Discrete and continuous state population models in a noisy world

Simple ecological models operate mostly with population densities using continuous variables. However, in reality densities could not change continuously, since the population itself consists of integer numbers of individuals. At first sight this discrepancy appears to be irrelevant, nevertheless, it can cause large deviations between the actual statistical behaviour of biological populations and that predicted by the corresponding models. We investigate the conditions under which simple models, operating with continuous numbers of individuals can be used to approximate the dynamics of populations consisting of integer numbers of individuals. Based on our definition for the (statistical) distance between the two models we show that the continuous approach is acceptable as long as sufficiently high biological noise is present, or, the dynamical behaviour is regular (non-chaotic). The concepts are illustrated with the Ricker model and tested on the Tribolium castaneum data series. Further, we demonstrate with the help of T. castaneum's model that if time series are not much larger than the possible population states (as in this practical case) the noisy discrete and continuous models can behave temporarily differently, almost independently of the noise level. In this case the noisy, discrete model is more accurate [OR has to be applied].     

Deterministic chaоs and the problem of predictability in population dynamics

The studies of the processes that can significantly influence the predictability in population dynamics are reviewed and the results of mathematical simulations of population dynamics are compared to the time series obtained in field observations. Considerable attention is given to the chaotic changes in population abundance. Some methods of numerical analysis of chaoticity and predictability of the time series are considered. The importance of comparing the results of mathematical simulation and observation data is tightly linked to problems in detecting chaos in the dynamics of natural populations and estimating the prevalence of chaotic regimes in nature. Insight into these problems can allow identification of the functional role of chaotic regimes in population dynamics.     

Structural perturbations to population skeletons: transient dynamics, coexistence of attractors and the rarity of chaos

BACKGROUND: Simple models of insect populations with non-overlapping generations have been instrumental in understanding the mechanisms behind population cycles, including wild (chaotic) fluctuations. The presence of deterministic chaos in natural populations, however, has never been unequivocally accepted. Recently, it has been proposed that the application of chaos control theory can be useful in unravelling the complexity observed in real population data. This approach is based on structural perturbations to simple population models (population skeletons). The mechanism behind such perturbations to control chaotic dynamics thus far is model dependent and constant (in size and direction) through time. In addition, the outcome of such structurally perturbed models is [almost] always equilibrium type, which fails to commensurate with the patterns observed in population data. METHODOLOGY/PRINCIPAL FINDINGS: We present a proportional feedback mechanism that is independent of model formulation and capable of perturbing population skeletons in an evolutionary way, as opposed to requiring constant feedbacks. We observe the same repertoire of patterns, from equilibrium states to non-chaotic aperiodic oscillations to chaotic behaviour, across different population models, in agreement with observations in real population data. Model outputs also indicate the existence of multiple attractors in some parameter regimes and this coexistence is found to depend on initial population densities or the duration of transient dynamics. Our results suggest that such a feedback mechanism may enable a better understanding of the regulatory processes in natural populations.     

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.     

Control of neural chaos by synaptic noise

We study neural automata - or neurobiologically inspired cellular automata - which exhibits chaotic itinerancy among the different stored patterns or memories. This is a consequence of activity-dependent synaptic fluctuations, which continuously destabilize the attractor and induce irregular hopping to other possible attractors. The nature of these irregularities depends on the dynamic details, namely, on the intensity of the synaptic noise and the number of sites of the network, which are synchronously updated at each time step. Varying these factors, different regimes occur, ranging from regular to chaotic dynamics. As a result, and in absence of external agents, the chaotic behavior may turn regular after tuning the noise intensity. It is argued that a similar mechanism might be on the basis of self-controlling chaos in natural systems.     

Intraseasonal predictability of natural phytoplankton population dynamics

It is difficult to make skillful predictions about the future dynamics of marine phytoplankton populations. Here, we use a 22‐year time series of monthly average abundances for 198 phytoplankton taxa from Station L4 in the Western English Channel (1992–2014) to test whether and how aggregating phytoplankton into multi‐species assemblages can improve predictability of their temporal dynamics. Using a non‐parametric framework to assess predictability, we demonstrate that the prediction skill is significantly affected by how species data are grouped into assemblages, the presence of noise, and stochastic behavior within species. Overall, we find that predictability one month into the future increases when species are aggregated together into assemblages with more species, compared with the predictability of individual taxa. However, predictability within dinoflagellates and larger phytoplankton (>12 μm cell radius) is low overall and does not increase by aggregating similar species together. High variability in the data, due to observational error (noise) or stochasticity in population growth rates, reduces the predictability of individual species more than the predictability of assemblages. These findings show that there is greater potential for univariate prediction of species assemblages or whole‐community metrics, such as total chlorophyll or biomass, than for the individual dynamics of phytoplankton species.     

