Sturdy cycles in the chaotic Tribolium castaneum data series

Because of the inherent discreteness of individuals, population dynamical models must be discrete variable systems. In case of strong nonlinearity, such systems interacting with noise can generate a great variety of patterns from nearly periodic behavior through complex combination of nearly periodic and chaotic patterns to noisy chaotic time series. The interaction of a population consisting of discrete individuals and demographic noise has been analyzed in laboratory population data Henson et al. (Science 294 (2001) 602; Proc. Roy. Soc. Ser. B 270 (2003) 1549). In this paper we point out that some of the cycles are fragile, i.e. they are sensitive to the discretization algorithm and to small variation of the model parameters, while others remain "sturdy" against the perturbations. We introduce a statistical algorithm to detect disjoint, nearly-periodic patterns in data series. We show that only the sturdy cycles of the discrete variable models appear in the data series significantly. Our analysis identified the quasiperiodic 11-cycle (emerging in the continuous model) to be present significantly only in one of the three experimental data series. Numerical simulations confirm that cycles can be detected only if noise is smaller than a certain critical level and population dynamics display the largest variety of nearly-periodic patterns if they are on the border of "grey" and "noisy" regions, defined in Domokos and Scheuring (J. Theor. Biol. 227 (2004) 535).     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

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.     

Experimental support of the scaling rule for demographic stochasticity

A scaling rule of ecological theory, accepted but lacking experimental confirmation, is that the magnitude of fluctuations in population densities due to demographic stochasticity scales inversely with the square root of population numbers. This supposition is based on analyses of models exhibiting exponential growth or stable equilibria. Using two quantitative measures, we extend the scaling rule to situations in which population densities fluctuate due to nonlinear deterministic dynamics. These measures are applied to populations of the flour beetle Tribolium castaneum that display chaotic dynamics in both 20-g and 60-g habitats. Populations cultured in the larger habitat exhibit a clarification of the deterministic dynamics, which follows the inverse square root rule. Lattice effects, a deterministic phenomenon caused by the discrete nature of individuals, can cause deviations from the scaling rule when population numbers are small. The scaling rule is robust to the probability distribution used to model demographic variation among individuals.     

Comparing a discrete and continuum model of the intestinal crypt

The integration of processes at different scales is a key problem in the modelling of cell populations. Owing to increased computational resources and the accumulation of data at the cellular and subcellular scales, the use of discrete, cell-level models, which are typically solved using numerical simulations, has become prominent. One of the merits of this approach is that important biological factors, such as cell heterogeneity and noise, can be easily incorporated. However, it can be difficult to efficiently draw generalizations from the simulation results, as, often, many simulation runs are required to investigate model behaviour in typically large parameter spaces. In some cases, discrete cell-level models can be coarse-grained, yielding continuum models whose analysis can lead to the development of insight into the underlying simulations. In this paper we apply such an approach to the case of a discrete model of cell dynamics in the intestinal crypt. An analysis of the resulting continuum model demonstrates that there is a limited region of parameter space within which steady-state (and hence biologically realistic) solutions exist. Continuum model predictions show good agreement with corresponding results from the underlying simulations and experimental data taken from murine intestinal crypts.     

Large amplification in stage-structured models: Arnol’d tongues revisited

The coexistence of periodic and point attractors has been confirmed for a range of stage-structured discrete time models. The periodic attractor cycles have large amplitude, with the populations cycling between extremely low and surprisingly high values when compared to the equilibrium level. In this situation a stable state can be shocked by noise of sufficient strength into a state of high volatility. We found that the source of these large amplitude cycles are Arnold tongues, special regions of parameter space where the system exhibits periodic behaviour. Most of these tongues lie entirely in that part of parameter space where the system is unstable, but there are exceptions and these exceptions are the tongues that lead to attractor coexistence. Similarity in the geometry of Arnold tongues over the range of models considered might suggest that this is a common feature of stage-structured models but in the absence of proof this can only be a useful working hypothesis. The analysis shows that although large amplitude cycles might exist mathematically they might not be accessible biologically if biological constraints, such as non-negativity of population densities and vital rates, are imposed. Accessibility is found to be highly sensitive to model structure even though the mathematical structure is not. This highlights the danger of drawing biological conclusions from particular models. Having a comprehensive view of the different mechanisms by which periodic states can arise in families of discrete time models is important in the debate on whether the causes of periodicity in particular ecological systems are intrinsic, environmental or trophic. This paper is a contribution to that continuing debate.     

