Can noise induce chaos?

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.     

Plasma membrane--order or chaos?

In the influential "fluid mosaic" model of plasmalemma, transmembrane proteins drift regardless of lipids. Recently researches widen this to a view in which membrane lipids are not randomly distributed but they form liquid-ordered regions with local heterogenity, called lipid rafts. Lipid rafts are subdomains of the plaSma membrane that contain high concentration of cholesterol and glycosphingolipids. They are 50-100 nm distinct liquid-ordered regions of the membrane that are resistant to extraction with nonionic detergents. They are proposed to function as dynamic lipid assemblies which serve as platforms for protein segregation and signaling, protein and lipid sorting during post-Golgi sorting, dynamic of plasmalemma and virial entry budding. Markers for the lipid rafts are flotillin, GPI - linked proteins, Src family kinases, EGF receptors and G proteins. The lifetime, biological relevance and properties of these domains in vivo are still unclear. However the answers will shape our views of signaling and membrane dynamics.     

Is there chaos in plankton dynamics?

A controversial issue in ecosystem modeling is whether the irregularfluctuations that one observes in nature are due solely to randomenvironmental factors or whether, at least partially, a deterministicmechanism is responsible for the unpredictable behavior. Thissecond alternative is called deterministic chaos and the issuein this paper is to decide if actual plankton time series canvindicate the hypothesis of chaotic dynamics. The near-neighborforecasting method is a recent technique for detecting determinismin a time series and we apply it to measurements of phytoplanktonand zooplankton biomass obtained at a single station in theMiddle Atlantic Bight. Although the results do not concludethe presence of chaos, they do give some support to the ideathat deterministic non-linear trophic dynamics may account forat least some of the variability that is seen in the data, particularlyin terms of inferring zooplankton oscillations from those ofphytoplankton.     

Ontoecogenophyloconstraints? The chaos of constraint terminology

The world of evolutionary biology today is being bombarded with all kinds of possible constraints to the process of natural selection. Are we witnessing the end of the neodarwinistic theory of evolution, as some may like to see it, or is it just another whim of giving new names to old things? Here, we attempt to unravel the meaning and name-giving of constraints in a small and nonrandom sample of the literature, and suggest a way out from the present confusion of usages.     

Missing chaos in global climate change data interpreting?

The main problem of ecological data modeling is their interpretation and its correct understanding. This problem cannot be solved solely by a big data collection. To sufficiently understand ecosystems we need to know how these processes behave and how they respond to internal and external factors. Similarly, we need to know the behavior of processes that are involved in the climate system and the biosphere of the earth. In order to characterize precisely the behavior of individual elements and ecosystems we need to use deterministic, stochastic and chaotic behavior. Unfortunately, the chaotic part of systems is typically completely ignored in almost all approaches. Ignoring of chaotical part leads to many biased outcomes. To overcome this gap we model chaotic system behavior by random iterated function system which provides a generic guideline for such data management. This also allows to replicate a complexity and chaos of ecosystem.     

Do birds differentiate between white noise and deterministic chaos?

Noisy, unpredictable sounds are often present in the vocalizations of fearful and stressed animals across many taxa. A variety of structural characteristics, called nonlinear acoustic phenomena, that include subharmonics, rapid frequency modulations, and deterministic chaos are responsible for the harsh sound quality of these vocalizations. Exposure to nonlinear sound can elicit increased arousal in birds and mammals. Past experiments have used white noise to test for effects of deterministic chaos on perceivers. However, deterministic chaos differs structurally from white noise (i.e., random signal with equal energy at all frequencies), and unlike white noise, may differ dramatically depending on how it is produced. In addition, the subtle structural variation of chaos may not be distinguishable in the environment due to the attenuation and degradation of sound over distance and different habitat types. We designed two experiments to clarify whether American robins (Turdus migratorius) and warbling vireos (Vireo gilvus) discriminate between white noise and deterministic chaos. We broadcast and re‐recorded white noise and two exemplars of deterministic chaos—one generated with a Chua oscillator and the other generated using a logistic equation—at 1, 10, 20, 30, 40, and 80 m across open and forested habitat and used spectrogram correlations to compare stimuli along this degradational gradient. We found that sounds degraded similarly in both habitats when compared to a reference distance of 1 m. Comparing pairs of stimuli across distances suggested that Chua chaos was more easily distinguishable from noise and logistic chaos. In addition, all stimuli became more distinctive over increased distance. The second experiment tested behavioral responses of robins and warbling vireos to control sounds of tropical kingbird (Quiscalus mexicanus), white noise, and two exemplars of deterministic chaos (Chua and logistic). Neither American robins nor warbling vireos responded differently to at least two types of deterministic chaos and white noise, validating previous playback studies that used white noise as a surrogate for deterministic chaos. Uniform responses to a variety of nonlinear features in these birds possibly reflect error management in alarm signal detection.     

