Emergence Of Spatiotemporal Patterns Research Articles

On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction–Diffusion Systems

Nonlinear coupled reaction–diffusion (NCRD) systems have played a crucial role in the emergence of spatiotemporal patterns across various scientific and engineering domains. The NCRD systems considered in this study encompass various models, such as linear, Gray–Scott, Brusselator, isothermal chemical, and Schnakenberg, with the aim of capturing the spatiotemporal patterns they generate. These models cover a diverse range of intricate spatiotemporal patterns found in nature, including spots, spot replication, stripes, hexagons, and more. A mixed-type modal discontinuous Galerkin approach is employed for solving one- and two-dimensional NCRD systems. This approach introduces a mathematical formulation to handle the occurrence of second-order derivatives in diffusion terms. For spatial discretization, hierarchical modal basis functions premised on orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge–Kutta algorithm. The spatiotemporal patterns generated with the present approach are comparable to those found in the literature. This approach can readily be expanded to handle large multi-dimensional problems that appear as model equations in developed biological and chemical applications.

Nonlinear Dynamics in Alloy Electrodissolution: Modeling and Simulations

The formation of spatiotemporal patterns on electrochemical systems during the electrodissolution of metals depends on the nature of the metal constituent of the electrodes and can present regions with different behaviors, such as bistability and oscillations, for a set of physical-chemical parameters under potentiostatic control. The understanding of the influence of the parameters that govern the dynamics of the electrodissolution system in the emergence of spatiotemporal patterns has already been well studied taking into account its geometry and arrangement of the reference electrode, usually simplifying or neglecting the kinetics of reaction in the counter-electrode. This study aims to expand the complexity of the problem by considering the nature of the metal, changing the composition of the electrodes using metal alloys. The modelling of metal electrodissolution was based on previous works in the literature, in which the potential and the hydronium activity are considered as dependent and representative variables of the system. In this electrochemical system, the following geometries are adopted for the system: the electrochemical cell is cylindrical and the working electrode and counter-electrode are one on each side of this cylinder, having a ring or disc shape and the reference electrode is considered as a point belonging to the system.The activation-passivation transition that occurs in the working electrode is described by a smooth function Heaviside dependent on the Flade potential, as a consequence on the hydronium activity, and double-layer potential. Therefore, the total current density is the sum of two terms: one faradic current density, that is represented by this transition and the other assuming the existence of a capacitive current. In the mass transport of the hydron ions near the electrode we consider a linear diffusional term and a migratory one. The kinetics of the counter-electrode potential has a parameter that depends on the nature of the metal or alloy, from which the electrodes are made, and provides a novel approach to this study.The model takes advantage of the given geometric arrangement and its symmetries to reduce the spatial dimensionality by placing the reference electrode on the cylinder wall. In the case of ring electrodes geometry, it can be imposed that the radial derivative is null on this surface, reducing the evaluation of three spatial variables for cylindrical system: radius (r), angle (θ) and height (z), respectively, to s and z, where s is the arc of circumference. The simulations start with a working electrode entirely passive, and a section of the working electrode is disturbed, calculating the potentials of the system by numerical integration for each time step. A rectangular grid of points was used, and the time integration was done using a Runge-Kutta fourth order method. The disturbance in the initial condition is done considering a small section as activated. This disturbance is reproduced experimentally by scratching away a small portion of a passivating oxide layer. Several assumptions were made regarding the different metal potential in the counter-eletrode kinetics, comparing pure metal and these different ratio of metals in alloys, looking for three different responses to the temporal evolution of the potentials in the system due to these initial conditions: a totally active state, a totally passive state, and a partial active state.In future works, it is intended to validate the model using experimental data during the electrodissolution of alloy electrodes, so that can be applied as a guide in the search for desired spatiotemporal patterns. Figure 1

Noise-driven cell differentiation and the emergence of spatiotemporal patterns.

The emergence of phenotypic diversity in a population of cells and their arrangement in space and time is one of the most fascinating features of living systems. In fact, understanding multicellularity is unthinkable without explaining the proximate and the ultimate causes of cell differentiation in time and space. Simpler forms of cell differentiation can be found in unicellular organisms, such as bacterial biofilm, where reversible cell differentiation results in phenotypically diverse populations. In this manuscript, we attempt to start with the simple case of reversible nongenetic phenotypic to construct a model of differentiation and pattern formation. Our model, which we refer to as noise-driven differentiation (NDD) model, is an attempt to consider the prevalence of noise in biological systems, alongside what is known about genetic switches and signaling, to create a simple model which generates spatiotemporal patterns from bottom-up. Our simulations indicate that the presence of noise in cells can lead to reversible differentiation and the addition of signaling can create spatiotemporal pattern.