A general population-genetic model for the production by population structure of spurious genotype-phenotype associations in discrete, admixed or spatially distributed populations

In linkage disequilibrium mapping of genetic variants causally associated with phenotypes, spurious associations can potentially be generated by any of a variety of types of population structure. However, mathematical theory of the production of spurious associations has largely been restricted to population structure models that involve the sampling of individuals from a collection of discrete subpopulations. Here, we introduce a general model of spurious association in structured populations, appropriate whether the population structure involves discrete groups, admixture among such groups, or continuous variation across space. Under the assumptions of the model, we find that a single common principle--applicable to both the discrete and admixed settings as well as to spatial populations--gives a necessary and sufficient condition for the occurrence of spurious associations. Using a mathematical connection between the discrete and admixed cases, we show that in admixed populations, spurious associations are less severe than in corresponding mixtures of discrete subpopulations, especially when the variance of admixture across individuals is small. This observation, together with the results of simulations that examine the relative influences of various model parameters, has important implications for the design and analysis of genetic association studies in structured populations.     

Demographic window to aging in the wild: constructing life tables and estimating survival functions from marked individuals of unknown age

Summary We address the problem of establishing a survival schedule for wild populations. A demographic key identity is established, leading to a method whereby age-specific survival and mortality can be deduced from a marked cohort life table established for individuals that are randomly sampled at unknown age and marked, with subsequent recording of time-to-death. This identity permits the construction of life tables from data where the birth date of subjects is unknown. An analogous key identity is established for the continuous case in which the survival schedule of the wild population is related to the density of the survival distribution in the marked cohort. These identities are explored for both life tables and continuous lifetime data. For the continuous case, they are implemented with statistical methods using non-parametric density estimation methods to obtain flexible estimates for the unknown survival distribution of the wild population. The analytical model provided here serves as a starting point to develop more complex models for residual demography, i.e. models for estimating survival of wild populations in which age-at-entry is unknown and using remaining information in randomly encountered individuals. This is a first step towards a broad new concept of 'expressed demographic information content of marked or captured individuals'.     

Statistical interpretation of the interplay between noise and chaos in the stochastic logistic map

A recurring problem in population biology - as well as other stochastic dynamical systems in biology, the physical and social sciences - is the distinction between the ‘true’ dynamics of a system and observational noise: i.e. can we from present data reliably infer e.g. biological mechanisms, or are signals swamped by noise.Here, we approach this problem using the canonical model for simple systems that exhibit complex behaviour, the logistic map. At each time-point noise is added, which allows us to study the long-term behaviour of a system which exhibits both non-linear dynamics and intrinsic noise.We show that the interplay between deterministic non-linear dynamics and simple Gaussian noise results in a perplexingly simple system when viewed statistically.In particular we show that for the case of Gaussian noise it is possible to derive at very reliable approximations for the time until the system has reached an absorbing state. This generic model allows us, for example, to study the life-time of molecular species involved in noisy feedback loops.     

