scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments'

01 Feb 1995-Ecological Monographs (Ecological Society of America)-Vol. 65, Iss: 3, pp 261-282
TL;DR: This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.
Abstract: Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2-cy- cles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

Content maybe subject to copyright    Report

Citations
More filters
Journal ArticleDOI
01 Sep 2004-Ecology
TL;DR: A change in approach is needed to determine whether pollen limitation reflects random fluctuations around a pollen–resource equilibrium, an adaptation to stochastic pollination environments, or a chronic syndrome caused by an environmental perturbation.
Abstract: Determining whether seed production is pollen limited has been an area of intensive empirical study over the last two decades. Yet current evidence does not allow satisfactory assessment of the causes or consequences of pollen limitation. Here, we critically evaluate existing theory and issues concerning pollen limitation. Our main conclusion is that a change in approach is needed to determine whether pollen limitation reflects random fluctuations around a pollen–resource equilibrium, an adaptation to stochastic pollination environments, or a chronic syndrome caused by an environmental perturbation. We formalize and extend D. Haig and M. Westoby's conceptual model, and illustrate its use in guiding research on the evolutionary consequences of pollen limitation, i.e., whether plants evolve or have evolved to ameliorate pollen limitation. This synthesis also reveals that we are only beginning to understand when and how pollen limitation at the plant level translates into effects on plant population dynamics...

1,128 citations


Cites methods from "Nonlinear demographic dynamics: mat..."

  • ...A framework for incorporating density dependence exists for animal models (Dennis et al. 1995, Vonesh and De la Cruz 2002), so incorporating positive density dependence from variation in pollination into plant-population models should be relatively straightforward (see also Morris and Doak 2002)....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors derived three properties of stochastic multispecies communities that measure different characteristics associated with community stability using first-order multivariate autoregressive (MAR(1)) models.
Abstract: Natural ecological communities are continuously buffeted by a varying environment, often making it difficult to measure the stability of communities using concepts requiring the existence of an equilibrium point. Instead of an equilibrium point, the equilibrial state of communities subject to environmental stochasticity is a stationary distribution, which is characterized by means, variances, and other statistical moments. Here, we derive three properties of stochastic multispecies communities that measure different characteristics associated with community stability. These properties can be estimated from multispecies time-series data using first-order multivariate autoregressive (MAR(1)) models. We demonstrate how to estimate the parameters of MAR(1) models and obtain confidence intervals for both parameters and the measures of stability. We also address the problem of estimation when there is observation (measurement) error. To illustrate these methods, we compare the stability of the planktonic commun...

478 citations

Journal ArticleDOI
TL;DR: The numerically integrated state-space (NISS) method as mentioned in this paper was proposed to fit models to time series of population abun- dances that incorporate both process noise and observation error in a likelihood framework.
Abstract: We evaluate a method for fitting models to time series of population abun- dances that incorporates both process noise and observation error in a likelihood framework. The method follows the probability logic of the Kalman filter, but whereas the Kalman filter applies to linear, Gaussian systems, we implement the full probability calculations numerically so that any nonlinear, non-Gaussian model can be used. We refer to the method as the "numerically integrated state-space (NISS) method" and compare it to two common methods used to analyze nonlinear time series in ecology: least squares with only process noise (LSPN) and least squares with only observation error (LSOE). We compare all three methods by fitting Beverton-Holt and Ricker models to many replicate model-generated time series of length 20 with several parameter choices. For the Ricker model we chose parameters for which the deterministic part of the model produces a stable equilibrium, a two-cycle, or a four-cycle. For each set of parameters we used three process-noise and observation-error scenarios: large standard deviation (0.2) for both, and large for one but small (0.05) for the other. The NISS method had lower estimator bias and variance than the other methods in nearly all cases. The only exceptions were for the Ricker model with stable-equilibrium parameters, in which case the LSPN and LSOE methods has lower bias when noise variances most closely met their assumptions. For the Beverton-Holt model, the NISS method was much less biased and more precise than the other methods. We also evaluated the utility of each method for model selection by fitting simulated data to both models and using information criteria for selection. The NISS and LSOE methods showed a strong bias toward selecting the Ricker over the Beverton-Holt, even when data were generated with the Beverton-Holt. It remains unclear whether the LSPN method is generally superior for model selection or has fortuitously better biases in this particular case. These results suggest that information criteria are best used with caution for nonlinear population models with short time series. Finally we evaluated the convergence of likelihood ratios to theoretical asymptotic distributions. Agreement with asymptotic distributions was very good for stable-point Rick- er parameters, less accurate for two-cycle and four-cycle Ricker parameters, and least accurate for the Beverton-Holt model. The numerically integrated state-space method has a number of advantages over least squares methods and offers a useful tool for connecting models and data and ecology.

