Lattice effects observed in chaotic dynamics of experimental populations.
19 Oct 2001-Science (American Association for the Advancement of Science)-Vol. 294, Iss: 5542, pp 602-605
TL;DR: This work compared the predictions of discrete-state and continuous-state population models and suggested that such lattice effects could be an important component of natural population fluctuations.
Abstract: Animals and many plants are counted in discrete units. The collection of possible values (state space) of population numbers is thus a nonnegative integer lattice. Despite this fact, many mathematical population models assume a continuum of system states. The complex dynamics, such as chaos, often displayed by such continuous-state models have stimulated much ecological research; yet discrete-state models with bounded population size can display only cyclic behavior. Motivated by data from a population experiment, we compared the predictions of discrete-state and continuous-state population models. Neither the discrete- nor continuous-state models completely account for the data. Rather, the observed dynamics are explained by a stochastic blending of the chaotic dynamics predicted by the continuous-state model and the cyclic dynamics predicted by the discrete-state models. We suggest that such lattice effects could be an important component of natural population fluctuations.
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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
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
Cites background from "Lattice effects observed in chaotic..."
...…three-cycles, invariant loops, and chaos), multiple attractors, saddle influences, sensitivity to initial conditions, and lattice effects (Costantino et al. 1995, 1997, 1998, Cushing et al. 1996, 1998a, b, 2001, Dennis et al. 1995, 1997, 2001, Henson et al. 1999, 2001, Desharnais et al. 2001)....
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TL;DR: A hierarchical approach is adapted to the problem of estimating population growth rates and their uncertainties when individuals vary and that variability cannot be assigned to specific causes, which shows that population growth models that rely on standard propagation of estimation error but ignore variability among individuals can misrepresent uncertainties in ways that erode credibility.
Abstract: Estimates of uncertainty are the basis for inference of population risk. Uncertainty is estimated from models fitted to data that typically include a deterministic model (e.g., population growth) and stochastic elements, which should accommodate errors in sampling and any sources of variability that affect observations. Prediction from fitted models (of, say, demography) to new variables (say, population growth) requires propagation of these stochastic elements. Ecological models ignore most forms of variability, because they make statistical models complex, and they pose computational challenges. Variability associated with space, time, and among individuals that is not accommodated by demographic models can make parameter estimates and growth rate predictions unrealistic.
I adapt a hierarchical approach to the problem of estimating population growth rates and their uncertainties when individuals vary and that variability cannot be assigned to specific causes. In contrast to an overfitted model that would assign a different parameter value to each individual, hierarchical models accommodate individual differences, but assume that those differences derive from an underlying distribution—they belong to a “population.” The hierarchical model can be implemented in classical (frequentist) and Bayesian frameworks (I demonstrate both) and analyzed using Markov chain Monte Carlo simulation. Results show that population growth models that rely on standard propagation of estimation error but ignore variability among individuals can misrepresent uncertainties in ways that erode credibility.
Corresponding Editor: O. N. Bjornstad.
183 citations
TL;DR: It is shown that endogenously generated variance in spread rates is remarkably high between replicated invasions of the flour beetle Tribolium castaneum in laboratory microcosms, which indicates inherent limitations to predictability in even the simplest ecological settings.
Abstract: Although mean rates of spread for invasive species have been intensively studied, variance in spread rates has been neglected. Variance in spread rates can be driven exogenously by environmental variability or endogenously by demographic or genetic stochasticity in reproduction, survival, and dispersal. Endogenous variability is likely to be important in spread but has not been studied empirically. We show that endogenously generated variance in spread rates is remarkably high between replicated invasions of the flour beetle Tribolium castaneum in laboratory microcosms. The observed variation between replicate invasions cannot be explained by demographic stochasticity alone, which indicates inherent limitations to predictability in even the simplest ecological settings.
162 citations
TL;DR: It is argued that the dynamics of many systems are a result of interactions between the deterministic nonlinear skeleton and noise.
Abstract: Population dynamics models remain largely deterministic, although the presence of random fluctuations in nature is well recognized. This deterministic approach is based on the implicit assumption that systems can be separated into a deterministic part that captures the essential features of the system and a random part that can be neglected. But is it possible, in general, to understand population dynamics without the explicit consideration of random fluctuations? Here, we suggest perhaps not, and argue that the dynamics of many systems are a result of interactions between the deterministic nonlinear skeleton and noise.
142 citations
Additional excerpts
...Until recently, it had been assumed there would be few consequences of working with mathematically tractable continuous-state models; however, recent research has challenged this view by demonstrating the importance of recognizing the discrete nature of ecological populations [ 32 ,33]....
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21 Aug 1973
TL;DR: Preface vii Preface to the Second Edition Biology Edition 1.
Abstract: Preface vii Preface to the Second Edition Biology Edition 1. Intoduction 3 2. Mathematical Models and Stability 13 3. Stability versus Complexity in Multispecies Models 4. Models with Few Species: Limit Cycles and Time Delays 79 5. Randomly Fluctuating Environments 109 6. Niche Overlap and Limiting Similarity 139 7. Speculations 172 Appendices 187 Afterthoughts for the Second Edition 211 Bibliography to Afterthoghts 234 Bibliography 241 Author Index 259 Subject Index 263
5,083 citations
TL;DR: Plotting net reproduction (reproductive potential of the adults obtained) against the density of stock which produced them, for a number of fish and invertebrate populations, gives a domed curve whose apex lies above the line representing replacement reproduction.
Abstract: Plotting net reproduction (reproductive potential of the adults obtained) against the density of stock which produced them, for a number of fish and invertebrate populations, gives a domed curve wh...
3,037 citations
TL;DR: This paper presents a dynamical regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur.
Abstract: Some of the simplest nonlinear difference equations describing the growth of biological populations with nonoverlapping generations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between 2 population points, to stable cycles with 4, 8, 16, . . . points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized analyses; its existence in such fully deterministic nonlinear difference equations is a fact of considerable mathematical and ecological interest.
1,456 citations
TL;DR: It is shown that as a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes".
Abstract: Many biological populations breed seasonally and have nonoverlapping generations, so that their dynamics are described by first-order difference equations, Nt+1 = F (Nt). In many cases, F(N) as a function of N will have a hump. We show, very generally, that as such a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes." We give a detailed account of the underlying mathematics of this process and review other situations (in two- and higher dimensional systems, or in differential equation systems) where apparently random dynamics can arise from bifurcation processes. This complicated behavior, in simple deterministic models, can have disturbing implications for the analysis and interpretation of biological data.
1,119 citations