Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

Abstract : The emerging discipline known as \"chaos theory\" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for \"chaos\" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.

Introduction of chaotic dynamical systems

Exotic species pose a threat to most ecosystems because of their potential to establish negative interactions with native biota. However, exotic species can also offer resources to native species, especially within highly modified environments such as urban ecosystems. We studied 17 exotic-native pairs of species with the potential to compete with one another, or in which one of the species could offer resources to the other, in an urban ecological reserve located within Mexico City. We used two-species occupancy models to analyze the potential association between the presence of the exotic species and the spatial distribution of the native species, as well as to assess if these species tend to avoid each other (negative spatial interaction) or to co-occur more often than expected under the hypothesis of independent occurrences (positive spatial interaction). Our results revealed few cases in which the exotic species influenced occupancy of the native species, and these spatial interactions were mainly positive, indicated by the fact that the occupancy of the native species was usually higher when the exotic species was also present. Seven of the eight observed non-independent patterns of co-occurrence were evident during the dry months of the year, when resources become scarce for most species. Our results also demonstrate that the observed patterns of species co-occurrence depend on the distance to the nearest urban structure and the amount of herb, shrub, and tree cover, indicating that these habitat features influence whether native species avoid or co-occur with exotic species. Our study represents an important contribution to the understanding of temporal dynamics in the co-occurrence between exotic and native species within urban ecological reserves.

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This town ain't big enough for both of us…or is it? Spatial co-occurrence between exotic and native species in an urban reserve.

Modelling Fluctuating Populations, originally published in 1982, is a classic textbook because primarily, this book takes a distinctive approach to population dynamics, by emphasizing from the earliest chapters that all populations fluctuate continuously. Traditional themes in theoretical ecology such as equilibrium and population stability are linked to analyses of the response of a population to environmental fluctuations and to extinction probabilities. Thus, the book's approach confronts head-on one common criticism of simple ecological models - the mismatch between the mathematical mechanisms studied and the questions of top ecological concern. Secondly, the book demonstrates the power of techniques based on linear mathematics.

Modelling Fluctuating Populations

Analysis and behavior control of a modified singular prey–predator model

Optimal harvesting strategy and stochastic analysis for a two species commensaling system

Stochastic effects on an ecosystem with predation, Commensalism, mutualism and neutralism

The article concentrates on the role of fluctuating parameters for removable population from the incubated class in a susceptible-incubated-infected model. The discrete analogous of this model is also investigated. Conditions for local asymptotic stability are derived for both the disease free and endemic cases. Numerical simulations are performed to validate the theoretical results.

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Fluctuating Role of Parameters in the Analysis of the Continues and Discrete Version of a Susceptible-Incubated-Infected Model

In the present study, we consider a two-dimensional nonlinear prey-predator model with harvesting on both the species and specifically with the mortality rate of the predator species. The purpose of this work is to offer mathematical analysis of the model and to discuss some of the vibrant qualitative results that are expected to arise from the inter-play of the biological internal and external forces. Here, all the possible equilibrium points of the model are identified and the criteria for the local as well as global stabilities are investigated. Apart from this, the existence of bio-economic equilibrium with optimal scheme has been critically discussed here. We studied the analytical estimates for the population intensities of fluctuations by Fourier transform methods through stochastic perturbations. We have also highlighted the diffusive stability of the system along with some numerical simulations.

Dynamics in harvested prey-predator mathematical model with noise and diffusion

In this paper we have studied the stability of three typical species syn-ecosystem. The system comprises of one commensal

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A. Sabarmathi

Papers

Optimal harvesting strategy and stochastic analysis for a two species commensaling system

Stochastic effects on an ecosystem with predation, Commensalism, mutualism and neutralism

Fluctuating Role of Parameters in the Analysis of the Continues and Discrete Version of a Susceptible-Incubated-Infected Model

Dynamics in harvested prey-predator mathematical model with noise and diffusion

An Optimal Harvesting Strategy of a Three Species Syn-ecosystem with Commensalism and Stochasticity