438 citations

Journal ArticleDOI
TL;DR: In this paper, the authors describe a discrete-time, stochastic population model with density dependence, environmental-type process noise, and lognormal observation or sampling error.
Abstract: We describe a discrete-time, stochastic population model with density depend ence, environmental-type process noise, and lognormal observation or sampling error. The model, a stochastic version of the Gompertz model, can be transformed into a linear Gaussian state-space model (Kaiman filter) for convenient fitting to time series data. The model has a multivariate normal likelihood function and is simple enough for a variety of uses ranging from theoretical study of parameter estimation issues to routine data analyses in population monitoring. A special case of the model is the discrete-time, stochastic exponential growth model (density independence) with environmental-type process error and lognormal observation error. We describe two methods for estimating parameters in the Gompertz state-space model, and we compare the statistical qualities of the methods with computer simulations. The methods are maximum likelihood based on observations and restricted maximum likelihood based on first differences. Both offer adequate statistical properties. Because the likelihood function is identical to a repeated-measures analysis of variance model with a random time effect, parameter estimates can be calculated using PROC MIXED of SAS. We use the model to analyze a data set from the Breeding Bird Survey. The fitted model suggests that over 70% of the noise in the population's growth rate is due to observation error. The model describes the autocovariance properties of the data especially well. While observation error and process noise variance parameters can both be estimated from one time series, multimodal likelihood functions can and do occur. For data arising from the model, the statistically consistent parameter estimates do not necessarily correspond to the global maximum in the likelihood function. Maximization, simulation, and bootstrapping programs must accommodate the phenomenon of multimodal likelihood functions to produce statistically valid results.

409 citations


Cites background or methods from "Nonlinear demographic dynamics: mat..."

  • ...The basic approach converts the deterministic models of ecology textbooks into statistical analysis tools by incorporating into the models terms representing the ubiquitous stochastic forces affecting populations (Dennis et al. 1995, Hilborn and Mangel 1997, Ives et al. 2003, Turchin 2003)....

    [...]

  • ...When process noise is added on the logarithmic scale, a model of environmental variability is produced (Dennis et al. 1995)....

    [...]

Journal ArticleDOI
17 Jan 1997-Science
TL;DR: A nonlinear demographic model was used to predict the population dynamics of the flour beetle Tribolium under laboratory conditions and to establish the experimental protocol that would reveal chaotic behavior.
Abstract: A nonlinear demographic model was used to predict the population dynamics of the flour beetle Tribolium under laboratory conditions and to establish the experimental protocol that would reveal chaotic behavior. With the adult mortality rate experimentally set high, the dynamics of animal abundance changed from equilibrium to quasiperiodic cycles to chaos as adult-stage recruitment rates were experimentally manipulated. These transitions in dynamics corresponded to those predicted by the mathematical model. Phase-space graphs of the data together with the deterministic model attractors provide convincing evidence of transitions to chaos.

395 citations

References
More filters
Book
01 Jan 1983
TL;DR: In this paper, a generalization of the analysis of variance is given for these models using log- likelihoods, illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables), and gamma (variance components).
Abstract: The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).

23,215 citations

Book ChapterDOI
01 Jan 1981

9,756 citations

Journal Article
07 Apr 2005

3,470 citations