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.     

Numerically evaluated functional equivalence between chaotic dynamics in neural networks and cellular automata under totalistic rules

Chaotic dynamics in a recurrent neural network model and in two-dimensional cellular automata, where both have finite but large degrees of freedom, are investigated from the viewpoint of harnessing chaos and are applied to motion control to indicate that both have potential capabilities for complex function control by simple rule(s). An important point is that chaotic dynamics generated in these two systems give us autonomous complex pattern dynamics itinerating through intermediate state points between embedded patterns (attractors) in high-dimensional state space. An application of these chaotic dynamics to complex controlling is proposed based on an idea that with the use of simple adaptive switching between a weakly chaotic regime and a strongly chaotic regime, complex problems can be solved. As an actual example, a two-dimensional maze, where it should be noted that the spatial structure of the maze is one of typical ill-posed problems, is solved with the use of chaos in both systems. Our computer simulations show that the success rate over 300 trials is much better, at least, than that of a random number generator. Our functional simulations indicate that both systems are almost equivalent from the viewpoint of functional aspects based on our idea, harnessing of chaos.     

When can noise induce chaos and why does it matter: a critique

Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ₀ that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.     

Time delay as a key factor of model plankton dynamics

Studies of the mechanisms underlying complex dynamics of ecological systems at various spatial and time scales bring increasing awareness that complexity is an intrinsic feature of ecological functioning. This paper is to investigate the role of such an ecologically significant parameter as the time delay due to maturation processes in the complex plankton dynamics. We show that the time lag T1, associated with the zooplankton maturation period can lead to essential changes in the plankton dynamics. Particularly, we show that the coexistence of limit cycle and chaotic attractor we have recently found to be typical of the system at T1 = 0 [A.B. Medvinsky, I.A. Tikhonova, R.R. Aliev, B.-L. Li, Z.-S. Lin, H. Malchow, Patchy environment as a factor of complex plankton dynamics, Phys. Rev. E 64 (2001) 021915] is replaced by pure chaotic plankton dynamics as T1 becomes more than a critical value. The results obtained imply that chaos is a rather common phenomenon in the plankton functioning.     

Hysteresis Dynamics,Bursting Oscillations and Evolution to Chaotic Regimes

This article describes new aspects of hysteresis dynamics which have been uncovered through computer experiments. There are several motivations to be interested in fast-slow dynamics. For instance, many physiological or biological systems display different time scales. The bursting oscillations which can be observed in neurons, beta-cells of the pancreas and population dynamics are essentially studied via bifurcation theory and analysis of fast-slow systems (Keener and Sneyd, 1998; Rinzel, 1987). Hysteresis is a possible mechanism to generate bursting oscillations. A first part of this article presents the computer techniques (the dotted-phase portrait, the bifurcation of the fast dynamics and the wave form) we have used to represent several patterns specific to hysteresis dynamics. This framework yields a natural generalization to the notion of bursting oscillations where, for instance, the active phase is chaotic and alternates with a quiescent phase. In a second part of the article, we emphasize the evolution to chaos which is often associated with bursting oscillations on the specific example of the Hindmarsh-Rose system. This evolution to chaos has already been studied with classical tools of dynamical systems but we give here numerical evidence on hysteresis dynamics and on some aspects of the wave form. The analytical proofs will be given elsewhere.     

Dynamic complexities in host-parasitoid interaction

In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of self-similarity and fractal basin boundaries), (c) intermittency, (d) supertransients, (e) chaotic attractors, and (f) "transient chaos". Because of these complexities minor changes in parameter or initial values may strikingly change the dynamic behavior of the system. All the phenomena presented in this paper should be kept in mind when examining and interpreting the dynamics of ecological systems. Copyright 1999 Academic Press.     

Synchrony &; chaos in patchy ecosystems

The apparent synchronisation of spatially discrete populations is a well documented phenomenon. However, it is not clear what the governing mechanisms are for this synchrony, and whether they are robust over a range of environmental conditions and patch specific population dynamic behaviours. In this paper, we explore two (possibly interacting) modes of coupling, and investigate their theoretically discernible, and perhaps even experimentally measurable, signatures. To aid us in this investigation we employ a planktonic example system, with direct application to plankton patchiness. Furthermore, we address the role of chaos in complex spatio-temporal dynamics; we find that chaos associated with funnel attractors can play a distinguished role, over dynamics less sensitive to small variations, in being more susceptible to generalised synchronisation (such as phase synchronisation) in the presence of small local parameter variation. This is in contrast to the case for coupled systems with identical dynamics, and suggests that non-identically coupled systems are more vulnerable to global extinction events when exhibiting funnel-type chaotic dynamics.     