Red environmental noise and the appearance of delayed density dependence in age-structured populations

Previous work suggests that red environmental noise can lead to the spurious appearance of delayed density dependence (DDD) in unstructured populations regulated only by direct density dependence. We analysed the effect of noise reddening on the pattern of spurious DDD in several variants of the density-dependent age-structured population model. We found patterns of spurious DDD in structured populations with either density-dependent fertility or density-dependent survival of the first age class, inconsistent with predictions from unstructured population models. Moreover, we found that nonspurious negative DDD always emerges in populations with deterministic chaotic dynamics, regardless of population structure or the type of environmental noise. The effect of noise reddening in generating spurious DDD is often negligible in the chaotic region of population deterministic dynamics. Our findings suggest that differences in species' life histories may exhibit different patterns of spurious DDD (owing to noise reddening) than predicted by unstructured models.     

The Effect of Loss of Immunity on Noise-Induced Sustained Oscillations in Epidemics

The effect of loss of immunity on sustained population oscillations about an endemic equilibrium is studied via a multiple scales analysis of a SIRS model. The analysis captures the key elements supporting the nearly regular oscillations of the infected and susceptible populations, namely, the interaction of the deterministic and stochastic dynamics together with the separation of time scales of the damping and the period of these oscillations. The derivation of a nonlinear stochastic amplitude equation describing the envelope of the oscillations yields two criteria providing explicit parameter ranges where they can be observed. These conditions are similar to those found for other applications in the context of coherence resonance, in which noise drives nearly regular oscillations in a system that is quiescent without noise. In this context the criteria indicate how loss of immunity and other factors can lead to a significant increase in the parameter range for prevalence of the sustained oscillations, without any external driving forces. Comparison of the power spectral densities of the full model and the approximation confirms that the multiple scales analysis captures nonlinear features of the oscillations.     

Respiratory influences on non-linear dynamics of heart rate variability in humans

 The goal of our study was to determine whether evidence for chaos in heart rate variability (HRV) can be observed when the respiratory input to the autonomic controller of heart rate is forced by the deterministic pattern associated with periodic breathing. We simultaneously recorded, in supine healthy volunteers, RR intervals and breathing volumes for 20 to 30 min (1024 data point series) during three protocols: rest (control), fixed breathing (15 breath/min) and voluntary periodic breathing (3 breaths with 2 s inspiration and 2 s expiration followed by an 8 s breath hold). On both the RR interval and breathing volume series we applied the non-linear prediction method (Sugihara and May algorithm) to the original time series and to distribution-conserved isospectral surrogate data. Our results showed that, in contrast to the control test, during both fixed and voluntary periodic breathing the variability of breathing volumes was clearly deterministic non-chaotic. During all the three protocols, the RR-interval series’ non-linear predictability was consistent with one of a chaotic series. However, at rest, no clear difference was observed between the RR-interval series and their surrogates, which means that no clear chaos was observed. During fixed breathing a difference appeared, and this difference seemed clearer during voluntary periodic breathing. We concluded that HRV dynamics were chaotic when respiration was forced with a deterministic non-chaotic pattern and that normal spontaneous respiratory influences might mask the normally chaotic pattern in HRV. Received: 7 August 1995 / Accepted in revised form: 20 March 1997     

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.     

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.     

A bit of sex stabilizes host-parasite dynamics

To date, only a few studies have focused on the effects of sex on population dynamics. Previous models have typically found that sexual reproduction dampens population fluctuations. Although asexual and sexual reproduction are just the two endpoints along a continuum of varying rates of sex, previous work has ignored the effects of intermediate degrees of sex on population dynamics. Here we study the effects of partial sexual reproduction (i.e. sex occurs only every few generations or with small probability in each generation) on the coupled population dynamics of a Nicholson-Bailey host-parasite model. We show that complex dynamics are simplified for high host population growth rates if the frequency of sex is sufficiently high in both host and parasite: sex decreases fluctuations in population density, and leads to non-chaotic dynamics for population growth rates that would result in chaotic dynamics in the absence of sexual reproduction. However, the simplification does not increase gradually with an increasing frequency of sex but appears abruptly at low-to-intermediate frequencies of sex. For some parameter settings, intermediate frequencies of sexual reproduction can simplify the dynamics more than lower or higher frequencies. Thus, in agreement with earlier results, sexual reproduction typically stabilizes complex population dynamics in our models. Additionally, our results suggest that low-to-intermediate frequencies of sex may often be as (or even more) stabilizing as high frequencies.     

Comparison of newtonian and special-relativistic trajectories with the general-relativistic trajectory for a low-speed weak-gravity system

We show, contrary to expectation, that the trajectory predicted by general-relativistic mechanics for a low-speed weak-gravity system is not always well-approximated by the trajectories predicted by special-relativistic and newtonian mechanics for the same parameters and initial conditions. If the system is dissipative, the breakdown of agreement occurs for chaotic trajectories only. If the system is non-dissipative, the breakdown of agreement occurs for chaotic trajectories and non-chaotic trajectories. The agreement breaks down slowly for non-chaotic trajectories but rapidly for chaotic trajectories. When the predictions are different, general-relativistic mechanics must therefore be used, instead of special-relativistic mechanics (newtonian mechanics), to correctly study the dynamics of a weak-gravity system (a low-speed weak-gravity system).     

