconstruction.

Bioluminescence tomography (BLT) is a promising pre-clinical imaging technique for a wide variety of biomedical applications, which can non-invasively reveal functional activities inside living animal bodies through the detection of visible or near-infrared light produced by bioluminescent reactions. Recently, reconstruction approaches based on deep learning have shown great potential in optical tomography modalities. However, these reports only generate data with stationary patterns of constant target number, shape, and size. The neural networks trained by these data sets are difficult to reconstruct the patterns outside the data sets. This will tremendously restrict the applications of deep learning in optical tomography reconstruction. To address this problem, a self-training strategy is proposed for BLT reconstruction in this paper. The proposed strategy can fast generate large-scale BLT data sets with random target numbers, shapes, and sizes through an algorithm named random seed growth algorithm and the neural network is automatically self-trained. In addition, the proposed strategy uses the neural network to build a map between photon densities on surface and inside the imaged object rather than an end-to-end neural network that directly infers the distribution of sources from the photon density on surface. The map of photon density is further converted into the distribution of sources through the multiplication with stiffness matrix. Simulation, phantom, and mouse studies are carried out. Results show the availability of the proposed self-training strategy.

It's about time: Linking dynamical systems with human neuroimaging to understand the brain.

Most human neuroscience research to date has focused on statistical approaches that describe stationary patterns of localized neural activity or blood flow. While these patterns are often interpreted in light of dynamic, information-processing concepts, the static, local, and inferential nature of the statistical approach makes it challenging to directly link neuroimaging results to plausible underlying neural mechanisms. Here, we argue that dynamical systems theory provides the crucial mechanistic framework for characterizing both the brain's time-varying quality and its partial stability in the face of perturbations, and hence, that this perspective can have a profound impact on the interpretation of human neuroimaging results and their relationship with behavior. After briefly reviewing some key terminology, we identify three key ways in which neuroimaging analyses can embrace a dynamical systems perspective: by shifting from a local to a more global perspective, by focusing on dynamics instead of static snapshots of neural activity, and by embracing modeling approaches that map neural dynamics using "forward" models. Through this approach, we envisage ample opportunities for neuroimaging researchers to enrich their understanding of the dynamic neural mechanisms that support a wide array of brain functions, both in health and in the setting of psychopathology.

Marangoni Patterns in a Non-Isothermal Liquid with Deformable Interface Covered by Insoluble Surfactant

Marangoni patterns are created by instabilities caused by thermocapillary and solutocapillary stresses on the deformable free surface of a thin liquid layer. In the present paper, we consider the influence of the insoluble surfactant on the selection and modulational instability of stationary Marangoni patterns near their onset threshold. The basic governing parameters of the problem are the Biot number characterizing the heat-transfer resistances of and at the surface, the Galileo number indicating the role of gravity via viscous forces, and the elasticity number specifying the influence of insoluble surfactant on the interfacial dynamics of the liquid. The paper includes a review of the previous results obtained in that problem as well as new ones.

Summer Midlatitude Stationary Wave Patterns Synchronize Northern Hemisphere Wildfire Occurrence

AbstractMidlatitude stationary waves are relatively persistent large‐scale longitudinal variations in atmospheric circulation. Although recent case studies have suggested a close connection between stationary waves and extreme weather events, little is known about the global‐scale linkage between stationary waves and wildfire activity, as well as the potential changes in this relationship in a warmer climate. Here, by analyzing the Community Earth System Model version 2 large ensemble, we show that a zonal wavenumber 5–6 stationary wave pattern tends to synchronize wildfire occurrences across the Northern Hemisphere midlatitudes. The alternation of upper‐troposphere ridges and troughs creates a hemispheric‐scale spatial pattern of alternating hot/dry and cold/wet conditions, which increases or decreases wildfire occurrence, respectively. More persistent high‐pressure conditions drastically increase wildfire probabilities. Even though the dynamics of these waves change little in response to anthropogenic global warming, the corresponding midlatitude wildfire variability is projected to intensify due to changes in climate background conditions.