Detection and computation of high codimension bifurcations in diffuse predator–prey systems

In this paper, we consider reaction–diffusion model with modified Leslie–Gower and Holling-type II functional response and by varying simultaneously three model parameters, our model exhibits different types of complex dynamics. This allowed us to delimit several bifurcation surfaces with higher codimension corresponding to: Turing, Turing-Transcritical, Turing–Bogdanov–Taken, Turing–Hopf–Andronov, Turing-Saddle–node. Moreover by varying at least three bifurcation parameters, we show that small variations in the ratio of the diffusion coefficients can significantly alter bifurcation structure. Finally, the paper ends with the emergence of spatio-temporal patterns via numerical simulations. These simultaneous parameter variations leaded to spatio-temporal local and global bifurcations which are always catastrophic. Our results give new insights about how simultaneous changes in environmental and life history parameters drive different distribution dynamics of predator–prey populations. This only underscores the importance of including global bifurcations in the analysis of food chain models. Such results can be used to improve ecological decision-making on species conservation.

Size-dependent diffusion promotes the emergence of spatiotemporal patterns.

Spatiotemporal patterns, indicating the spatiotemporal variability of individual abundance, are a pronounced scenario in ecological interactions. Most of the existing models for spatiotemporal patterns treat species as homogeneous groups of individuals with average characteristics by ignoring intraspecific physiological variations at the individual level. Here we explore the impacts of size variation within species resulting from individual ontogeny, on the emergence of spatiotemporal patterns in a fully size-structured population model. We found that size dependency of animal's diffusivity greatly promotes the formation of spatiotemporal patterns, by creating regular spatiotemporal patterns out of temporal chaos. We also found that size-dependent diffusion can substitute large-amplitude base harmonics with spatiotemporal patterns with lower amplitude oscillations but with enriched harmonics. Finally, we found that the single-generation cycle is more likely to drive spatiotemporal patterns compared to predator-prey cycles, meaning that the mechanism of Hopf bifurcation might be more common than hitherto appreciated since the former cycle is more widespread than the latter in case of interacting populations. Due to the ubiquity of individual ontogeny in natural ecosystems we conclude that diffusion variability within populations is a significant driving force for the emergence of spatiotemporal patterns. Our results offer a perspective on self-organized phenomena, and pave a way to understand such phenomena in systems organized as complex ecological networks.

Spatiotemporal patterns provoked by environmental variability in a predator–prey model

The emergence of spatiotemporal patterns in the distribution of species is one of the most striking phenomena in ecology and nonlinear science. Since it is known that spatial inhomogeneities can significantly affect the dynamics of ecological populations, in the present paper we investigate the impact of environmental variability on the formation of patterns in a spatially extended predator–prey model. In particular, we utilize a predator–prey system with a Holling III functional response and introduce random spatial variations of the kinetic parameter signifying the intrinsic growth rate of the prey, reflecting the impact of a heterogeneous environment. Our results reveal that in the proximity of the Hopf bifurcation environmental variability is able to provoke pattern formation, whereby the coherence of the patterns exhibits a resonance-like dependence on the variability strength. Furthermore, we show that the phenomenon can only be observed if the spatial heterogeneities exhibit large enough regions with high growth rates of the prey. Our findings thus indicate that variability could be an essential pattern formation mechanism of the populations.

Emergence of spatio-temporal patterns in forest-fire sequences

Temporal variation of spatial clustering in fire data recorded from 1997 to 2003 in Tuscany region, central Italy, has been investigated using the Voronoï polygon area, the Morishita index and the fractal dimension. Our findings reveal that the spatial clustering of fire events changes with time, showing an enhancement of the clustering degree before the largest events.

Thermal patterns in simple models of cylindrical reactors

The propagation of fronts and the emergence of spatiotemporal patterns on a cylindrically shaped thin catalytic reactor is simulated with a homogeneous model of a fixed catalytic bed, with characteristically large Lewis and Peclet numbers, and a first-order Arrhenius kinetics (i.e., thermokinetic model) which may be coupled with slow changes of catalytic activity (i.e., oscillatory kinetics). Planar fronts of the thermokinetic model may undergo symmetry breaking in the transversal direction only at relatively low Lewis number, but for high Le the front remains flat. Patterns due to oscillatory kinetics in reactors of high Le are shown, for the first time, to undergo symmetry breaking in the azimuthal direction when the perimeter is sufficiently large. The generic regular patterns simulated then are rotating multi-wave patterns of constant rotation-speed and oscillatory-‘firing’ ones, and theirs selection is highly sensitive to governing parameters and initial conditions. The results are organized in bifurcation diagrams showing the coexisting two-dimensional solutions with varying perimeter. Increasing convective velocity or reactor radius leads to symmetry breaking of regular patterns and the system may switch to chaos.

A. Introductory and inorganic oscillations. Thermodynamic aspects and bifurcation analysis of spatio-temporal dissipative structures

The thermodynamic prerequisites for the emergence of spatio-temporal patterns of organization in nonlinear chemical systems are reviewed. Analytic expressions for steady-state, time-periodic and travelling wave like solution of reaction-diffusion equations are reported. A comparison with the results based on computer simulations is outlined.