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Author: Wen Jin Publisher: ISBN: Category : Population Languages : en Pages : 69
Book Description
Persistence theory provides a mathematically rigorous answer to the question of population survival by establishing an initial-condition- independent positive lower bound for the long-term value of the population size. This study focuses on the persistence of discrete semiflows in infinite-dimensional state spaces that model the year-to-year dynamics of structured populations. The map which encapsulates the population development from one year to the next is approximated at the origin (the extinction state) by a linear or homogeneous map. The (cone) spectral radius of this approximating map is the threshold between extinction and persistence. General persistence results are applied to three particular models: a size-structured plant population model, a diffusion model (with both Neumann and Dirichlet boundary conditions) for a dispersing population of males and females that only mate and reproduce once during a very short season, and a rank-structured model for a population of males and females.
Author: Wen Jin Publisher: ISBN: Category : Population Languages : en Pages : 69
Book Description
Persistence theory provides a mathematically rigorous answer to the question of population survival by establishing an initial-condition- independent positive lower bound for the long-term value of the population size. This study focuses on the persistence of discrete semiflows in infinite-dimensional state spaces that model the year-to-year dynamics of structured populations. The map which encapsulates the population development from one year to the next is approximated at the origin (the extinction state) by a linear or homogeneous map. The (cone) spectral radius of this approximating map is the threshold between extinction and persistence. General persistence results are applied to three particular models: a size-structured plant population model, a diffusion model (with both Neumann and Dirichlet boundary conditions) for a dispersing population of males and females that only mate and reproduce once during a very short season, and a rank-structured model for a population of males and females.
Author: Hal L. Smith Publisher: American Mathematical Soc. ISBN: 082184945X Category : Mathematics Languages : en Pages : 426
Book Description
Providing a self-contained treatment of persistence theory that is accessible to graduate students, this monograph includes chapters on infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bioreactor, and an age-structured model of cells growing in a chemostat.
Author: Hal L. Smith Publisher: American Mathematical Soc. ISBN: 0821884220 Category : Mathematics Languages : en Pages : 426
Book Description
"The mathematical theory of persistence answers questions such as which species, in a mathematical model of interacting species, will survive over the long term. It applies to infinite-dimensional as well as to finite-dimensional dynamical systems, and to discrete-time as well as to continuous-time semiflows. This monograph provides a self-contained treatment of persistence theory that is accessible to graduate students. The key results for deterministic autonomous systems are proved in full detail such as the acyclicity theorem and the tripartition of a global compact attractor. Suitable conditions are given for persistence to imply strong persistence even for nonautonomous semiflows, and time-heterogeneous persistence results are developed using so-called 'average Lyapunov functions'. Applications play a large role in the monograph from the beginning. These include ODE models such as an SEIRS infectious disease in a meta-population and discrete-time nonlinear matrix models of demographic dynamics. Entire chapters are devoted to infinite-dimensional examples including an SI epidemic model with variable infectivity, microbial growth in a tubular bioreactor, and an age-structured model of cells growing in a chemostat."--Publisher's description.
Author: Adrian Ziessler Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
One central goal in the analysis of dynamical systems is the characterization of long term behavior of the system state. To this end, the so-called global attractor is of interest, that is, an invariant set that attracts all the trajectories of the underlying dynamical system. Over the last 20 years so-called set-oriented numerical methods have been developed that allow to compute approximations of invariant sets. The basic idea is to cover the objects of interest, for instance attractors or unstable manifolds, by outer approximations which are created via subdivision techniques. However, the applicability of those techniques is restricted to finite dimensional dynamical systems, i.e., ordinary differential equations or discrete dynamical systems. In this thesis, we extend the set-oriented numerical methods to the infinite dimensional context. With those extensions we will be able to compute finite dimensional invariant sets for infinite dimensional dynamical systems, e.g., for delay and partial differential equations. The idea is to utilize infinite dimensional embedding techniques in our numerical treatment. This will allow us to construct a finite dimensional dynamical system, the core dynamical system (CDS), on an appropriately chosen observation space. Using the CDS, we then can approximate finite dimensional embedded attractors or embedded unstable manifolds. Furthermore, we will be able to compute approximations of the embedded invariant measure in the observation space which gives a statistical description of the dynamical behavior of the infinite dimensional dynamical system. We present numerical realizations of the CDS for delay and partial differential equations and illustrate the efficiency of our approach in several examples. Furthermore, we present modifications for the set-oriented subdivision and continuation method. ... ; eng
Author: Roger Temam Publisher: Springer Science & Business Media ISBN: 1468403133 Category : Mathematics Languages : en Pages : 517
Book Description
This is the first attempt at a systematic study of infinite dimensional dynamical systems generated by dissipative evolution partial differential equations arising in mechanics and physics. Other areas of science and technology are included where appropriate. The relation between infinite and finite dimensional systems is presented from a synthetic viewpoint and equations considered include reaction-diffusion, Navier-Stokes and other fluid mechanics equations, magnetohydrodynamics, thermohydraulics, pattern formation, Ginzburg-Landau, damped wave and an introduction to inertial manifolds.