Chaos in a periodically forced chemostat with algal mortality

We study the possibility of chaotic dynamics in the externally driven Droop model. This model describes a phytoplankton population in a chemostat under periodic nutrient supply. Previously, it has been proven under very general assumptions, that such systems are not able to exhibit chaotic dynamics. We show that the simple introduction of algal mortality may lead to chaotic oscillations of algal density in the forced chemostat. Our numerical simulations show that the existence of chaos is intimately related to plankton overshooting in the unforced model. We provide a simple measure, based on stability analysis, for estimating the amount of overshooting. These findings are not restricted to the Droop model but also hold for other chemostat models with mortality. Our results suggest periodically driven chemostats as a simple model system for the experimental verification of chaos in ecology.     

Rotifer Population Dynamics in Two Coupled Habitats: Invasion of Chaos

We study the role of interactions between habitats in rotifer dynamics. We use a simple discrete-time model to simulate the interactions between neighboring habitats with different intrinsic dynamics. Being uncoupled, one habitat shows periodical oscillations of the rotifer biomass while the other one demonstrates chaotic oscillations. As a result of the exchange of rotifer biomass, chaos replaces regular oscillations. As a result, the rotifer dynamics becomes chaotic in both habitats. We show that the invasion of chaos is followed by the synchronization of the chaotic regimes of both habitats, and this synchronization increases as coupling between the habitats is increased. We also demonstrate that the biological invasion of the rotifer species, which show chaotic dynamics, to a neighboring habitat with intrinsically regular plankton dynamics leads to the invasion of chaos and the synchronization of chaotic oscillations of the plankton biomass in both the habitats.     

Chaos and the (un)predictability of evolution in a changing environment

Among the factors that may reduce the predictability of evolution, chaos, characterized by a strong dependence on initial conditions, has received much less attention than randomness due to genetic drift or environmental stochasticity. It was recently shown that chaos in phenotypic evolution arises commonly under frequency‐dependent selection caused by competitive interactions mediated by many traits. This result has been used to argue that chaos should often make evolutionary dynamics unpredictable. However, populations also evolve largely in response to external changing environments, and such environmental forcing is likely to influence the outcome of evolution in systems prone to chaos. We investigate how a changing environment causing oscillations of an optimal phenotype interacts with the internal dynamics of an eco‐evolutionary system that would be chaotic in a constant environment. We show that strong environmental forcing can improve the predictability of evolution by reducing the probability of chaos arising, and by dampening the magnitude of chaotic oscillations. In contrast, weak forcing can increase the probability of chaos, but it also causes evolutionary trajectories to track the environment more closely. Overall, our results indicate that, although chaos may occur in evolution, it does not necessarily undermine its predictability.     

Only noise can induce chaos in discrete populations

Motivated by the papers from Ellner and Turchin 2005 and Dennis et al. 2003 we investigate the possibility to detect chaos in noisy ecological systems. One message of our paper is that if a dynamic model is available and if this model predicts chaotic behaviour, one should consider its discrete-state, noisy version when fitting numerical predictions to observations. We emphasize that deterministic discrete-state models behave periodically, thus only the interaction of these deterministic skeletons with random noise can produce non-regular dynamics. We detect and describe a relatively sharply defined range of the noise (the grey zone) where the gradual transition from periodic to chaotic behaviour happens. This zone, the upper border of which can be predicted analytically, is identified in experimental data as well as in numerical simulations. In the grey zone the global, statistical behaviour will approach the statistics produced by the chaotic, continuous model, and in this sense we claim that noise can produce chaos.     

Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.     