A statistical approach to quasi-extinction forecasting

Forecasting population decline to a certain critical threshold (the quasi-extinction risk) is one of the central objectives of population viability analysis (PVA), and such predictions figure prominently in the decisions of major conservation organizations. In this paper, we argue that accurate forecasting of a population's quasi-extinction risk does not necessarily require knowledge of the underlying biological mechanisms. Because of the stochastic and multiplicative nature of population growth, the ensemble behaviour of population trajectories converges to common statistical forms across a wide variety of stochastic population processes. This paper provides a theoretical basis for this argument. We show that the quasi-extinction surfaces of a variety of complex stochastic population processes (including age-structured, density-dependent and spatially structured populations) can be modelled by a simple stochastic approximation: the stochastic exponential growth process overlaid with Gaussian errors. Using simulated and real data, we show that this model can be estimated with 20-30 years of data and can provide relatively unbiased quasi-extinction risk with confidence intervals considerably smaller than (0,1). This was found to be true even for simulated data derived from some of the noisiest population processes (density-dependent feedback, species interactions and strong age-structure cycling). A key advantage of statistical models is that their parameters and the uncertainty of those parameters can be estimated from time series data using standard statistical methods. In contrast for most species of conservation concern, biologically realistic models must often be specified rather than estimated because of the limited data available for all the various parameters. Biologically realistic models will always have a prominent place in PVA for evaluating specific management options which affect a single segment of a population, a single demographic rate, or different geographic areas. However, for forecasting quasi-extinction risk, statistical models that are based on the convergent statistical properties of population processes offer many advantages over biologically realistic models.     

Extrapolating toxic effects on individuals to the population level: the role of dynamic energy budgets

The interest of environmental management is in the long-term health of populations and ecosystems. However, toxicity is usually assessed in short-term experiments with individuals. Modelling based on dynamic energy budget (DEB) theory aids the extraction of mechanistic information from the data, which in turn supports educated extrapolation to the population level. To illustrate the use of DEB models in this extrapolation, we analyse a dataset for life cycle toxicity of copper in the earthworm Dendrobaena octaedra. We compare four approaches for the analysis of the toxicity data: no model, a simple DEB model without reserves and maturation (the Kooijman-Metz formulation), a more complex one with static reserves and simplified maturation (as used in the DEBtox software) and a full-scale DEB model (DEB3) with explicit calculation of reserves and maturation. For the population prediction, we compare two simple demographic approaches (discrete time matrix model and continuous time Euler-Lotka equation). In our case, the difference between DEB approaches and population models turned out to be small. However, differences between DEB models increased when extrapolating to more field-relevant conditions. The DEB3 model allows for a completely consistent assessment of toxic effects and therefore greater confidence in extrapolating, but poses greater demands on the available data.     

Spatial coupling in cyclic population dynamics: models and data

We use a dynamic random field to model a spatial collection of coupled oscillators with discrete time stochastic dynamics. At each time step the phase of each cyclic local population is subject to random noise, incremented by a common dynamic, and pulled by a coupling force in the direction of some collective mean phase. We define asynchrony and derive expressions for its measurement in this model. We describe robust methods for phase estimation of cyclic population time series, for estimating strength of coupling between local populations, and for measuring variance of locally acting noise from field data. Proposed methods allow intermittently acting phase synchronizing events operating over large spatial scales to be distinguished from more continuous and possibly locally acting coupling, both of which could result in elevated levels of phase synchronization. We demonstrate the utility of this approach by applying it to classical spatial time series data of Canadian lynx. Analysis confirms findings of previous studies and reveals evidence to suggest that interpopulation coupling was weaker over the 20th century than for the 1800s. Analysis supports the notion that synchrony in these populations is maintained by a continuous and locally acting coupling between adjacent regions with large phase adjustments occurring only infrequently. When this coupling is absent, asynchrony develops between populations.     

Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models

 High dimensional Leslie matrix models have long been viewed as discretizations of McKendrick PDE models. However, these two fundamental classes of models can be linked in a completely different way. For populations with periodic birth pulses, Leslie models of any dimension can be viewed as “stroboscopic snapshots” (in time) of an associated impulsive McKendrick model; that is, the solution of the discrete model matches the solution of the corresponding continuous model at every discrete time step. In application, McKendrick models of populations with birth pulses can be used to identify the state of the population between the discrete census times of the associated Leslie model. Furthermore, McKendrick models describing populations with near-synchronous birth pulses can be viewed as realistic perturbations of the associated Leslie model. Received: 7 August 1997 / Revised version: 15 January 1998     

A comparative evaluation of models of cinnabar moth dynamics

Summary 1. A complex model of cinnabar moth dynamics proposed by Dempster and Lakhani (1979) with 23 parameters is reduced to a single equation with five parameters, and the behaviour of the reduced model shown to explain most features of the full model. 2. The efficiency of the full model is compared with the reduced model and with two even simpler models (the two parameter discrete logistic and a four parameter model based on a step-function for mortality) in their abilities to describe time series data of cinnabar moth population densities from Weeting Heath. Models with more parameters were not significantly better than few-parameter models in describing population trajectories. 3. Models that included a driving variable (in this case observed rainfall data) were no better at describing the data than simpler models without driving variables. It appears, therefore, that the routine inclusion of driving variables may be counterproductive, unless there is compelling empirical or theoretical evidence of their importance and the mode of action of the driving variables can be modelled mechanistically. For example, the regression model used to describe the relationship between rainfall and plant biomass in Dempster and Lakhani (1979), breaks down if rainfall is assumed to be constant, because there is no explicit model for the regulation of plant biomass. 4. The parameter values of the cinnabar-ragwort interaction suggest that cinnabar moth dynamics may be chaotic. Whether or not field data exhibit chaos or environmental stochasticity (or a mixture of both) is impossible to determine from inspection of time series data on population density. There is an urgent need for experimental and theoretical protocols to disentangle these two sources of population fluctuation.     

Some population consequences of variation in preference among individual predators

A number of foraging studies have demonstrated that populations of predators rarely consist of individuals with identical preferences for particular types of prey. Variation among predators can lead to frequency-dependent changes in population preference, because those predators mat prefer the rarer type of prey generally have the greatest influence on population preference. In this study we develop a series of theoretical models which demonstrate how anti-apostatic selection (i.e. selection against the rare form) can arise out of (a) bimodal and (b) normal variation in preference among individuals of the same species. We show that population level anti-apostatic selection can occur even when individual predators show pro-apostatic selection (i.e. selection against the common form). Furthermore, patterns of population prey selection that arise out of variation in preference can potentially be pro-apostatic over one range of relative densities and anti-apostatic over another range of relative densities. Finally, we examine a case study involving predation by female waterboatmen Notonecta glauca and show that the variation in preference in this species is large enough to generate higher anti-apostatic selection than would be expected from the diet selected by the average individual.     

Methods of aggregation of variables in population dynamics

In ecology, we are faced with modelling complex systems involving many variables corresponding to interacting populations structured in different compartmental classes, ages and spatial patches. Models that incorporate such a variety of aspects would lead to systems of equations with many variables and parameters. Mathematical analysis of these models would, in general, be impossible. In many real cases, the dynamics of the system corresponds to two or more time scales. For example, individual decisions can be rapid in comparison to growth of the populations. In that case, it is possible to perform aggregation methods that allow one to build a reduced model that governs the dynamics of a lower dimensional system, at a slow time scale. In this article, we present a review of aggregation methods for time continuous systems as well as for discrete models. We also present applications in population dynamics. A first example concerns a continuous time model of a single population distributed on a system of two connected patches (a logistic source and a sink), by fast migration. It is shown that under a certain condition, the total equilibrium population can be larger than the carrying capacity of the logistic source. A second example concerns a discrete model of a population distributed on two patches, still a source and a sink, connected by fast migration. The use of aggregation methods permits us to conclude that density-dependent migration can stabilize the total population.     