Eckhaus instability of stationary patterns in hyperbolic reaction–diffusion models on large finite domains

We have theoretically investigated the phenomenon of Eckhaus instability of stationary patterns arising in hyperbolic reaction–diffusion models on large finite domains, in both supercritical and subcritical regime. Adopting multiple-scale weakly-nonlinear analysis, we have deduced the cubic and cubic–quintic real Ginzburg–Landau equations ruling the evolution of pattern amplitude close to criticality. Starting from these envelope equations, we have provided the explicit expressions of the most relevant dynamical features characterizing primary and secondary quantized branches of any order: stationary amplitude, existence and stability thresholds and linear growth rate. Particular emphasis is given on the subcritical regime, where cubic and cubic–quintic Ginzburg–Landau equations predict qualitatively different dynamical pictures. As an illustrative example, we have compared the above-mentioned analytical predictions to numerical simulations carried out on the hyperbolic modified Klausmeier model, a conceptual tool used to describe the generation of stationary vegetation stripes over flat arid environments. Our analysis has also allowed to elucidate the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips. In particular, we have inspected the functional dependence of time and location at which wavelength adjustment takes place as well as the possibility to control these quantities, independently of each other.

Nonlinear coupling in an asymmetric pendulum

We investigate the nonlinear behavior of a simple pendulum fixed to an elastic rod, which can only vibrate horizontally. When released without initial angular momentum, the plane of the pendulum rotates, and the bob traces a delicate stationary pattern. We explain this effect as amplitude modulation due to the nonlinear coupling between the two degrees of freedom. We construct a theoretical model and approximately solve the equations of motion analytically (using the method of multiple scales); we also solve these equations numerically. In the analytical solution, the modulation period depends not only on the dynamical parameters but also on the pendulum's initial release position, which is typical of nonlinear systems. Finally, we build a simple apparatus and conduct a quantitative experiment. The approximate analytical solutions exhibit the same trends as the numerical results and experimental data. After adding linear dissipation, the numerical and experimental results match fairly well. This system can serve as an instructive demonstration as well as a nonlinear dynamics research project for undergraduate students.

Turing and wave instabilities in hyperbolic reaction–diffusion systems: The role of second-order time derivatives and cross-diffusion terms on pattern formation

Hyperbolic reaction–diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction–diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction–diffusion systems which are distinct from the well studied Turing case.

Stationary and non-stationary pattern formation over fragmented habitat

Spatio-temporal pattern formation over the square and rectangular domain has received significant attention from researchers. A wide range of stationary and non-stationary patterns produced by two interacting populations is abundant in the literature. Fragmented habitats are widespread in reality due to the irregularity of the landscape. This work considers a prey-predator model capable of producing a wide range of stationary and time-varying patterns over a complex habitat. The complex habitat is assumed to have consisted of two rectangular patches connected through a corridor. Our main aim is to explain how the shape and size of the fragmented habitat regulate the spatio-temporal pattern formation at the initial time. The analytical conditions are derived to ensure the existence of a stationary pattern and illustrate the role of the most unstable eigenmodes in determining the number of patches for the stationary pattern. Exhaustive numerical simulations help to explain the effect of the spatial domain size and shape on the transient patterns and the duration of the transients.

Stability and metastability in a chemotaxis model

This work studies the stability and metastability of stationary patterns in a diffusion-chemotaxis model without cell proliferation. We first establish the interval of unstable wave modes of the homogeneous steady state, and show that the chemotactic flux is the key mechanism for pattern formation. Then, we treat the chemotaxis coefficient as a bifurcation parameter to obtain the asymptotic expressions of steady states. Based on this, we derive the sufficient conditions for the stability of one-step pattern, and prove the metastability of two or more step patterns. All the analytical results are demonstrated by numerical simulations.

Malleability of time through progress bars and throbbers

Compared to a stationary pattern, a moving pattern dilates the perception of time. However, when it comes to comparing only moving stimulus, the exact dilation effects are less clear. The time dilation may be attributed to either speed of motion, temporal and spatial frequency, stimulus complexity, or the number of changes in the stimulus pattern. In the present study, we used progress bars and throbbers for inducing impressions of fast and slow “apparent” motions while the speed of motion and distance covered was actually equivalent across all conditions. The results indicate that higher number of steps produced the impression of a faster progression leading to an underestimation of time, whereas a progression in large fewer steps, produced slower apparent progression, creating the illusion of dilated time. We suggest that the perception of time depends on the nature of the stimulus rather than the speed of motion or the distance covered by the stimulus.