Transport properties of rigid and flexible macromolecules by brownian dynamics simulation

The method of Ermak and McCammon [Ermak, D. L. & McCammon, J. A. (1978) J. Chem. Phys. 69 , 1352–1360] is used to simulate the Brownian dynamics of a system of identical, interacting beads. In the present study, we use the method to obtain transport coefficients for a variety of rigid and flexible structures modeled as arrays of spherical subunits. Constraints are enforced using the SHAKE algorithm or a modification, SHAKE-HI, that is described for the first time. In SHAKE-HI, hydrodynamic interactions between subunits are taken into account when the constraints are enforced. Use of SHAKE-HI yields transport coefficients that are in perfect agreement with those obtained by other methods. The primary advantage of the present method is its generality. We also propose that multistep Brownian dynamics may be important in simulating actual experiments (such as fluorescence depolarization) on well-defined model systems that possess an arbitrary degree of internal flexibility.     

Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.     

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.     

马尾松毛虫种群动态的时间序列分析及复杂性动态研究

自从May(1974)指出即使是简单的种群模型也能揭示混沌动态以来,自然种群是否存在混沌一直具有争论,如何检测自然种群的混沌行为也成为种群动态研究的一个难点,通过时间序列分析和反应面模型建模的8方法分析了马尾松毛虫的复杂性动态,用自相关函数对马尾松毛虫发生的时间动态分析的结果认为动态是平衡的,其周期性不显著,而具有一定的复杂性,这种类型可以是减幅波动,有限周期或弱混沌,波动主要由系统内因引起,进一步采用反应面模型估计全局李雅普若夫指数和局域李雅普若夫指数结果均为负,显示马尾松毛虫种群动态不存在混沌现象,但是在增加一个小的噪音以后,局域李雅普若夫指数变为在0以上的波动,说明系统对噪音非常敏感,噪音对松毛虫种群动态具有很大的影响,可以将其从非混沌状态变为混沌,研究结果认为全局郴雅普若夫指数λ是一定时间内两个变动轨迹的总平均偏差,而随着种群动态的波动,指数也是波动的,所以对于检测自然种群的混沌来说不是一个好的指标,局域李雅普若夫指数λM能更好地表示自然种群混沌的存在和产生混沌的条件,对害虫管理来说对种群暴发初期的预测是尤其重要的,而此时又最难于预测,所以对种群动态的监测就尤为重要,由于马尾松毛虫的代间种群动态为第一级密度相关,前一代的虫口密度与下一代的虫口密度相关性最强,所以前一代预测下一代是最可靠的。     

Long food chains are in general chaotic

The question whether chaos exists in nature is much debated. In this paper we prove that chaotic parameter regions exist generically in food chains of length greater than three. While nonchaotic dynamics is also possible, the presence of chaotic parameter regions indicates that chaotic dynamics is likely. We show that the chaotic regions survive even at high exponents of closure. Our results have been obtained using a general food chain model that describes a large class of different food chains. The existence of chaos in models of such generality can be deduced from the presence of certain bifurcations of higher codimension.     

Bifurcations and chaos in a predator-prey system with the Allee effect

It is known from many theoretical studies that ecological chaos may have numerous significant impacts on the population and community dynamics. Therefore, identification of the factors potentially enhancing or suppressing chaos is a challenging problem. In this paper, we show that chaos can be enhanced by the Allee effect. More specifically, we show by means of computer simulations that in a time-continuous predator-prey system with the Allee effect the temporal population oscillations can become chaotic even when the spatial distribution of the species remains regular. By contrast, in a similar system without the Allee effect, regular species distribution corresponds to periodic/quasi-periodic oscillations. We investigate the routes to chaos and show that in the spatially regular predator-prey system with the Allee effect, chaos appears as a result of series of period-doubling bifurcations. We also show that this system exhibits period-locking behaviour: a small variation of parameters can lead to alternating regular and chaotic dynamics.     

Why metabolic systems are rarely chaotic

One of the mysteries surrounding the phenomenon of chaos is that it can rarely be found in biological systems. This has led to many discussions of the possible presence and interpretation of chaos in biological signals. It has caused empirical biologists to be very sceptical of models that have chaotic properties or even employ chaos for problem solving tasks. In this paper, it is demonstrated that there exists a possible mechanism that is part of the catalytical reaction mechanisms which may be responsible for controlling enzymatic reactions such that they do not become chaotic. It is proposed that where these mechanisms are not present or not effective, chaos may still occur in biological systems.