Metabolic network dynamics in open chaotic flow

We have analyzed the dynamics of metabolically coupled replicators in open chaotic flows. Replicators contribute to a common metabolism producing energy-rich monomers necessary for replication. The flow and the biological processes take place on a rectangular grid. There can be at most one molecule on each grid cell, and replication can occur only at localities where all the necessary replicators (metabolic enzymes) are present within a certain neighborhood distance. Due to this finite metabolic neighborhood size and imperfect mixing along the fractal filaments produced by the flow, replicators can coexist in this fluid system, even though coexistence is impossible in the mean-field approximation of the model. We have shown numerically that coexistence mainly depends on the metabolic neighborhood size, the kinetic parameters, and the number of replicators coupled through metabolism. Selfish parasite replicators cannot destroy the system of coexisting metabolic replicators, but they frequently remain persistent in the system. (c) 2002 American Institute of Physics.     

Dynamical role of the degree of intraspecific cooperation: A simple model for prebiotic replicators and ecosystems

We present a simple mean field model to analyze the dynamics of competition between two populations of replicators in terms of the degree of intraspecific cooperation (i.e., autocatalysis) in one of these populations. The first population can only replicate with Malthusian kinetics while the second one can reproduce with Malthusian or autocatalytic replication or with a combination of both reproducing strategies. The model consists of two coupled, nonlinear, autonomous ordinary differential equations. We investigate analytically and numerically the phase plane dynamics and the bifurcation scenarios of this ecologically coupled system, focusing on the outcome of competition for several degrees of intraspecific cooperation, σ, in the second population of replicators. We demonstrate that the dynamics of both populations can not be governed by a limit cycle, and also that once cooperation is considered, the topology of phase space does not allow for coexistence. Even for low values of the degree of intraspecific cooperation, for large enough autocatalytic replication rates, the second population of replicators is able to outcompete the first one, having a wide basin of attraction in state space. We characterize the same power law dependence between the outcompetition extinction times, τ, and the degree of intraspecific cooperation for both populations, given by τ∼c_iσ⁻¹. Our results suggest that, under some kinetic conditions, the appearance of autocatalysis might be favorable in a population of replicators growing with Malthusian kinetics competing with another population also reproducing exponentially.     

Evolutionary dynamics of RNA-like replicator systems: A bioinformatic approach to the origin of life

We review computational studies on prebiotic evolution, focusing on informatic processes in RNA-like replicator systems. In particular, we consider the following processes: the maintenance of information by replicators with and without interactions, the acquisition of information by replicators having a complex genotype–phenotype map, the generation of information by replicators having a complex genotype–phenotype–interaction map, and the storage of information by replicators serving as dedicated templates. Focusing on these informatic aspects, we review studies on quasi-species, error threshold, RNA-folding genotype–phenotype map, hypercycle, multilevel selection (including spatial self-organization, classical group selection, and compartmentalization), and the origin of DNA-like replicators. In conclusion, we pose a future question for theoretical studies on the origin of life.     

Phase transitions in cellular automata models of spatial susceptible--infected--resistant--susceptible epidemics

Spatially explicit models have become widely used in today's mathematical ecology and epidemiology to study the persistence of populations. For simplicity, population dynamics is often analysed by using ordinary differential equations (ODEs) or partial differential equations (PDEs) in the one-dimensional (1D) space. An important question is to predict species extinction or persistence rate by mean of computer simulation based on the spatial model. Recently, it has been reported that stable turbulent and regular waves are persistent based on the spatial susceptible--infected--resistant--susceptible (SIRS) model by using the cellular automata (CA) method in the two-dimensional (2D) space [Proc. Natl. Acad. Sci. USA 101, 18246 (2004)]. In this paper, we address other important issues relevant to phase transitions of epidemic persistence. We are interested in assessing the significance of the risk of extinction in 1D space. Our results show that the 2D space can considerably increase the possibility of persistence of spread of epidemics when the degree distribution of the individuals is uniform, i.e. the pattern of 2D spatial persistence corresponding to extinction in a 1D system with the same parameters. The trade-offs of extinction and persistence between the infection period and infection rate are observed in the 1D case. Moreover, near the trade-off (phase transition) line, an independent estimation of the dynamic exponent can be performed, and it is in excellent agreement with the result obtained by using the conjectured relationship of directed percolation. We find that the introduction of a short-range diffusion and a long-range diffusion among the neighbourhoods can enhance the persistence and global disease spread in the space.     

