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This book provides a crash course on various methods from the
bifurcation theory of Functional Differential Equations (FDEs).
FDEs arise very naturally in economics, life sciences and
engineering and the study of FDEs has been a major source of
inspiration for advancement in nonlinear analysis and infinite
dimensional dynamical systems. The book summarizes some practical
and general approaches and frameworks for the investigation of
bifurcation phenomena of FDEs depending on parameters with chap.
This well illustrated book aims to be self contained so the readers
will find in this book all relevant materials in bifurcation,
dynamical systems with symmetry, functional differential equations,
normal forms and center manifold reduction. This material was used
in graduate courses on functional differential equations at Hunan
University (China) and York University (Canada).
This collection covers a wide range of topics of infinite
dimensional dynamical systems generated by parabolic partial
differential equations, hyperbolic partial differential equations,
solitary equations, lattice differential equations, delay
differential equations, and stochastic differential equations.
Infinite dimensional dynamical systems are generated by
evolutionary equations describing the evolutions in time of systems
whose status must be depicted in infinite dimensional phase spaces.
Studying the long-term behaviors of such systems is important in
our understanding of their spatiotemporal pattern formation and
global continuation, and has been among major sources of motivation
and applications of new developments of nonlinear analysis and
other mathematical theories. Theories of the infinite dimensional
dynamical systems have also found more and more important
applications in physical, chemical, and life sciences. This book
collects 19 papers from 48 invited lecturers to the International
Conference on Infinite Dimensional Dynamical Systems held at York
University, Toronto, in September of 2008. As the conference was
dedicated to Professor George Sell from University of Minnesota on
the occasion of his 70th birthday, this collection reflects the
pioneering work and influence of Professor Sell in a few core areas
of dynamical systems, including non-autonomous dynamical systems,
skew-product flows, invariant manifolds theory, infinite
dimensional dynamical systems, approximation dynamics, and fluid
flows.
This book provides a crash course on various methods from the
bifurcation theory of Functional Differential Equations (FDEs).
FDEs arise very naturally in economics, life sciences and
engineering and the study of FDEs has been a major source of
inspiration for advancement in nonlinear analysis and infinite
dimensional dynamical systems. The book summarizes some practical
and general approaches and frameworks for the investigation of
bifurcation phenomena of FDEs depending on parameters with chap.
This well illustrated book aims to be self contained so the readers
will find in this book all relevant materials in bifurcation,
dynamical systems with symmetry, functional differential equations,
normal forms and center manifold reduction. This material was used
in graduate courses on functional differential equations at Hunan
University (China) and York University (Canada).
Abstract semilinear functional differential equations arise from
many biological, chemical, and physical systems which are
characterized by both spatial and temporal variables and exhibit
various spatio-temporal patterns. The aim of this book is to
provide an introduction of the qualitative theory and applications
of these equations from the dynamical systems point of view. The
required prerequisites for that book are at a level of a graduate
student. The style of presentation will be appealing to people
trained and interested in qualitative theory of ordinary and
functional differential equations.
The book is devoted to the study of limit theorems and stability of
evolving biologieal systems of "particles" in random environment.
Here the term "particle" is used broadly to include moleculas in
the infected individuals considered in epidemie models, species in
logistie growth models, age classes of population in demographics
models, to name a few. The evolution of these biological systems is
usually described by difference or differential equations in a
given space X of the following type and dxt/dt = g(Xt, y), here,
the vector x describes the state of the considered system, 9
specifies how the system's states are evolved in time (discrete or
continuous), and the parameter y describes the change ofthe
environment. For example, in the discrete-time logistic growth
model or the continuous-time logistic growth model dNt/dt =
r(y)Nt(l-Nt/K(y)), N or Nt is the population of the species at time
n or t, r(y) is the per capita n birth rate, and K(y) is the
carrying capacity of the environment, we naturally have X = R, X ==
Nn(X == Nt), g(x, y) = r(y)x(l-xl K(y)) , xE X. Note that n t for a
predator-prey model and for some epidemie models, we will have that
X = 2 3 R and X = R , respectively. In th case of logistic growth
models, parameters r(y) and K(y) normaIly depend on some random
variable y.
The book is devoted to the study of limit theorems and stability of
evolving biologieal systems of "particles" in random environment.
Here the term "particle" is used broadly to include moleculas in
the infected individuals considered in epidemie models, species in
logistie growth models, age classes of population in demographics
models, to name a few. The evolution of these biological systems is
usually described by difference or differential equations in a
given space X of the following type and dxt/dt = g(Xt, y), here,
the vector x describes the state of the considered system, 9
specifies how the system's states are evolved in time (discrete or
continuous), and the parameter y describes the change ofthe
environment. For example, in the discrete-time logistic growth
model or the continuous-time logistic growth model dNt/dt =
r(y)Nt(l-Nt/K(y)), N or Nt is the population of the species at time
n or t, r(y) is the per capita n birth rate, and K(y) is the
carrying capacity of the environment, we naturally have X = R, X ==
Nn(X == Nt), g(x, y) = r(y)x(l-xl K(y)) , xE X. Note that n t for a
predator-prey model and for some epidemie models, we will have that
X = 2 3 R and X = R , respectively. In th case of logistic growth
models, parameters r(y) and K(y) normaIly depend on some random
variable y.
