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

On the rate of genetic adaptation under natural selection

https://math.arizona.edu/~cushing/LifeStages_InteractionsAndSpatialPatterns%20BMB2012.pdf

Life stages: interactions and spatial patterns.

The study of population dynamics has opened the way to new explanations of data, to new procedures for data analyses, and will surely lead to a new phase of research in which experiments are focused directly on phenomena such as stable equilibria, bifurcations, multiple attractors, limit cycles, life not lived, and stationary distributions. In this final chapter we speculate on the emerging discipline of population dynamics and the role of Tribolium as an experimental animal model.

Dynamics and the Tribolium Model

An experimental check of fitness entropy vs. selective delay

In the first experiment in which flour beetles of the genus Tribolium were used for the study of populations, Chapman (1928) made an important observation: when the beetles were grown in a continuous culture, the number of animals did not increase indefinitely but rather obtained a steady-state density. The projection of Malthusian growth was simply not appropriate. It is not surprising, then, that Gause (1931) used Chapman's data to "prove" the general applicability of the Verhulst-Pearl equation. But this exercise did not identify the biological basis for the regulation of animal numbers. Chapman (1928) proposed cannibalism: by eating eggs and pupae, the adult and larval stages control population density. Others agreed and also suggested that fecundity is reduced with an increase in animal numbers (Allee 1931; MacLagan 1932; Park 1932). Although the total number of beetles, in particular, the number of adults, appeared to approach an upper asymptote, the numbers of eggs and larvae displayed large oscillations that were, according to Chapman (1933), similar to those forecast by Volterra (1931) for predator and prey. Indeed, Chapman and Whang (1934) concluded that larval consumption of eggs caused the population fluctuations in these two life stages. During the next half-century, both experimentalists and theorists agreed with this interpretation (Stanley 1932; Landahl 1955a,b; Mertz 1969; Taylor 1971; Fujii 1978). Despite the general agreement, it was still not possible to determine the effect of such factors as the initial numbers of eggs and larvae, the magnitude of larval cannibalism, and the lengths of the egg and larval stages on the observed oscillations in egg and larval numbers. Although empirical data were abundant and a corresponding number of computations amply demonstrated these cycles, no central, unifying model or framework to study the effects of various factors on the population cycles was available. Recognizing this difficulty, and benefiting from the earlier work, we suggest that the biology of Tribolium leads, quite straightforwardly, to a simple model consisting of a system of nonlinear Volterra integral equations that can be studied using

https://www.jstor.org/stable/info/2461879

R. F. Costantino

Papers

On the rate of genetic adaptation under natural selection

Life stages: interactions and spatial patterns.

Dynamics and the Tribolium Model

An experimental check of fitness entropy vs. selective delay

Cannibalistic egg-larva interactions in tribolium: