Synchronization and oscillator death in oscillatory media with stirring

Optimal Stretching in Advection-Reaction-Diffusion Systems

We investigate growth of the excitable Belousov-Zhabotinsky reaction in chaotic, time-varying flows. In slow flows, reacted regions tend to lie near vortex edges, whereas fast flows restrict reacted regions to vortex cores. We show that reacted regions travel toward vortex centers faster as flow speed increases, but nonreactive scalars do not. For either slow or fast flows, reaction is promoted by the same optimal range of the local advective stretching, but stronger stretching causes reaction blowout and can hinder reaction from spreading. We hypothesize that optimal stretching and blowout occur in many advection-diffusion-reaction systems, perhaps creating ecological niches for phytoplankton in the ocean.

Restoration of rhythmicity in diffusively coupled dynamical networks

Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks.

How turbulence regulates biodiversity in systems with cyclic competition

Cyclic, nonhierarchical interactions among biological species represent a general mechanism by which ecosystems are able to maintain high levels of biodiversity. However, species coexistence is often possible only in spatially extended systems with a limited range of dispersal, whereas in well-mixed environments models for cyclic competition often lead to a loss of biodiversity. Here we consider the dispersal of biological species in a fluid environment, where mixing is achieved by a combination of advection and diffusion. In particular, we perform a detailed numerical analysis of a model composed of turbulent advection, diffusive transport, and cyclic interactions among biological species in two spatial dimensions and discuss the circumstances under which biodiversity is maintained when external environmental conditions, such as resource supply, are uniform in space. Cyclic interactions are represented by a model with three competitors, resembling the children's game of rock-paper-scissors, whereas the flow field is obtained from a direct numerical simulation of two-dimensional turbulence with hyperviscosity. It is shown that the space-averaged dynamics undergoes bifurcations as the relative strengths of advection and diffusion compared to biological interactions are varied.

Amplitude death of coupled hair bundles with stochastic channel noise

Hair cells conduct auditory transduction in vertebrates. In lower vertebrates such as frogs and turtles, due to the active mechanism in hair cells, hair bundles (stereocilia) can be spontaneously oscillating or quiescent. Recently an amplitude death phenomenon has been proposed [K.-H. Ahn, J. R. Soc. Interface, 10, 20130525 (2013)] as a mechanism for auditory transduction in frog hair-cell bundles, where sudden cessation of the oscillations arises due to the coupling between nonidentical hair bundles. The gating of the ion channel is intrinsically stochastic due to the stochastic nature of the configuration change of the channel. The strength of the noise due to the channel gating can be comparable to the thermal Brownian noise of hair bundles. Thus, we perform stochastic simulations of the elastically coupled hair bundles. In spite of stray noisy fluctuations due to its stochastic dynamics, our simulation shows the transition from collective oscillation to amplitude death as interbundle coupling strength increases. In its stochastic dynamics, the formation of the amplitude death state of coupled hair bundles can be seen as a sudden suppression of the displacement fluctuation of the hair bundles as the coupling strength increases. The enhancement of the signal-to-noise ratio through the amplitude death phenomenon is clearly seen in the stochastic dynamics. Our numerical results demonstrate that the multiple number of transduction channels per hair bundle is an important factor to the amplitude death phenomenon, because the phenomenon may disappear for a small number of transduction channels due to strong gating noise.

Stirring effects in models of oceanic plankton populations

We present an overview and extend previous results on the effects of large scale oceanic transport processes on plankton population dynamics, considering different types of ecosystem models. We find that increasing stirring rate in an environment where the carrying capacity is non-uniformly distributed leads to an overall decrease of the effective carrying capacity of the system. This may lead to sharp regime shifts induced by stirring in systems with multiple steady states. In prey-predator type systems, stirring leads to resonant response of the population dynamics to fluctuations enhancing the spatial variability-patchiness-in a certain range of stirring rates. Oscillatory population models produce strongly heterogeneous patchy distribution of plankton blooms when the stirring is weak, while strong stirring may either synchronise the oscillatory dynamics, when the inhomogeneity is relatively weak, or suppress oscillations completely (oscillator death) by reducing the effective carrying capacity below the bifurcation point.

Parameter mismatches and oscillation death in coupled oscillators

We use a set of qualitatively different models of coupled oscillators (genetic, membrane, Ca-metabolism, and chemical oscillators) to study dynamical regimes in the presence of small detuning. In particular, we focus on a distinct oscillation quenching mechanism, the oscillation death phenomenon. Using bifurcation analysis in general, we demonstrate that under strong coupling via slow variable detuning can eliminate standard oscillatory solutions from a large region of the parameter space, establishing the dominance of oscillation death. We argue furthermore that the oscillation death dominance effect provides a reliable dynamical control mechanism in the general case of N coupled oscillators.

