One-Dimensional Maps.- Two-Dimensional Maps.- Chaos.- Fractals.- Chaos in Two-Dimensional Maps.- Chaotic Attractors.- Differential Equations.- Periodic Orbits and Limit Sets.- Chaos in Differential Equations.- Stable Manifolds and Crises.- Bifurcations.- Cascades.- State Reconstruction from Data.

Chaos: An Introduction to Dynamical Systems

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.

Ecological Models and Data in R

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.

Spatial Analysis A Guide for Ecologists

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.

http://eaton.math.rpi.edu/CSUMS/Papers/Ecostability/ivesstabilitydiversityeco.pdf

Stability and Diversity of Ecosystems

Nonlinear dynamics deals with more-or-less regular fluctuations in system variables caused by feedback intrinsic to the system (as opposed to external forces). Chaos is the most exotic form of nonlinear dynamics, in which deterministic interactions produce apparently irregular fluctuations, and small changes in the initial state of the system are magnified through time.7


Keywords:

chaos;
population cycles;
population dynamics;
nonlinear time-series analysis

Nonlinear Dynamics and Chaos

We develop a model of larval cannibalism of Tribolium eggs that is both analytically and numerically tractable and that is consistent with available data on Tribolium dynamics. The model consists of a system of nonlinear integral equations amenable to a bifurcation analysis. We are thus able to predict the presence of multiple attractors for the same parameter values in the model. Our model lets us determine the effect of varying cannibalism rates, fecundity, survival, and lengths of egg and larval stages on stability. In particular, intermediate lengths for larval stages are least stable, and increasing the length of the egg stage is always destabilizing. The model thus provides a conceptual framework for further studies of Tribolium dynamics.

Cannibalistic Egg-Larva Interactions in Tribolium: An Explanation for the Oscillations in Population Numbers

We used small perturbations in adult numbers to control large fluctuations in the chaotic demographic dynamics of laboratory populations of the flour beetle Tribolium castaneum. A nonlinear mathematical model was used to identify a sensitive region of phase space where the addition of a few adult insects would result in a dampening of the life stage fluctuations. Three experimental treatments were applied: one in which perturbations were made whenever the populations were inside the sensitive region (“in-box treatment”), another where perturbations were made whenever the populations were outside the sensitive region (“out-box treatment”), and an unperturbed control. The in-box treatment caused a stabilization of insect densities at numbers well below the peak values exhibited by the out-box and control populations. This study demonstrates how small perturbations can be used to influence the chaotic dynamics of an ecological system.

Chaos and population control of insect outbreaks

Stochasticity alters the nonlinear dynamics of inherently cycling populations. The power spectrum can describe and explain the impacts of stochasticity. We fitted models to short observed time series of flour beetle populations in the frequency domain, then used a well fitting stochastic mechanistic model to generate detailed predictions of population spectra. Some predicted spectral peaks represent periodic phenomena induced or modified by stochasticity and were experimentally confirmed. For one experimental treatment, linearization theory explained that these peaks represent overcompensatory decay of deviations from deterministic oscillation. In another treatment, stochasticity caused frequent directional phase shifting around a cyclic attractor. This directional phase shifting was not explained by linearization theory and modified the periodicity of the system. If field systems exhibit directional phase shifting, then changing the intensity of demographic or environmental noise while holding constant the structure of the noise can change the main frequency of population fluctuations.

/pdf/power-spectra-reveal-the-influence-of-stochasticity-on-203wbfpd2b.pdf

R. F. Costantino

Papers

Cannibalistic Egg-Larva Interactions in Tribolium: An Explanation for the Oscillations in Population Numbers

Chaos and population control of insect outbreaks

Power spectra reveal the influence of stochasticity on nonlinear population dynamics

Population Dynamics and the Tribolium Model: Genetics and Demography

An interdisciplinary approach to understanding nonlinear ecological dynamics