The comparative analysis of prebiological evolution models

Several models of prebiological systems are described and analyzed. The following models are characterized: a quasispecies model, a hypercycle model, a syser model (the term "syser" is an abbreviation of SYstem of SElf-Reproduction), a stochastic corrector model, a model of the origin of a primordial genome through spontaneous symmetry breaking. The quasispecies model analyzes the Darwinian evolution of information chains; this evolution is similar to the evolution of RNA molecules. Rather general estimates of the speed and efficiency of evolutionary processes can be obtained in the framework of the quasispecies model. We briefly describe the method for obtaining these estimates and the corresponding results. The hypercycle model considers the interaction of RNA chains and enzymes. The syser model characterizes a rather general scheme of the self-reproducing system, which is similar to the self-reproducing systems of biological cells. Syser includes a polynucleotide sequence, a replication enzyme, a translation enzyme, and other enzymes; these macromolecules are located inside the protocell. The stochastic corrector model describes the process of using a relatively small number of molecules of competing and cooperating replicators in protocells. The model of the origin of a primordial genome through spontaneous symmetry breaking characterizes an interesting and important process of the appearance of genotypes in protocells. This model was proposed and investigated by Takeuchi, Hogeweg, and Kaneko in 2017; we call it further “the THK model.” The current article characterizes and compares all these models.     

Homoclinic bifurcations in radiating diffusion flames

We analyse the dynamics of a model describing a planar diffusion flame with radiative heat losses incorporating a single step kinetic using timestepping techniques for Lewis number equal to one. We construct the full bifurcation diagram with respect to the Damköhler number including the branches of oscillating solutions. Based on this analysis we found, for the first time, homoclinic bifurcations that mark the abrupt disappearance of the nonlinear oscillations near extinction as reported in experiments.     

Replication in one-dimensional cellular automata

In a cellular automaton (CA), replication is the ability to indefinitely generate copies of a finite collection of patterns, starting from finite seeds. A transparent feature of additive CA, replication mechanisms are less clear in the absence of additivity; this paper investigates such dynamics through several examples. For the 1 Or 2 rule and its generalizations, replication is inevitable and we investigate self-organization properties. In the Perturbed Exactly 1 rule we study frequency of replicators, and the new phenomenon is called quasireplication. The last CA is the Extended 1 Or 3 rule, which allows for replication on different backgrounds. We employ a mixture of rigorous and empirical techniques.     

Kinetic effects of NO addition on n-dodecane cool and warm diffusion flames

The kinetic effects of NO addition on the flame dynamics and burning limits of n-dodecane cool and warm diffusion flames are investigated experimentally and computationally using a counterflow system. The results show that NO plays different roles in cool and warm flames due to their different reaction pathway sensitivities to the flame temperature and interactions with NO. We observe that NO addition decreases the cool flame extinction limit, delays the extinction transition from warm flame to cool flame, and promotes the ignition transition from warm flame to hot flame. In addition, jet-stirred reactor (JSR) experiments of n-dodecane oxidation with and without NO addition are also performed to develop and validate a n-dodecane/NO_x kinetic model. Reaction pathway and sensitivity analyses reveal that, for cool flames, NO addition inhibits the low-temperature oxidation of n-dodecane and reduces the flame temperature due to the consumption of RO₂ via NO+RO₂?NO₂+RO, which competes with the isomerization reaction that continues the peroxy radical branching sequence. The model prediction captures well the experimental trend of the inhibiting effect of NO on the cool flame extinction limit. For warm flames, two different kinds of warm flame transitions, the warm flame extinction transition to cool flame and the warm flame reignition transition to hot flame, were observed. The results suggest that warm extinction transition to cool flame is suppressed by NO addition while the warm flame reignition transition to hot flame is promoted. The kinetic model developed captures well the experimentally observed warm flame transitions to cool flame but fails to predict the warm flame reignition to hot flame at similar experimental conditions.     

Coexistence of Languages is possible

In this work we study the dynamics of language competition. In Abrams and Strogatz [Modeling the dynamics of language death, Nature 424 (2003) 900], the extinction of one of the competing languages is predicted, although in some case the coexistence occurs. The preservation of both languages was explained by Patriarca and Leppanen [Modeling language competition, Physica A 338 (2004) 296] by introducing the existence of two disjoint zones where each language is predominant. However, their results cannot explain the survivance of both languages in only one zone of competition. In this work we discuss their results and propose a new alternative model of Lotka–Volterra type in order to explain the coexistence of two languages.     