Quantized point vortex equilibria in a one-way interaction model with a Liouville-type background vorticity on a curved torus

We construct point vortex equilibria with strengths quantized by multiples of 2π in a fixed background vorticity field on the surface of a curved torus. The background vorticity consists of two terms: first, a term exponentially related to the stream function and a second term arising from the curvature of the torus, which leads to a Liouville-type equation for the stream function. By using a stereographic projection of the torus onto an annulus in a complex plane, the Liouville-type equation admits a class of exact solutions, given in terms of a loxodromic function on the annulus. We show that appropriate choices of the loxodromic function in the solution lead to stationary vortex patterns with 4n̂ point vortices of identical strengths, n̂∈N. The quantized point vortices are stationary in the sense that they are equilibria of a “one-way interaction” model where the evolution of point vortices is subject to the continuous background vorticity, while the background vorticity distribution is not affected by the velocity field induced by the point vortices. By choosing loxodromic functions continuously dependent on a parameter and taking appropriate limits with respect to this parameter, we show that there are solutions with inhomogeneous point vortex strengths, in which the exponential part of the background vorticity disappears. The point vortices are always located at the innermost and outermost rings of the torus owing to the curvature effects. The topological features of the streamlines are found to change as the modulus of the torus changes.

Improved functional–weight approach to oscillatory patterns in excitable networks

Studies of sustained oscillations on complex networks with excitable node dynamics received much interest in recent years. Although an individual unit is non-oscillatory, they may organize to form various collective oscillatory patterns through networked connections. An excitable network usually possesses a number of oscillatory modes dominated by different Winfree loops and numerous spatiotemporal patterns organized by different propagation path distributions. The traditional approach of the so-called dominant phase-advanced drive method has been well applied to the study of stationary oscillation patterns on a network. In this paper, we develop the functional–weight approach that has been successfully used in studies of sustained oscillations in gene-regulated networks by an extension to the high-dimensional node dynamics. This approach can be well applied to the study of sustained oscillations in coupled excitable units. We tested this scheme for different networks, such as homogeneous random networks, small-world networks, and scale-free networks and found it can accurately dig out the oscillation source and the propagation path. The present approach is believed to have the potential in studies competitive non-stationary dynamics.

Stationary pattern and bifurcation of a Leslie–Gower predator–prey model with prey-taxis

In this paper, we consider a Leslie–Gower predator–prey model with prey-taxis subject to the homogeneous Neumann boundary condition. First, the stability of positive constant solution for corresponding ODE and PDE models is studied, respectively. It turns out that small prey-taxis can cause Turing instability and produce spatio-temporal patterns; large prey-taxis can stabilize the instability caused by diffusion. Then, in spatial one dimension, we use the center manifold theory to analyze the direction of Hopf bifurcation and stability of the bifurcation periodic solutions. By regarding prey-tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of non-constant solutions bifurcating from the positive equilibrium and find the stability criterion of such bifurcation solutions. Finally, some numerical simulations are shown to verify and expand our theoretical findings.

Perceptual scale for transparency: Common fate overrides geometrical and color cues.

Objects that pass light through are considered transparent, and we generally expect that the light coming out will match the color of the object. However, when the object is placed on a colored surface, the light coming back to our eyes becomes a composite of surface, illumination, and transparency properties. Despite that, we can often perceive separate overlaid and overlaying layers differing in colors. How neurons separate the information to extract the transparent layer remains unknown, but the physical characteristics of transparent filters generate geometrical and color features in retinal images, which could provide cues for separating layers. We estimated the relative importance of such cues in a perceptual scale for transparency, using stimuli in which X- or T-junctions, different relative motions, and consistent or inconsistent colors cooperated or competed in forced-preference psychophysics experiments. Maximum-likelihood Thurstone scaling revealed that motion increased transparency for X-junctions, but decreased transparency for T-junctions by creating the percept of an opaque patch. However, if the motion of a filter uncovered a dynamically changing but stationary pattern, sharing a common fate with the surround but forming T-junctions, the probability of seeing transparency was almost as high as for moving X-junctions, despite the stimulus being physically improbable. In addition, geometric cues overrode color inconsistency to a great degree. Finally, a linear model of transparency perception as a function of relative motions between filter, overlay, and surround layers, contour continuation, and color consistency, quantified a hierarchy of latent influences on when the filter is seen as a separate transparent layer.

