Formation Of Stationary Patterns Research Articles

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.

Excitable FitzHugh-Nagumo model with cross-diffusion: close and far-from-equilibrium coherent structures

In this paper, we shall study the formation of stationary patterns for a reaction-diffusion system in which the FitzHugh-Nagumo (FHN) kinetics, in its excitable regime, is coupled to linear cross-diffusion terms. In (Gambino et al. in Excitable Fitzhugh-Nagumo model with cross-diffusion: long-range activation instabilities, 2023), we proved that the model supports the emergence of cross-Turing patterns, i.e., close-to-equilibrium structures occurring as an effect of cross-diffusion. Here, we shall construct the cross-Turing patterns close to equilibrium on 1-D and 2-D rectangular domains. Through this analysis, we shall show that the species are out-of-phase spatially distributed and derive the amplitude equations that govern the pattern dynamics close to criticality. Moreover, we shall classify the bifurcation in the parameter space, distinguishing between super-and sub-critical transitions. In the final part of the paper, we shall numerically investigate the impact of the cross-diffusion terms on large-amplitude pulse-like solutions existing outside the cross-Turing regime, showing their emergence also in the case of lateral activation and short-range inhibition.

Stationary Pattern Formation in Iron Electrodissolution with Two Cells: Effects of Serial or Parallel Coupling

We investigate the stationary pattern formation for two coupled cells at constant circuit potential. Two configurations are explored: two cells in serial and parallel. In each cell the anode is iron electrodissolution and the cathode is hydrogen ion reductions on a large surface area Pt electrode with sulfuric acid electrolyte. Iron electrodissolution in sulfuric acid exhibits negative differential resistance due to electrode passivation by the surface oxide where the stability of the surface oxide is dependent on electrode potential and pH at the electrode interface. In the presence of ohmic potential drops (e.g., due to solution or external resistance), bistability can occur where the active and passive states co-exist at a given circuit potential. First, we investigated the patterns that emerge for two serially coupled iron electrodissolution in sulfuric acid. Investigation of the forward scan potentiodynamic I-V curves reveals sequential passivation of electrodes in no particular order and a potential region is identified where one electrode is passive and the other is inactive, and thus a stationary pattern is formed. Numerical simulations and equivalent circuit network modeling of the experimental set-up reveals a bidirectional negative coupling topology as a consequence of serial configuration, providing insight into the pattern formation. The patterns are interpreted with a kinetic model which shows that the coupled NDR system can exhibit symmetry breaking transitions in the presence of negative coupling. Next, we studied stationary pattern formation in two parallel cells. In contrast to the serial placement, the potentiodynamic I-V curves reveal simultaneous passivation of the two electrodes. The electrode potential versus circuit potential plot displays a uniform state of electrodes throughout the scan. The corresponding modeling showed that the lack of sequential passivation and the presence of uniform states were due to the inherent bidirectional positive coupling in the system. The experiments demonstrate the rich dynamical response of a coupled negative differential resistance systems to generate stationary patterns similar to Turing patterns that are used to interpret morphogenesis in biological systems. The theoretical understanding and insight gained from modeling and simulating observed patterns of relatively smaller electrochemical cells could be crucial in the accelerated understanding, design, and control of battery packs that utilize serial and parallel coupled electrodes.

Spatio-temporal pattern selection in a prey–predator model with hunting cooperation and Allee effect in prey

In this work, we have studied the temporal as well as spatio-temporal dynamics of a prey–predator model with additive Allee effect in prey growth and hunting cooperation among the specialist predators. Hunting cooperation impacts both the stability and the existence scenario of the coexistence equilibrium of the temporal model. The temporal system is capable of exhibiting a wide range of local bifurcations such as transcritical, saddle–node, Hopf, Bogdanov–Takens bifurcations and a global Homoclinic bifurcation. For the diffusive system, the well-posedness is first proved. Then Turing instability is investigated to understand the relationship between the diffusion and the hunting cooperation behind the stationary pattern formation. By employing weakly nonlinear analysis, the amplitude equation of stationary solution is derived, and using the stability of amplitude equation, the pattern selection of several stationary patterns such as spot, stripe, and their combinations have been discussed under various parameter regimes. Extensive numerical simulations are carried out to verify the analytical results. Main contribution of this work is the identification of all possible stationary patterns depending upon the signs of the coefficients involved in the amplitude equation. In addition, some time-dependent nonhomogeneous irregular solutions are observed for parameter values in the Hopf domain or Turing–Hopf domain. With the help of time-series analysis of the obtained numerical results, we have established that the time and space-varying solution is spatio-temporal chaos.

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.

Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

<p style='text-indent:20px;'>We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the existence of a double bifurcation threshold of the inhibitor/activator diffusivity ratio for the onset of patterning instabilities: for large values of inhibitor/activator diffusivity ratio, classical Turing patterns emerge where the two species are in-phase, while, for small values of the diffusion ratio, the analysis predicts the formation of out-of-phase spatial structures (named <i>cross-Turing patterns</i>). In addition, for increasingly large values of the inhibitor cross-diffusion, the upper and lower bifurcation thresholds merge, so that the instability develops independently on the value of the diffusion ratio, whose magnitude selects Turing or cross-Turing patterns. Finally, the pattern selection problem is addressed through a weakly nonlinear analysis.</p>

Self-organized structures of two-component laser fields and their active control in a cold Rydberg atomic gas

We investigate the formation and control of stationary optical patterns in a cold Rydberg atomic gas via double electromagnetically induced transparency. We show that, through the modulational instability of plane-wave state of a laser field with two polarization components, the system undergoes a spontaneous symmetry breaking and hence the emergence of plentiful self-organized spatial optical structures, which can be manipulated by the ratio between the cross- and self-Kerr nonlinearities, the nonlocality degree of the Kerr nonlinearities, and the populations initially prepared in the two atomic ground states. Interestingly, a crossover from mixture to separation in space (optical phase separation) of the two polarization components occurs when the ratio between the cross- and self-Kerr nonlinearities exceeds a critical value. We also show that the system supports nonlocal two-component spatial optical solitons and vortices when the parameters of the system are selected suitably. The rich diversity and active controllability of the self-organized optical structures reported here provide a way for realizing novel optical patterns and solitons and their structural phase transitions based on Rydberg atomic gases.

Pattern Formation in a Three-Species Cyclic Competition Model.

In nature, different species compete among themselves for common resources and favorable habitat. Therefore, it becomes really important to determine the key factors in maintaining the bio-diversity. Also, some competing species follow cyclic competition in real world where the competitive dominance is characterized by a cyclic ordering. In this paper, we study the formation of a wide variety of spatiotemporal patterns including stationary, periodic, quasi-periodic and chaotic population distributions for a diffusive Lotka-Volterra type three-species cyclic competition model with two different types of cyclic ordering. For both types of cyclic ordering, the temporal dynamics of the corresponding non-spatial system show the extinction of two species through global bifurcations such as homoclinic and heteroclinic bifurcations. For the spatial system, we show that the existence of Turing patterns is possible for a particular cyclic ordering, while it is not the case for the other cyclic ordering through both the analytical and numerical methods. Further, we illustrate an interesting scenario of short-range invasion as opposed to the usual invasion phenomenon over the entire habitat. Also, our study reveals that both the stationary and dynamic population distributions can coexist in different parts of a habitat. Finally, we extend the spatial system by incorporating nonlocal intra-specific competition terms for all the three competing species. Our study shows that the introduction of nonlocality in intra-specific competitions stabilizes the system dynamics by transforming a dynamic population distribution to stationary. Surprisingly, this nonlocality-induced stationary pattern formation leads to the extinction of one species and hence, gives rise to the loss of bio-diversity for intermediate ranges of nonlocality. However, the bio-diversity can be restored for sufficiently large extent of nonlocality.

Energetic and entropic cost due to overlapping of Turing-Hopf instabilities in the presence of cross diffusion.

A systematic introduction to nonequilibrium thermodynamics of dynamical instabilities are considered for an open nonlinear system beyond conventional Turing pattern in presence of cross diffusion. An altered condition of Turing instability in presence of cross diffusion is best reflected through a critical control parameter and wave number containing both the self- and cross-diffusion coefficients. Our main focus is on entropic and energetic cost of Turing-Hopf interplay in stationary pattern formation. Depending on the relative dispositions of Turing-Hopf codimensional instabilities from the reaction-diffusion equation it clarifies two aspects: energy cost of pattern formation, especially how Hopf instability can be utilized to dictate a stationary concentration profile, and the possibility of revealing nonequilibrium phase transition. In the Brusselator model, to understand these phenomena, we have analyzed through the relevant complex Ginzberg-Landau equation using multiscale Krylov-Bogolyubov averaging method. Due to Hopf instability it is observed that the cross-diffusion parameters can be a source of huge change in free-energy and concentration profiles.

Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations

Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a one-dimensional (1-D) model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a two-dimensional (2-D) model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov–Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns. Our work clarifies the effect of geometry and dimensionality on behaviors observed in mass-conserved cell polarity models.

Emergent spatiotemporal instabilities in reactive spatially extended systems by thermodiffusion.

Thermodiffusion or thermophoresis or Soret effect, i.e., mass-transport induced by thermal gradient, has immense application in segregation of species in two or multicomponent gaseous, liquid, or colloidal mixtures. Here, we show that an external thermal gradient can be effectively utilized in creation and modification of patterns in spatially extended systems. We consider Brusselator and chlorine-dioxide iodine malonic acid (CDIMA) reaction-diffusion systems, which follow activator-inhibitor kinetics subjected to an external thermal gradient. We find that the conspicuous interaction of emergent thermodiffusive flux with reaction kinetics and diffusion can lead to various spatiotemporal instabilities in these two models. Specifically, our result reveals formation of Turing-like spatial patterns even for equal diffusivities of the activator and inhibitor components in the Brusselator model under the influence of differential thermodiffusion, whereas formation of such stationary patterns in the CDIMA system from a homogenous stable steady state, which is also stable under differential diffusion, requires the same sign and magnitude of Soret coefficients. However, with equal diffusivities of the components of the CDIMA system and without starch in the medium, our result identifies formation of drifting spiral waves which finally disappears at longer times under the influence of thermodiffusion. We also show formation of propagating patterns of spotlike or stripelike heterogeneity in both the model systems under appropriate conditions. Our study provides a route to pattern formation beyond Turing space and reveals remarkable influence of thermodiffusion to modify the pattern types just by employing an external thermal gradient which also opens up the possibility to set up new related experiments.

Spatial Memory and Taxis-Driven Pattern Formation in Model Ecosystems

Mathematical models of spatial population dynamics typically focus on the interplay between dispersal events and birth/death processes. However, for many animal communities, significant arrangement in space can occur on shorter timescales, where births and deaths are negligible. This phenomenon is particularly prevalent in populations of larger, vertebrate animals who often reproduce only once per year or less. To understand spatial arrangements of animal communities on such timescales, we use a class of diffusion–taxis equations for modelling inter-population movement responses between N ge 2 populations. These systems of equations incorporate the effect on animal movement of both the current presence of other populations and the memory of past presence encoded either in the environment or in the minds of animals. We give general criteria for the spontaneous formation of both stationary and oscillatory patterns, via linear pattern formation analysis. For N=2, we classify completely the pattern formation properties using a combination of linear analysis and nonlinear energy functionals. In this case, the only patterns that can occur asymptotically in time are stationary. However, for N ge 3, oscillatory patterns can occur asymptotically, giving rise to a sequence of period-doubling bifurcations leading to patterns with no obvious regularity, a hallmark of chaos. Our study highlights the importance of understanding between-population animal movement for understanding spatial species distributions, something that is typically ignored in species distribution modelling, and so develops a new paradigm for spatial population dynamics.

Pattern formation of a diffusive predator–prey model with strong Allee effect and nonconstant death rate

This paper is concerned with the pattern formation of a diffusive predator–prey model with strong Allee effect in the prey population and nonconstant death rate for the predator. The necessary and sufficient conditions of Turing instability are explicitly obtained. By regarding the diffusion coefficient d2 of the predator as bifurcation parameter, we demonstrate that at the critical value of the bifurcation parameter d2 a Turing bifurcation occurs (i.e. a pattern arises). The conditions for the stability of the pattern are also derived in detail. Our results show that strong Allee effect in our model plays a crucial role in the formation of stationary pattern, which is a strong contrast to the case without strong Allee effect.

Pattern formation for a two-dimensional reaction-diffusion model with chemotaxis

This paper is devoted to study the formation of stationary patterns for a chemotaxis model with nonlinear diffusion and volume-filling effect over a bounded rectangular domain. By using linear stability analysis around the homogeneous steady states we establish conditions for the existence of unstable mode bands that lead to the formation of spatial patterns. We derive the Stuart-Landau equations for the pattern amplitudes by means of weakly nonlinear multiple scales analysis and Fredholm theory. In particular, we find asymptotic expressions for a wide range of patterns sustained by the system. These patterns include mixed-mode, square, hexagonal, and roll stationary configurations. Our analytical results are corroborated by direct simulations of the underlying chemotaxis system.