Persistence of cluster synchronization under the influence of advection

We present a study on the emergence of spatial structure in plankton dynamics under the influence of stirring and mixing. A distribution of plankton is represented as a lattice of nonidentical, interacting, oscillatory plankton populations. Each population evolves according to (i) the internal biological dynamics represented by an NPZ model with population-specific phytoplankton growth rate, (ii) sub-grid-cell stirring and mixing parameterized by a nearest-neighbor coupling, and (iii) explicit advection resulting from a constant horizontal shear. Using the methods of synchronization theory, the emergent spatial structure of the simulation is investigated as a function of the coupling strength and rate of advection. Previous work using similar methods has neglected the effects of explicit stirring (i.e., at scales larger than the grid cell), leaving as an open question the relevance of the work to real marine systems. Here, we show that persistent spatial structure emerges for a range of coupling strengths for all realistic levels of surface ocean shear. Spatially, this corresponds to the formation of temporally evolving clusters of local synchronization. Increasing shear alters the spatial characteristics of this clustering by stretching and narrowing patches of synchronized dynamics. These patches are not stretched into stripes of synchronized abundance aligned with the flow, as may be expected, but instead lie at an angle to the flow. This study shows that advection does not diminish the relevance of conclusions from previous studies of spatial structure in plankton simulations. In fact, the inclusion of advection adds characteristic filamental structure, as observed in real-world plankton distributions. The results also show that the ability of coupled oscillators to synchronize depends strongly on the spatial arrangement of oscillator natural frequencies; under the influence of advection, therefore, the impact of the coupling strength on the emergent spatial structure of a biophysical simulation is time-dependent.

Emergence of collective behavior in groups of excitable catalyst-loaded particles: spatiotemporal dynamical quorum sensing

Spontaneous spatiotemporal wave activity occurs in groups of excitable particles for groups larger than a critical size. Experiments are carried out with particles loaded with the catalyst of the Belousov-Zhabotinsky reaction that are immersed in catalyst-free reaction mixture. The particles diffusively exchange activator and inhibitor species with the surrounding solution. All particles are nonoscillatory when separated from the other particles; however, target and spiral waves are exhibited in sufficiently large groups. A cellular particle model of the system also exhibits transitions from excitable steady state behavior to spatiotemporal wave activity with increasing group size.

Solitary Marangoni-driven convective structures in bistable chemical systems

Bistable chemical fronts can be deformed by Marangoni-driven convective flows induced by gradients of surface tension across the front. We investigate here the nonlinear dynamics of such a system by simulations of two-dimensional Navier-Stokes equations coupled to a reaction-diffusion-convection equation for a surface-active chemical species present in the bulk of the solution and involved in a bistable kinetics. We show that Marangoni flows cannot only alter the shape and speed of the front but also change the relative stability of the two stable steady states, reversing in some cases the direction of propagation of the front with regard to the pure reaction-diffusion situation. A detailed parametric study discusses the properties of the asymptotic dynamics as a function of the Marangoni number M and of a kinetic parameter d.

Active media under rotational forcing

The bubble-free Belousov-Zhabotinsky reaction has been used to study the effects of centrifugal forces on autowave propagation. The reaction parameters were chosen such that the system oscillates naturally creating target waves. In the present study, the system was forced to rotate with a constant velocity around a central axis. In studying the effects of such a forcing on the system, we focused on target dynamics. The system reacts to this forcing in different ways, the most spectacular being a dramatic increase in the period of the target, the effect growing stronger as we move away from the center of rotation. A numerical study was carried out using the two-variable Oregonator model, modified to include convective effects through the diffusion coefficient. The numerical results showed a good qualitative agreement with those of the experiments.

Resonant pattern formation in active media driven by time-dependent flows

The effect of a time-dependent flow in an oscillatory chemical system supporting front propagation is studied. Resonant target patterns depend on the strength and frequency of the time-dependent flow. The flow time scale needed to entrain the system to the resonant target period of oscillation depends on the closeness to the natural oscillation frequency of the medium. The flow strength needed to obtain these patterns is interpreted in terms of mixing optimization, and we give conditions for the flow that guarantee the best mixing with the Bernoulli property.

Collective Behavior of a Population of Chemically Coupled Oscillators

Experiments are performed in which a large number (approximately 10(4)) of relaxation oscillators are globally coupled through the concentration of chemicals in the surrounding solution. Each oscillator consists of a microscopic catalyst-loaded particle that displays oscillations in the concentrations of chemical species when suspended in catalyst-free Belousov-Zhabotinsky (BZ) reaction solution. In the absence of stirring, the uncoupled particles display a range of oscillatory frequencies. In the well-stirred system, oscillations appear in the surrounding solution for greater than a critical number density of particles (n(crit)). There is a growth in the amplitude of oscillations with increasing n, accompanied by a slight increase or no change in frequency. A model is proposed to account for the behavior, in which the transfer of activator and inhibitor to and from the bulk medium is considered for each particle. We demonstrate that the appearance and subsequent growth in the amplitude of oscillations may be associated with partial synchronization of the oscillators.