This book provides an introduction to the qualitative theory and
applications of partial functional differential equations from the
viewpoint of dynamical systems. Many fundamental results and
methods scattered throughout research journals are described,
various applications to population growth in a heterogeneous
environment are presented and a comprehensive bibliography from
both mathematical and biological sources is provided. The main
emphasis of the book is on reaction-diffusion equations with
delayed nonlinear reaction terms and on the joint effect of the
time delay and spatial diffusion on the spatial-temporal patterns
of the considered systems. The presentation is self-contained and
accessible to the nonspecialist. The book should be of value to
graduate students and researchers in dynamical systems,
differential equations, semigroup theory, nonlinear analysis and
mathematical biology. The style of the presentation appeals
especially to people trained and interested in the qualitative
theory of ordinary/functional/partial differential equations.
This monograph introduces some current developments in the
modelling of the spread of tick-borne diseases. Effective modelling
requires the integration of multiple frameworks. Here, particular
attention is given to the previously neglected issues of tick
developmental and behavioral diapause, tick-borne pathogen
co-feeding transmission, and their interactions. An introduction to
the required basics of structured population formulations and delay
differential equations is given, and topics for future study are
suggested. The described techniques will also be useful in the
study of other vector-borne diseases. The ultimate aim of this
project is to develop a general qualitative framework leading to
tick-borne disease risk predictive tools and a decision support
system. The target audience is mathematical biologists interested
in modelling tick population dynamics and tick-borne disease
transmission, and developing computational tools for disease
prevention and control.
Curated by the Fields Institute for Research in Mathematical
Sciences from their COVID-19 Math Modelling Seminars, this first in
a series of volumes on the mathematics of public health allows
readers to access the dominant ideas and techniques being used in
this area, while indicating problems for further research. This
work brings together experts in mathematical modelling from across
Canada and the world, presenting the latest modelling methods as
they relate to the COVID-19 pandemic. A primary aim of this book is
to make the content accessible so that researchers share the core
methods that may be applied elsewhere. The mathematical theories
and technologies in this book can be used to support decision
makers on critical issues such as projecting outbreak trajectories,
evaluating public health interventions for infection prevention and
control, developing optimal strategies to return to a new normal,
and designing vaccine candidates and informing mass immunization
program. Topical coverage includes: basic
susceptible-exposed-infectious-recovered (SEIR) modelling framework
modified and applied to COVID-19 disease transmission dynamics;
nearcasting and forecasting for needs of critical medical resources
including personal protective equipment (PPE); predicting COVID-19
mortality; evaluating effectiveness of convalescent plasma
treatment and the logistic implementation challenges; estimating
impact of delays in contact tracing; quantifying heterogeneity in
contact mixing and its evaluation with social distancing; modelling
point of care diagnostics of COVID-19; and understanding
non-reporting and underestimation. Further, readers will have the
opportunity to learn about current modelling methodologies and
technologies for emerging infectious disease outbreaks, pandemic
mitigation rapid response, and the mathematics behind them. The
volume will help the general audience and experts to better
understand the important role that mathematics has been playing
during this on-going crisis in supporting critical decision-making
by governments and public health agencies.
This collection covers a wide range of topics of infinite
dimensional dynamical systems generated by parabolic partial
differential equations, hyperbolic partial differential equations,
solitary equations, lattice differential equations, delay
differential equations, and stochastic differential equations.
Infinite dimensional dynamical systems are generated by
evolutionary equations describing the evolutions in time of systems
whose status must be depicted in infinite dimensional phase spaces.
Studying the long-term behaviors of such systems is important in
our understanding of their spatiotemporal pattern formation and
global continuation, and has been among major sources of motivation
and applications of new developments of nonlinear analysis and
other mathematical theories. Theories of the infinite dimensional
dynamical systems have also found more and more important
applications in physical, chemical, and life sciences. This book
collects 19 papers from 48 invited lecturers to the International
Conference on Infinite Dimensional Dynamical Systems held at York
University, Toronto, in September of 2008. As the conference was
dedicated to Professor George Sell from University of Minnesota on
the occasion of his 70th birthday, this collection reflects the
pioneering work and influence of Professor Sell in a few core areas
of dynamical systems, including non-autonomous dynamical systems,
skew-product flows, invariant manifolds theory, infinite
dimensional dynamical systems, approximation dynamics, and fluid
flows. "
In the design of a neural network, either for biological modeling,
cognitive simulation, numerical computation or engineering
applications, it is important to investigate the network's
computational performance which is usually described by the
long-term behaviors, called dynamics, of the model equations. The
purpose of this book is to give an introduction to the mathematical
modeling and analysis of networks of neurons from the viewpoint of
dynamical systems.
Data clustering, also known as cluster analysis, is an unsupervised
process that divides a set of objects into homogeneous groups.
Since the publication of the first edition of this monograph in
2007, development in the area has exploded, especially in
clustering algorithms for big data and open-source software for
cluster analysis. This second edition reflects these new
developments. Data Clustering: Theory, Algorithms, and
Applications, Second Edition: covers the basics of data clustering,
includes a list of popular clustering algorithms, and provides
program code that helps users implement clustering algorithms.
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