Modeling of Multifactor Taxis in a Predator–Prey System

A model of predator–prey dynamics in a spatially heterogeneous range is considered using a system of two nonlinear equations of the diffusion–advection reaction. The pursuit–evasion activity because of the nonuniform distribution of the predator and the resource is taken into account. A numerical study of the effect of directed migration on the formation of population structures was carried out on the basis of the method of lines and staggered grids. The role of taxis in population growth in the area was analyzed, and migration parameters, which are used for the formation of stationary and oscillatory spatiotemporal patterns, were calculated.

Nonequilibrium transition and pattern formation in a linear reaction-diffusion system with self-regulated kinetics.

We consider a reaction-diffusion system with linear, stochastic activator-inhibitor kinetics where the time evolution of concentration of a species at any spatial location depends on the relative average concentration of its neighbors. This self-regulating nature of kinetics brings in spatial correlation between the activator and the inhibitor. An interplay of this correlation in kinetics and disparity of diffusivities of the two species leads to symmetry breaking non-equilibrium transition resulting in stationary pattern formation. The role of initial noise strength and the linear reaction terms has been analyzed for pattern selection.

Turing-like structures in a functional model of cortical spreading depression.

Cortical spreading depression (CSD) along with migraine waves and spreading depolarization events with stroke or injures are the front-line examples of extreme physiological behaviors of the brain cortex which manifest themselves via the onset and spreading of localized areas of neuronal hyperactivity followed by their depression. While much is known about the physiological pathways involved, the dynamical mechanisms of the formation and evolution of complex spatiotemporal patterns during CSD are still poorly understood, in spite of the number of modeling studies that have been already performed. Recently we have proposed a relatively simple mathematical model of cortical spreading depression which counts the effects of neurovascular coupling and cerebral blood flow redistribution during CSD. In the present study, we address the main dynamical consequences of newly included pathways, namely, the changes in the formation and propagation speed of the CSD front and the pattern formation features in two dimensions. Our most notable finding is that the combination of vascular-mediated spatial coupling with local regulatory mechanisms results in the formation of stationary Turing-like patterns during a CSD event.

Stationary patterns in star networks of bistable units: Theory and application to chemical reactions.

We present theoretical and experimental studies on pattern formation with bistable dynamical units coupled in a star network configuration. By applying a localized perturbation to the central or the peripheral elements, we demonstrate the subsequent spreading, pinning, or retraction of the activations; such analysis enables the characterization of the formation of stationary patterns of localized activity. The results are interpreted with a theoretical analysis of a simplified bistable reaction-diffusion model. Weak coupling results in trivial pinned states where the activation cannot propagate. At strong coupling, a uniform state is expected with active or inactive elements at small or large degree networks, respectively. A nontrivial stationary spatial pattern, corresponding to an activation pinning, is predicted to occur at an intermediate number of peripheral elements and at intermediate coupling strengths, where the central activation of the network is pinned, but the peripheral activation propagates toward the center. The results are confirmed in experiments with star networks of bistable electrochemical reactions. The experiments confirm the existence of the stationary spatial patterns and the dependence of coupling strength on the number of peripheral elements for transitions between pinned and retreating or spreading fronts in forced network configurations (where the central or periphery elements are forced to maintain their states).

Nonlocal interaction driven pattern formation in a prey–predator model

A widely observed scenario in ecological systems is that populations interact not only with those living in the same spatial location but also with those in spatially adjacent locations, a phenomenon called nonlocal interaction. In this paper, we explore the role of nonlocal interaction in the emergence of spatial patterns in a prey–predator model under the reaction–diffusion framework, which is described by two coupled integro-differential equations. We first prove the existence and uniqueness of the global solution by means of the contraction mapping theory and then conduct stability analysis of the positive equilibrium. We find that nonlocal interaction can induce Turing bifurcation and drive the formation of stationary spatial patterns. Finally we carry out numerical simulations to demonstrate our analytical findings.

Spatial patterns through Turing instability in a reaction–diffusion predator–prey model

Pattern formation in nonlinear complex systems is one of the central problems of the natural, social and technological sciences. In this paper, we consider a mathematical model of predator–prey interaction subject to self as well as cross-diffusion, arising in processes described by a system of reaction–diffusion equations (coupled to a system of ordinary differential equations) exhibiting diffusion-driven instability. Spatial patterns through Turing instability in a reaction–diffusion predator–prey model around the unique positive interior equilibrium of the model are discussed. Furthermore, we present numerical simulations of time evolution of patterns subject to self as well as cross-diffusion in the proposed spatial model and find that the model dynamics exhibits complex pattern replication in the two-dimensional space. The obtained results unveil that the effect of self as well as cross-diffusion plays an important role on the stationary pattern formation of the predator–prey model which concerns the influence of intra-species competition among predators.