Author

# Shandelle M. Henson

Other affiliations: College of William & Mary, University of Arizona, University of Tennessee

Bio: Shandelle M. Henson is an academic researcher from Andrews University. The author has contributed to research in topics: Population & Larus glaucescens. The author has an hindex of 26, co-authored 80 publications receiving 2500 citations. Previous affiliations of Shandelle M. Henson include College of William & Mary & University of Arizona.

Topics: Population, Larus glaucescens, Cannibalism, Seabird, Population model

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TL;DR: In this article, it was shown that the sensitivity of a stochastic Lyapunov exponent (SLE) to the initial conditions of a deterministic model is not a sign of chaos.

Abstract: 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.

264 citations

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TL;DR: In this article, a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium.

Abstract: A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, non- linear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attri- buting the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a model- predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic vari- ability and obtained conditional least-squares estimates for the model parameters. We gen- erated 2000 ''bootstrapped data sets'' with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distri- butions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, in- cluding treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.

192 citations

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TL;DR: In this paper, the authors show that the Leslie/Gower model has the same dynamic scenarios as the Lotka/Volterra (differential equation) competition model and give an example of a competition model that has non-Lotka/volterra dynamics that are consistent with the anomalous case.

Abstract: A difference equation model, called that Leslie/Gower model, played a key historical role in laboratory experiments that helped establish the “competitive exclusion principle” in ecology. We show that this model has the same dynamic scenarios as the famous Lotka/Volterra (differential equation) competition model. It is less well known that some anomalous results from the experiments seem to contradict the exclusion principle and Lotka/Volterra dynamics. We give an example of a competition model that has non-Lotka/Volterra dynamics that are consistent with the anomalous case.

191 citations

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29 Oct 2002

TL;DR: Introduction.

Abstract: Introduction Models Bifurcations Chaos Patterns in Chaos What We Learned Bibliography Appendix

182 citations

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TL;DR: In this paper, the Beverton-Holt equation is modified for population dynamics, where the constant carrying capacity of a population is replaced by a periodic sequence of positive carrying capacities.

Abstract: has a unique positive equilibrium K and all solutions with x0 . 0 approach K as t !1: This equation (known as the Beverton–Holt equation) arises in applications to population dynamics, and in that context K is the “carrying capacity” and r is the “inherent growth rate”. A modification of this equation that arises in the study of populations living in a periodically (seasonally) fluctuating environment replaces the constant carrying capacity K by a periodic sequence Kt of positive carrying capacities.

110 citations

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2,618 citations

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21 Jul 2008

TL;DR: In step-by-step detail, Benjamin Bolker teaches ecology graduate students and researchers everything they need to know in order to use maximum likelihood, information-theoretic, and Bayesian techniques to analyze their own data using the programming language R.

Abstract: Ecological Models and Data in R is the first truly practical introduction to modern statistical methods for ecology. In step-by-step detail, the book teaches ecology graduate students and researchers everything they need to know in order to use maximum likelihood, information-theoretic, and Bayesian techniques to analyze their own data using the programming language R. Drawing on extensive experience teaching these techniques to graduate students in ecology, Benjamin Bolker shows how to choose among and construct statistical models for data, estimate their parameters and confidence limits, and interpret the results. The book also covers statistical frameworks, the philosophy of statistical modeling, and critical mathematical functions and probability distributions. It requires no programming background--only basic calculus and statistics.

1,626 citations

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01 Jan 2005TL;DR: In this paper, the authors present a spatial analysis of complete point location data, including points, lines, and graphs, and a multiscale analysis of the data set, including spatial diversity analysis and spatial autocorrelation.

Abstract: Preface 1. Spatial concepts and notions 2. Ecological and spatial processes 3. Points, lines and graphs 4. Spatial analysis of complete point location data 5. Contiguous units analysis 6. Spatial analysis of sample data 7. Spatial relationship and multiscale analysis 8. Spatial autocorrelation and inferential tests 9. Spatial partitioning: spatial clusters and boundary detection 10. Spatial diversity analysis 11. Spatio-temporal analysis 12. Closing comments and future directions References Index.

1,250 citations

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TL;DR: This work states that because anthropogenic changes often affect stability and diversity simultaneously, diversity-stability relationships cannot be understood outside the context of the environmental drivers affecting both.

Abstract: Understanding the relationship between diversity and stability requires a knowledge of how species interact with each other and how each is affected by the environment. The relationship is also complex, because the concept of stability is multifaceted; different types of stability describing different properties of ecosystems lead to multiple diversity-stability relationships. A growing number of empirical studies demonstrate positive diversity-stability relationships. These studies, however, have emphasized only a few types of stability, and they rarely uncover the mechanisms responsible for stability. Because anthropogenic changes often affect stability and diversity simultaneously, diversity-stability relationships cannot be understood outside the context of the environmental drivers affecting both. This shifts attention away from diversity-stability relationships toward the multiple factors, including diversity, that dictate the stability of ecosystems.

1,247 citations

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Colorado State University

^{1}, National Park Service^{2}, University of Rhode Island^{3}, United States Environmental Protection Agency^{4}, Emory University^{5}, University of Maryland, College Park^{6}, University of North Carolina at Chapel Hill^{7}, University of Wisconsin-Madison^{8}, Duke University^{9}, The Nature Conservancy^{10}TL;DR: The scope of the thresholds concept in ecological science is defined and methods for identifying and investigating thresholds using a variety of examples from terrestrial and aquatic environments, at ecosystem, landscape and regional scales are discussed.

Abstract: An ecological threshold is the point at which there is an abrupt change in an ecosystem quality, property or phenomenon, or where small changes in an environmental driver produce large responses in the ecosystem. Analysis of thresholds is complicated by nonlinear dynamics and by multiple factor controls that operate at diverse spatial and temporal scales. These complexities have challenged the use and utility of threshold concepts in environmental management despite great concern about preventing dramatic state changes in valued ecosystems, the need for determining critical pollutant loads and the ubiquity of other threshold-based environmental problems. In this paper we define the scope of the thresholds concept in ecological science and discuss methods for identifying and investigating thresholds using a variety of examples from terrestrial and aquatic environments, at ecosystem, landscape and regional scales. We end with a discussion of key research needs in this area.

1,049 citations