The Role of Density Regulation in Extinction Processes and Population Viability Analysis

We review the role of density dependence in the stochastic extinction of populations and the role density dependence has played in population viability analysis (PVA) case studies. In total, 32 approaches have been used to model density regulation in theoretical or applied extinction models, 29 of them are mathematical functions of density dependence, and one approach uses empirical relationships between density and survival, reproduction, or growth rates. In addition, quasi-extinction levels are sometimes applied as a substitute for density dependence at low population size. Density dependence further has been modelled via explicit individual spacing behaviour and/or dispersal. We briefly summarise the features of density dependence available in standard PVA software, provide summary statistics about the use of density dependence in PVA case studies, and discuss the effects of density dependence on extinction probability. The introduction of an upper limit for population size has the effect that the probability of ultimate extinction becomes 1. Mean time to extinction increases with carrying capacity if populations start at high density, but carrying capacity often does not have any effect if populations start at low numbers. In contrast, the Allee effect is usually strong when populations start at low densities but has only a limited influence on persistence when populations start at high numbers. Contrary to previous opinions, other forms of density dependence may lead to increased or decreased persistence, depending on the type and strength of density dependence, the degree of environmental variability, and the growth rate. Furthermore, effects may be reversed for different quasi-extinction levels, making the use of arbitrary quasi-extinction levels problematic. Few systematic comparisons of the effects on persistence between different models of density dependence are available. These effects can be strikingly different among models. Our understanding of the effects of density dependence on extinction of metapopulations is rudimentary, but even opposite effects of density dependence can occur when metapopulations and single populations are contrasted. We argue that spatially explicit models hold particular promise for analysing the effects of density dependence on population viability provided a good knowledge of the biology of the species under consideration exists. Since the results of PVAs may critically depend on the way density dependence is modelled, combined efforts to advance statistical methods, field sampling, and modelling are urgently needed to elucidate the relationships between density, vital rates, and extinction probability.     

Estimation and inference of R0 of an infectious pathogen by a removal method

The basic reproductive ratio, R0, is a central quantity in the investigation and management of infectious pathogens. The standard model for describing stochastic epidemics is the continuous time epidemic birth-and-death process. The incidence data used to fit this model tend to be collected in discrete units (days, weeks, etc.), which makes model fitting, and estimation of R0 difficult. Discrete time epidemic models better match the time scale of data collection but make simplistic assumptions about the stochastic epidemic process. By investigating the nature of the assumptions of a discrete time epidemic model, we derive a bias corrected maximum likelihood estimate of R0 based on the chain binomial model. The resulting 'removal' estimators provide estimates of R0 and the initial susceptible population size from time series of infectious case counts. We illustrate the performance of the estimators on both simulated data and real epidemics. Lastly, we discuss methods to address data collected with observation error.     

From local interactions to population dynamics in site-based models of ecology

A central problem in ecology is relating the interactions of individuals-described in terms of competition, predation, interference, etc.-to the dynamics of the populations of these individuals-in terms of change in numbers of individuals over time. Here, we address this problem for a class of site-based ecological models, where local interactions between individuals take place at a finite number of discrete resource sites over non-overlapping generations and, between generations, individuals move randomly between sites over the entire system. Such site-based models have previously been applied to a wide range of ecological systems: from those involving contest or scramble competition for resources to host-parasite interactions and meta-populations. We show how the population dynamics of site-based models can be accurately approximated by and understood through deterministic and stochastic difference equations. Conversely, we use the inverse of this approximation to show what implicit assumptions are made about individual interactions by modelling of population dynamics in terms of difference equations. To this end, we prove a useful and general theorem: that any model in our class of site-based models has a corresponding stochastic difference equation population model, by which it can be approximated. This theorem allows us to calculate long-term population dynamics, evolutionary stable strategies and, by extending our theory to account for large deviations, extinction probabilities for a wide range of site-based systems. Our methodology is then illustrated to various examples of between species competition, predator-prey interactions and co-operation.