Synchrony of spatial populations induced by colored environmental noise and dispersal

Spatial synchrony of oscillating populations has been observed in many ecological systems, and its influences and causes have attracted the interest of ecologists. Spatially correlated environmental noises, dispersal, and trophic interactions have been considered as the causes of spatial synchrony. In this study, we develop a spatially structured population model, which is described by coupled-map lattices and incorporates both dispersal and colored environmental noise. A method for generating time series with desired spatial correlation and color is introduced. Then, we use these generated time series to analyze the influence of noise color on synchrony in population dynamics. The noise color refers to the temporal correlation in the time series data of the noise, and is expressed as the degree of (first-order) autocorrelation for autoregressive noise. Patterns of spatial synchrony are considered for stable, periodic and chaotic population dynamics. Numerical simulations verify that environmental noise color has a major influence on the level of synchrony, which depends strongly on how noise is introduced into the model. Furthermore, the influence of noise color also depends on patterns of dispersal between local populations. In addition, the desynchronizing effect of reddened noise is always weaker than that of white noise. From our results, we notice that the role of reddened environmental noise on spatial synchrony should be treated carefully and cautiously, especially for the spatially structured populations linked by dispersal.     

The effect of seasonal harvesting on stage-structured population models

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose an exploited single-species model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Since birth pulse populations are often characterized with a discrete time dynamical system determined by its Poincaré map, we explore the consequences of harvest timing to equilibrium population sizes under seasonal dependence and obtain threshold conditions for their stability, and show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. Moreover, our results imply that the population can sustain much higher harvest rates if the mature fish is removed as early in the season (after the birth pulse) as possible. Further, the effects of harvesting effort and harvest timing on the dynamical complexity are also investigated. Bifurcation diagrams are constructed with the birth rate (or harvesting effort or harvest timing) as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, non-unique dynamics (meaning that several attractors coexist) and attractor crisis. This suggests that birth pulse, in effect, provides a natural period or cyclicity that makes the dynamical behavior more complex.This work is supported by National Natural Science Foundation of China (10171106)     

Nonlinear time-series modeling of vole population fluctuations

A central goal of population ecology is to understand and predict fluctuations in population numbers. Until recently, much of the debate focused on the issue of population regulation by density-dependent factors. In this paper, I describe an approach to nonlinear modeling of time-series data that is designed to go beyond this question by investigating the possibility of complex population dynamics, characterized by lags in regulation and periodic or chaotic oscillations. The questions motivating this approach are: what are relative contributions of endogenous vs. exogenous components of dynamics? Is the irregular component in fluctuations entirely due to exogenous noise, or do nonlinearities contribute to it, too? I describe the philosophy and the technical details of the nonlinear modeling approach, and then apply it to a collection of time-series data on vole population fluctuations in northern Europe. The results suggest that population dynamics of European voles undergo a latitudinal shift from stability to chaos. Dynamics in northern Fennoscandia are characterized by positive Lyapunov exponent estimates, and a high degree of short-term (one year ahead) predictability, suggesting a strong endogenous component. In more southerly populations estimated Lyapunov exponents are negative, and there is no one-step ahead predictability, suggesting that fluctuations are driven by exogenous factors.     

Nonlinear population dynamics: complication in the age structure influences transition-to-chaos scenarios

The work continues a series of studies on the evolution of a natural population of explicitly seasonal organisms. Model analyses have revealed relationships between the duration of ontogenesis and the pattern of temporal dynamics in size of an isolated population (i.e., the structure and dimensionality of the chaotic attractors). For nonlinear models of age-structured population dynamics (under long-lasting ontogenesis), increase in the reproductive potential is shown to result in the chaotic attractors whose structure and dimensionality changes in response to variations in the model parameters. When the ontogenesis becomes longer and more complicated, it does not, "on the average", augment the level of chaos in the attractors observed. There are wide enough regions in the space of the birth and death parameter values that provide for windows in the chaotic dynamics where the total or partial regularization occurs.     

Spatial synchrony in population fluctuations: extending the Moran theorem to cope with spatially heterogeneous dynamics

The recent interest in the spatial structure and dynamics of populations motivated numerous theoretical and empirical studies of spatial synchrony, the tendency of populations to fluctuate in unison over regional areas. The first comprehensive framework applied to spatial synchrony was probably the one elaborated by P. A. P. Moran back in 1953. He suggested that if two populations have the same linear density-dependent structure, the correlation between them will be equal to that between the local density-independent conditions. Surprisingly, the consequences of violating the assumption that the dynamics of the populations are identical has received little attention. In this paper, making the assumption that population dynamics can be described by linear and stationary autoregressive processes, I show that the observed spatial synchrony between two populations can be decomposed into two multiplicative components: the demographic component depending on the values of the autoregressive coefficients, and the correlation of the environmental noise. The Moran theorem corresponds to the special case where the demographic component equals unity. Using published data, I show that the spatial variability in population dynamics may substantially contribute to the spatial variability of population synchrony, and thus should not be neglected in future studies.     

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.