Hybrid phase transitions of spreading dynamics in multiplex networks

In this paper, we study the spreading dynamics of social behaviors and focus on heterogenous responses of individuals depending on whether they realize the spreading or not. We model the system with a two-layer multiplex network, in which one layer describes the spreading of social behaviors and the other layer describes the diffusion of the awareness about the spreading. We use the susceptible-infected-susceptible (SIS) model to describe the dynamics of an individual if it is unaware of the spreading of the behavior. While when an individual is aware of the spreading of the social behavior its dynamics will follow the threshold model, in which an individual will adopt a behavior only when the fraction of its neighbors who have adopted the behavior is above a certain threshold. We find that such heterogenous reactions can induce intriguing dynamical properties. The dynamics of the whole network may exhibit hybrid phase transitions with the coexistence of continuous phase transition and bi-stable states. Detailed study of how the diffusion of the awareness influences the spreading dynamics of social behavior is provided. The results are supported by theoretical analysis.     

Scaling laws of single polymer dynamics near attractive surfaces

We present a molecular dynamics study of a generic model for single polymer diffusion on surfaces, which have variable atomic-scale corrugation but no artificial, impenetrable obstacles. The diffusion coefficient D scales as D is proportional to (-3/2) with the degree of polymerization N for strongly adsorbed, linear polymers on solid substrates in good solvents. Weaker scaling, i.e., D is proportional to (-1), is found if (i) the substrate is a fluid, e.g., a membrane, (ii) the polymer is a ring polymer, and (iii) the polymer is commensurate with the substrate. In poor solvents, diffusion on solids slows exponentially fast with N. Reptation is not observed in any of the simulations presented here.     

First-order quantum phase transitions: Test ground for emergent chaoticity,regularity and persisting symmetries

We present a comprehensive analysis of the emerging order and chaos and enduring symmetries, accompanying a generic (high-barrier) first-order quantum phase transition (QPT). The interacting boson model Hamiltonian employed, describes a QPT between spherical and deformed shapes, associated with its U(5) and SU(3) dynamical symmetry limits. A classical analysis of the intrinsic dynamics reveals a rich but simply-divided phase space structure with a Hénon–Heiles type of chaotic dynamics ascribed to the spherical minimum and a robustly regular dynamics ascribed to the deformed minimum. The simple pattern of mixed but well-separated dynamics persists in the coexistence region and traces the crossing of the two minima in the Landau potential. A quantum analysis discloses a number of regular low-energy U(5)-like multiplets in the spherical region, and regular SU(3)-like rotational bands extending to high energies and angular momenta, in the deformed region. These two kinds of regular subsets of states retain their identity amidst a complicated environment of other states and both occur in the coexistence region. A symmetry analysis of their wave functions shows that they are associated with partial U(5) dynamical symmetry (PDS) and SU(3) quasi-dynamical symmetry (QDS), respectively. The pattern of mixed but well-separated dynamics and the PDS or QDS characterization of the remaining regularity, appear to be robust throughout the QPT. Effects of kinetic collective rotational terms, which may disrupt this simple pattern, are considered.     

Dynamics of a creep-slip model of earthquake faults

Starting off from the relationship between time-dependent friction and velocity softening we present a generalization of the continuous, one-dimensional homogeneous Burridge–Knopoff (BK) model by allowing for displacements by plastic creep and rigid sliding. The evolution equations describe the coupled dynamics of an order parameter-like field variable (the sliding rate) and a control parameter field (the driving force). In addition to the velocity-softening instability and deterministic chaos known from the BK model, the model exhibits a velocity-strengthening regime at low displacement rates which is characterized by anomalous diffusion and which may be interpreted as a continuum analogue of self-organized criticality (SOC). The governing evolution equations for both regimes (a generalized time-dependent Ginzburg–Landau equation and a non-linear diffusion equation, respectively) are derived and implications with regard to fault dynamics and power-law scaling of event-size distributions are discussed. Since the model accounts for memory friction and since it combines features of deterministic chaos and SOC it displays interesting implications as to (i) material aspects of fault friction, (ii) the origin of scaling, (iii) questions related to precursor events, aftershocks and afterslip, and (iv) the problem of earthquake predictability. Moreover, by appropriate re-interpretation of the dynamical variables the model applies to other SOC systems, e.g. sandpiles.     

Error propagation in the hypercycle

We study analytically the steady-state regime of a network of n error-prone self-replicating templates forming an asymmetric hypercycle and its error tail. We show that the existence of a master template with a higher noncatalyzed self-replicative productivity a than the error tail ensures the stability of chains in which m < n-1 templates coexist with the master species. The stability of these chains against the error tail is guaranteed for catalytic coupling strengths K on the order of a. We find that the hypercycle becomes more stable than the chains only if K is on the order of a2. Furthermore, we show that the minimal replication accuracy per template needed to maintain the hypercycle, the so-called error threshold, vanishes as square root of n/K for large K and N < or = 4.     