Author: Marcel de Jeu Publisher: Birkhäuser ISBN: 3319278428 Category : Mathematics Languages : en Pages : 516
Book Description
This book presents the proceedings of Positivity VII, held from 22-26 July 2013, in Leiden, the Netherlands. Positivity is the mathematical field concerned with ordered structures and their applications in the broadest sense of the word. A biyearly series of conferences is devoted to presenting the latest developments in this lively and growing discipline. The lectures at the conference covered a broad spectrum of topics, ranging from order-theoretic approaches to stochastic processes, positive solutions of evolution equations and positive operators on vector lattices, to order structures in the context of algebras of operators on Hilbert spaces. The contributions in the book reflect this variety and appeal to university researchers in functional analysis, operator theory, measure and integration theory and operator algebras. Positivity VII was also the Zaanen Centennial Conference to mark the 100th birth year of Adriaan Cornelis Zaanen, who held the chair of Analysis in Leiden for more than 25 years and was one of the leaders in the field during his lifetime.
Author: Christian Pötzsche Publisher: Springer ISBN: 3642142583 Category : Mathematics Languages : en Pages : 422
Book Description
Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to seasonal, modulation, controlling or even random effects. Our goal is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes). These dynamical systems are generated by implicit difference equations, which explicitly depend on time. Compactness and dissipativity conditions are provided for such problems in order to have attractors using the natural concept of pullback convergence. Concerning a necessary linear theory, our hyperbolicity concept is based on exponential dichotomies and splittings. This concept is in turn used to construct nonautonomous invariant manifolds, so-called fiber bundles, and deduce linearization theorems. The results are illustrated using temporal and full discretizations of evolutionary differential equations.
Author: El Hassan Zerrik Publisher: Springer Nature ISBN: 3030686000 Category : Technology & Engineering Languages : en Pages : 323
Book Description
This book deals with the stabilization issue of infinite dimensional dynamical systems both at the theoretical and applications levels. Systems theory is a branch of applied mathematics, which is interdisciplinary and develops activities in fundamental research which are at the frontier of mathematics, automation and engineering sciences. It is everywhere, innumerable and daily, and moreover is there something which is not system: it is present in medicine, commerce, economy, psychology, biological sciences, finance, architecture (construction of towers, bridges, etc.), weather forecast, robotics, automobile, aeronautics, localization systems and so on. These are the few fields of application that are useful and even essential to our society. It is a question of studying the behavior of systems and acting on their evolution. Among the most important notions in system theory, which has attracted the most attention, is stability. The existing literature on systems stability is quite important, but disparate, and the purpose of this book is to bring together in one document the essential results on the stability of infinite dimensional dynamical systems. In addition, as such systems evolve in time and space, explorations and research on their stability have been mainly focused on the whole domain in which the system evolved. The authors have strongly felt that, in this sense, important considerations are missing: those which consist in considering that the system of interest may be unstable on the whole domain, but stable in a certain region of the whole domain. This is the case in many applications ranging from engineering sciences to living science. For this reason, the authors have dedicated this book to extension of classical results on stability to the regional case. This book considers a very important issue, which is that it should be accessible to mathematicians and to graduate engineering with a minimal background in functional analysis. Moreover, for the majority of the students, this would be their only acquaintance with infinite dimensional system. Accordingly, it is organized by following increasing difficulty order. The two first chapters deal with stability and stabilization of infinite dimensional linear systems described by partial differential equations. The following chapters concern original and innovative aspects of stability and stabilization of certain classes of systems motivated by real applications, that is to say bilinear and semi-linear systems. The stability of these systems has been considered from a global and regional point of view. A particular aspect concerning the stability of the gradient has also been considered for various classes of systems. This book is aimed at students of doctoral and master’s degrees, engineering students and researchers interested in the stability of infinite dimensional dynamical systems, in various aspects.
Author: Saber Elaydi Publisher: Springer Nature ISBN: 303125225X Category : Mathematics Languages : en Pages : 534
Book Description
This book comprises selected papers of the 26th International Conference on Difference Equations and Applications, ICDEA 2021, held virtually at the University of Sarajevo, Bosnia and Herzegovina, in July 2021. The book includes the latest and significant research and achievements in difference equations, discrete dynamical systems, and their applications in various scientific disciplines. The book is interesting for Ph.D. students and researchers who want to keep up to date with the latest research, developments, and achievements in difference equations, discrete dynamical systems, and their applications, the real-world problems.
Author: Wen Jin
Author: Wen Jin
Author: Hal L. Smith
Author: Hal L. Smith
Author: Adrian Ziessler
Author: Roger Temam
Author: Marcel de Jeu
Author: Igor' D. Čuešov
Author: Christian Pötzsche
Author: El Hassan Zerrik
Author: Saber Elaydi