Evolution and diversity in the legume-rhizobium symbiosis: chaos theory?

This paper examines the general biology of mycorrhizal associations alongside the wide range of alternative trophic adaptations which higher plants may employ when competing for limited resources of specific nutrients within an ecosystem. All examples described come from highly nutrient-impoverished heathlands or open woodlands of the kwongan of southwest Australia. An account is given of the general patterns of rooting morphology and their association with various mycorrhizal and non-mycorrhizal nutrient-acquiring strategies, including various forms of parasitism, epiparasitism, autotrophy with or without mycorrhizal association. Taxonomic affinities of each grouping are examined alongside growth and life form characteristics. A case study of patterns of utilization of a specific nutrient, nitrogen in a Banksia woodland ecosystem is presented to illustrate how a multifaceted approach can be used for studying species responses and interactions. The study categorizes species according to nitrate-utilizing ability and suggests how ¹⁵N natural abundance of soil and plant components and organic solutes of nitrogen is xylem might be utilized to separate species into different trophic categories. Response of the ecosystem to fire is examined in respect of the nutritional interrelationships of component species as the ecosystem changes from being nitrate dominant immediately after fire to increasingly ammonium-producing thereafter. The paper concludes by examining generally trophic relationships within whole ecosystems and outlines some of the challenges for future research in this connection.     

Dissemination of spores in a glasshouse: Pattern or chaos?

The dissemination of particles in a glasshouse with three united compartments was observed from experiments withLycopodium sp. The experiments were performed in a glasshouse with open and closed ventilation openings, spore sources on two heights above soil surface, and in two consecutive time intervals. The spores were caught with self constructed spore traps placed at regular distances from each other. Air circulation was observed using smoke-puffs. High ventilation caused a rapid cleaning of the air while in a closed glasshouse spores remained suspended for quite a long time. Though the three compartments of the glasshouse were not separated by walls, a so-called cell structure of the individual compartments could be indicated.     

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Mechanisms and scenarios of pattern formation in predator–prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of ‘traditional’ predator–prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator–prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator–prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing–Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing–Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.     

Quantitative analysis of electroencephalograms: is there chaos in the future?

The history of quantitative, computerized electroencephalogram (EEG) analysis is reviewed. It is shown that, until very recently, the basic approach to EEG analysis involved the assumption that the EEG is stochastic. Consequently, statistical pattern recognition techniques, segmentation procedures, syntactic methods, knowledge-based approaches, and even artificial neural network methods have been developed with different levels of success. A fundamentally different approach to computerized EEG analysis, however, is making its way into the laboratories. The basic idea, inspired by recent advances in the area of non-linear dynamics, and especially the theory of chaos, is to view an EEG as the output of a deterministic system of relatively simple complexity, but containing non-linearities. This suggests that studying the geometrical dynamics of EEGs, and the development of neurophysiologically realistic models of EEG generation may produce more successful automated EEG analysis techniques than the classical, stochastic methods. Evidence supporting the non-linear dynamics paradigm is reviewed, and possible research paths are indicated.     

Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?

A constant dilemma in theoretical ecology is knowing whether model predictions corrspond to real phenomena or whether they are artifacts of the modelling framework. The frequent absence of detailed ecological data against which models can be tested gives this issue particular importance. We address this question in the specific case of invasion in a predator-prey system with oscillatory population kinetics, in which both species exhibit local random movement. Given only these two basic qualitative features, we consider whether we can deduce any properties of the behaviour following invasion. To do this we study four different types of mathematical model, which have no formal relationship, but which all reflect our two qualitative ingredients. The models are: reaction-diffusion equations, coupled map lattices, deterministic cellular automata, and integrodifference equations. We present results of numerical simulations of the invasion of prey by predators for each model, and show that although there are certain differences, the main qualitative features of the behaviour behind invasion are the same for all the models. Specifically, there are either irregular spatiotemporal oscillations behind the invasion, or regular spatiotemporal oscillations with the form of a periodic travelling ''wake'', depending on parameter values. The observation of this behaviour in all types of model strongly suggests that it is a direct consequence of our basic qualitative assumptions, and as such is an ecological reality which will always occur behind invasion in actual oscillatory predator-prey systems.     

The World-Wide Web information system: chemical chaos or global scientific enabler?