Combustion fronts in non-diffusing disordered premixtures. I: Single-channel curved flames

We consider curved flames in model solid-like premixtures, when the initial reactant content and/or reactivity are maxima along the axis of straight channels. Hydrodynamics and fuel diffusion are neglected. The one-step reaction has an explicit dependence on coordinates, and follows such a generalized Arrhenius law that the flat flame problem in homogeneous media is solved exactly for any activation exponent, n. In the large-n, Arrhenius-like, limit we produce a PDE for the flame shape and speed, U. This indicates that: (i) 2D Lorentzian transverse reactivity profiles yield uniform reaction temperatures for any effective channel widths (L ₁, L ₂), and give elliptic–paraboloidal fronts; (ii) too small L _i s induce extinction, at known n-dependent values; and (iii) reactivity and initial-composition gradients play similar roles if a certain combination thereof is maintained fixed. For (1) values of n, we revisit the problem via a δ-function model – tailored to reproduce as many exact results as possible. By means of generalized elliptic coordinates, we analytically confirm and sharpen items (i)–(iii).

Finite-difference direct numerical simulations, validated through comparisons with linear stability analyses, are finally developed. They also confirm (i)–(iii) for moderate ns and show how accurate the δ-model is, even then. More sensitives rates (n > 5) yield a whole hierarchy of instabilities close to extinction: Hopf bifurcation, travelling waves then hot spots, period doubling, premature extinction; yet the time-averaged U stays close to what the δ-model gave. Non-Lorentzian (e.g. Gaussian) reactivity profiles lead to nearly identical conclusions. Open problems, and implications as to larger-scale propagations in disordered media, are evoked.     

Double-spray counterflow diffusion flame model

In this spray model we consider two gaseous streams approaching each other from opposite directions in a counterflow. The two opposed streams each carry a distribution of liquid droplets. The sprays vaporize, and the vaporized fuel and oxidizer gases diffuse and convect toward a chemical reaction region near the stagnation plane, at which the reactants burn. A set of steady-state ordinary differential equations is derived to describe the temperature of the gas flow and the mass fractions of each reactant. We solve the differential equations in three consequent cases, each more complicated than the previous one: (i) fast vaporization and fast chemistry; (ii) finite-rate vaporization and fast chemistry; and (iii) finite-rate vaporization and finite-rate chemistry. Comparisons are made of our model results to previous fuel-spray-only and purely gaseous counterflow diffusion flame models. The parametric dependences of vaporization-zone movement, flame movement, temperature rise and degree of reactant leakage through the flame are examined. In addition, the strain rate dependence of these quantities is examined up to and including extinction.     

Effects of diffusion in fluctuating media: A noise-induced phase transition

An example of a noise-induced phase transition in a chemical system with diffusion is studied. It is shown that effects of diffusion can radically change the classical results on extinction and coexistence in the models of Lotka-Volterra type. Fluctuation phenomena near the point of the noise-induced phase transition are discussed.     

Basins of coexistence and extinction in spatially extended ecosystems of cyclically competing species

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system.     

Large Scale Dynamics of the Persistent Turning Walker Model of Fish Behavior

This paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results.     

On the existence of steady-state gaseous microgravity spherical diffusion flames in the presence of radiation heat loss

Whether steady-state gaseous microgravity spherical diffusion exist in the presence of radiation heat loss is an important fundamental question and has important implications for spacecraft fire safety. In this work, experiments aboard the International Space Station and a transient numerical model are used to investigate the existence of steady-state microgravity spherical diffusion flames. Gaseous spherical diffusion flames stabilized on a porous spherical burner are employed in normal (i.e., fuel flowing into an ambient oxidizer) and inverse (i.e., oxidizer flowing into an ambient fuel) flame configurations. The fuel is ethylene and the oxidizer oxygen, both diluted with nitrogen. The flow rate of the reactant gas from the burner is held constant. It is found that steady-state gaseous microgravity spherical diffusion flames can exist in the presence of radiation heat loss, provided that the steady-state flame size is less than the flame size for radiative extinction, and the flame develops fast enough that radiation heat loss does not drop the flame temperature below the critical temperature for radiative extinction (1130 K). A simple model is provided that allows for the identification of initial conditions that can lead to steady-state spherical diffusion flames. In the spherical, infinite domain configuration, the characteristic time for the diffusion-controlled system to effectively reach steady-state is found to be on the order of 100,000 s. Despite a narrow range of attainable conditions, flames that exhibit steady-state behavior are observed aboard the ISS for up to 870 s, even with the constraint of a finite boundary. Steady-state flames are simulated using the numerical model for over 100,000 s.