Synchronization of oscillating reactions in an extended fluid system

We present experiments on the synchronization of a dynamical, chemical process in an extended, flowing, fluid system. The oscillatory Belousov-Zhabotinsky chemical reaction is the process studied, and the flow is an annular chain of counterrotating vortices. Azimuthal motion of the vortices is controlled externally, enabling us to vary the type of transport. We find that oscillations of the Belousov-Zhabotinsky reaction synchronize throughout the extended fluid system only if transport in the flow is superdiffusive, with tracers in the flow undergoing rapid, distant jumps called Lévy flights.

Front propagation and mode-locking in an advection-reaction-diffusion system

Experiments are presented on chemical front propagation in an oscillating chain of vortices in which the mixing of passive impurities is chaotic. The excitable ruthenium-catalyzed Belousov-Zhabotinsky reaction is used in these studies. Velocities of the propagating fronts are measured as a function of the frequency and amplitude of external forcing. Mode locking is observed where the front propagates an integer number of vortices in an integer number of drive periods. Arnol'd tongues are mapped out for two of the locking regimes. These two tongues are shown to form a region of overlap where the velocity of the propagating front switches erratically between two locked values. The experimental results agree with numerical predictions of mode locking in a simplified model of the flow.

Chaos-induced coherence in two independent food chains

Coherence evolution of two food web models can be obtained under the stirring effect of chaotic advection. Each food web model sustains a three-level trophic system composed of interacting predators, consumers, and vegetation. These populations compete for a common limiting resource in open flows with chaotic advection dynamics. Here we show that two species (the top predators) of different colonies chaotically advected by a jetlike flow can synchronize their evolution even without migration interaction. The evolution is characterized as a phase synchronization. The phase differences (determined through the Hilbert transform) of the variables representing those species show a coherent evolution.

Rock-scissors-paper game in a chaotic flow: The effect of dispersion on the cyclic competition of microorganisms

Laboratory experiments and numerical simulations have shown that the outcome of cyclic competition is significantly affected by the spatial distribution of the competitors. Short-range interaction and limited dispersion allows for coexistence of competing species that cannot coexist in a well-mixed environment. In order to elucidate the mechanisms that destroy species diversity we study the intermediate situation of imperfect mixing, typical in aquatic media, in a model of cyclic competition between toxin producing, sensitive and resistant phenotypes. It is found, that chaotic mixing, by changing the character of the spatial distribution, induces coherent oscillations in the populations. The magnitude of the oscillations increases with the strength of mixing, leading to the extinction of some species beyond a critical mixing rate. When mixing is non-uniform in space, coexistence can be sustained at much stronger mixing by the formation of partially isolated regions, that prevent global extinction. The heterogeneity of mixing may enable toxin producing and sensitive strains to coexist for very long time at strong mixing.

Bifurcations in reaction-diffusion systems in chaotic flows

We study the behavior of reacting tracers in a chaotic flow. In particular, we look at an autocatalytic reaction and at a bistable system which are subjected to stirring by a chaotic flow. The impact of the chaotic advection is described by a one-dimensional phenomenological model. We use a nonperturbative technique to describe the behavior near a saddle node bifurcation. We also find an approximation of the solution far away from the bifurcation point. The results are confirmed by numerical simulations and show good agreement.

Experimental studies of pattern formation in a reaction-advection-diffusion system

Experiments are presented on pattern formation in the Belousov-Zhabotinsky (BZ) reaction in a blinking vortex flow. Mixing in this flow is chaotic, with nearby tracers separating exponentially with time. The patterns that form in this flow with the BZ reaction mimic chaotic mixing structures seen in passive transport. The behavior is analyzed in terms of a mixing time taum and a characteristic decorrelation time TBZ for the BZ system. Flows with taum comparable to or smaller than TBZ generate large-scale patterns whose features are captured by simulations of mixing fields for the flow.

Effects of time-delayed feedback on chaotic oscillators

We study the effects of time-delayed feedback on chaotic systems where the delay time is both fixed (static case) and varying (dynamic case) in time. For the static case, typical phase coherent and incoherent chaotic oscillators are investigated. Detailed phase diagrams are investigated in the parameter space of feedback gain ( K ) and delay time ( tau ). Linear stability analysis, by assuming the time-delayed perturbation, varies as e(lambdat) where lambda is the eigenvalue, gives the boundaries of the stability islands and critical feedback gains ( K(c) ) for both Rössler oscillators and Lorenz oscillators. We also found that the stability island are found when the delay time is about tau= (n+ 1 / 2 ) T , where n is an integer and T is the average period of the chaotic oscillator. It is shown that these analytical predictions agree well with the numerical results. For the dynamic case, we investigate Rössler oscillator with periodically modulated delay time. Stability regimes are found for parameter space of feedback gain and modulation frequency in which it was impossible to be stabilized for a fixed delay time. We also trace the detailed routes to the stability near the island boundaries for both cases by investigating bifurcation diagrams.