A proposed Internet based standard based on primary chemical MIME types used in conjunction with the World-Wide Web information delivery systems allows molecular structural and spectroscopic information to be transparently integrated into scientific publications, providing unparalleled access and control for the user. Examples of “hyperactive molecules” from the areas of organic synthesis, quantitative structural modelling, molecular dynamics, crystallography, NMR and mass spectrometry are discussed. The implications of these mechanisms for scientific publication in general, the impact on the quality and reproducibility of published experimental data and the enhancement of serendipitous discoveries are discussed.     

Mendelian segregation: a choice between “order” and “chaos”

Research supported in part by a US-Israel Binational Science Foundation grant 85-00021 and by NIH grants GM 28016 and GM 10452     

Immune network behavior—II. From oscillations to chaos and stationary states

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.     

Phylogenetic distribution of catalase-peroxidases: are there patches of order in chaos?

Hydrogen peroxide features in many biological oxidative processes and must be continuously degraded enzymatically either via a catalatic or a peroxidatic mechanism. For this purpose ancestral bacteria evolved a battery of different heme and non-heme enzymes, among which heme-containing catalase-peroxidases (CP) are one of the most widespread representatives. They are unique since they can follow both H(2)O(2)-degrading mechanisms, the catalase activity being clearly dominant. With the fast increasing amount of genomic data available, we were able to perform an extensive search for CP and found almost 300 sequences covering a large range of microorganisms. Most of them were encoded by bacterial genomes, but we could also find some in eukaryotic organisms other than fungi, which has never been shown until now. Our screen also reveals that approximately 60% of the bacteria do not possess CP genes. Chaotic distribution among species and incongruous phylogenetic reconstruction indicated existence of numerous lateral gene transfers in addition to duplication events and regular speciation. The results obtained show an impressively complex gene transmission pattern, and give some new insights about the role of CP and the origin of life on earth. Finally, we propose for the first time bacterial candidates that may have participated in the transfer of CP from bacteria to eukaryotes.     

Evolution of genome size: multilevel selection, mutation bias or dynamical chaos?

In the past two years, new data on conceptual aspects of the evolution of eukaryotic genome size have appeared, including the adaptivity of genome enlargement, the mechanisms of genome size change and the relation of genome size to organismal complexity. New data on the hypotheses of "selfish DNA" and "mutational equilibrium" have been recently obtained. A relationship is emerging between the intragenomic distribution of noncoding DNA and differential gene expression, which suggests that noncoding DNA is involved in epigenetic organization of the genome and organismal complexity. The standpoint of dynamical chaos, which integrates multilevel selection and mutation biases, may provide a framework for studying the evolution of genome size.     

Time-dependent regimes of a tourism-based social–ecological system: Period-doubling route to chaos

The period-doubling route to chaos has occupied a prominent position and it is still object of great interest among the different complex phenomena observed in nonlinear dynamical systems. The reason of such interest is that such route to chaos has been observed in many physical, chemical and ecological models when they change over from simple periodic to complex aperiodic motion. In interlinked social–ecological systems (SESs) there might be an apparent great ability to cope with change and adapt if analysed only in their social dimension. However, such an adaptation may be at the expense of changes in the capacity of ecosystems to sustain the adaptation and it could affect the quality of ecosystem goods and services since it could degrade natural renewable and non-renewable resources and generate traps and breakpoints in the whole SES eventually leading to chaotic behaviour. This paper is rooted in previous results on modelling tourism-based SESs, only recently object of theoretical investigations, focusing on the dynamics of the coexistence between mass-tourists and eco-tourists. Here we describe a finer scale analysis of time-dependent regimes in the ranges of the degradation coefficient (bifurcation parameter), for which the system can exhibit coexistence. This bifurcation parameter is determined by objective changes in the real world in the quality of ecosystem goods and services together with whether and how such changes are perceived by different tourist typologies. Varying the bifurcation parameter, the dynamical system may in fact evolve toward an aperiodical dynamical state in many ways, showing that there could be different scenarios for the transition to chaos. This paper provides a further evidence for the period-doubling route to chaos with reference to tourism-based socio-ecological models, and for a period locking behaviour, where a small variation in the bifurcation parameter can lead to alternating regular and chaotic dynamics. Moreover, for many models undergoing chaos via period-doubling, it has been showed that structural perturbations with real ecological justification, may break and reverse the expected period-doublings, hence inhibiting chaos. This feature may be of a certain relevance also in the context of adaptive management of tourism-based SESs: these period-doubling reversals might in fact be used to control chaos, since they potentially act in way to suppress possibly dangerous fluctuations.