Nonlocal interactions between vegetation induce spatial patterning

Vegetation pattern provides useful signals for vegetation protection and can be identified as an early warning of desertification. In some arid or semi-arid regions, vegetation absorbs water through nonlocal interaction of roots. In this study, we present a vegetation model with nonlocal interaction which is characterized by an integral term with a kernel function. Mathematical analysis provides the conditions for the generation of stationary pattern. Numerical simulations exhibit different spatial distributions of vegetation. Densities of vegetation and water show an inverse relationship at same spatial locations due to water transport mechanism. The results reveal that the interaction intensity and the shape of the kernel function can cause the transition of vegetation pattern. Specifically, the vegetation biomass increases as the interaction intensity decreases or as the nonlocal interaction distance increases. We demonstrate that the nonlocal interactions between roots of vegetation is a key mechanism for the formation of vegetation pattern, which provides a theoretical basis for the preservation and restoration of vegetation.

Hunting cooperation among slowly diffusing specialist predators can induce stationary Turing patterns

In this work we consider the spatio-temporal extension of a class of prey–predator models with hunting cooperation in the specialist predator population. Three different types of parametrization of functional response are used and the corresponding pattern formation scenario has been studied analytically with supportive numerical simulations. The main outcome of this work is that lower diffusivity of predator than the prey can lead to generation of stationary Turing patterns like spots, stripes and mixture of the two. Other non-Turing patterns like spirals and spatio-temporal chaos are also generated for parameter values outside the Turing domain and vividly discussed. Dispersal of the specialist predator lower than the diffusivity of the prey supports the formation of stationary patches as the hunting cooperation among the predators can support the success in predation.

Stationary localised patterns without Turing instability

Since the pioneering work of Turing, it has been known that diffusion can destablise a homogeneous solution that is stable in the underlying model in the absence of diffusion. The destabilisation of the homogeneous solutions leads to the generation of patterns. In recent years, techniques have been developed to analyse so‐called localised spatial structures. These are solutions in which the spatial structure occurs in a localised region. Unlike Turing patterns, they do not spread out across the whole domain. We investigate the existence of localised structures that occur in two predator‐prey models. The functionalities chosen have been widely used in the literature. The existence of localised spatial structures has not investigated previously. Indeed, it is easy to show that these models cannot exhibit the Turing instability. This has perhaps led earlier researchers to conclude that interesting spatial solutions can therefore not occur for these models. The novelty of our paper is that we show the existence of stationary localised patterns in systems which do not undergo the Turing instability. The mathematical tools used are a combination of Linear and weakly nonlinear analysis with supporting numerical methods. By combining these methods, we are able to identify conditions for a wide range of increasing exotic behaviour. This includes the Belyckov‐Devaney transition, a codimension two spatial instability point and the formation of localised patterns. The combination of spectral computations and numerical simulations reveals the crucial role played by the Hopf bifurcation in mediating the stability of localised spatial solutions. Finally, numerical solutions in two spatial dimensions confirms the onset of intricate spatio‐temporal patterns within the parameter regions identified within one spatial dimension.

Introducing a new tool to derive animal behaviour from GPS data without ancillary data: The red-footed falcon in Italy as a case study

Animal monitoring is a key stage in the process of understanding individual behaviour and social dynamics that characterize populations and species. Recent technological advances allow to collect large data sets on animal movement. Using such data to compute activity patterns is very challenging without expensive field surveys, video analyses and/or accelerometer sensors.We introduce here a new tool (UABE: Unsupervised Animal Behaviour Examiner) that makes use of a set of methodologies (if-then-else rules, sensitivity analyses, Markov chains) to interpret GPS data in behavioural terms under worst case conditions, i.e. in absence of ancillary data. The UABE is halfway between unsupervised inductive (clustering or segmentation) and deductive (expert-based) methods. As a case study, we applied the UABE to the investigation of the red-footed falcons' activity at nest at the two largest breeding colonies in Italy during the chick-rearing period.By using accurate GPS data-loggers, we tracked four adults in June–July 2019 and 2020, and collected 5840 GPS points. Activity at nest occupied one fourth of the red-footed falcons' day, with substantial differences between months and between daytime and night-time. On an hourly basis, clear temporal patterns of activity at nest emerged. Through the combined use of frequency and continuity of the studied behaviour, the UABE also allowed to raise plausible hypotheses for the reasons behind birds' activity at nest at specific times of day.The UABE supports the creation of unsupervised ethograms of both stationary (e.g., standing at nest or roost, hovering/perching) and nonstationary (e.g., flight, walking) activity patterns. It can be applied to any species regardless of the tracking instrument used.