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This book analyzes the impact of quiescent phases on biological
models. Quiescence arises, for example, when moving individuals
stop moving, hunting predators take a rest, infected individuals
are isolated, or cells enter the quiescent compartment of the cell
cycle. In the first chapter of Topics in Mathematical Biology
general principles about coupled and quiescent systems are derived,
including results on shrinking periodic orbits and stabilization of
oscillations via quiescence. In subsequent chapters classical
biological models are presented in detail and challenged by the
introduction of quiescence. These models include delay equations,
demographic models, age structured models, Lotka-Volterra systems,
replicator systems, genetic models, game theory, Nash equilibria,
evolutionary stable strategies, ecological models, epidemiological
models, random walks and reaction-diffusion models. In each case we
find new and interesting results such as stability of fixed points
and/or periodic orbits, excitability of steady states, epidemic
outbreaks, survival of the fittest, and speeds of invading fronts.
The textbook is intended for graduate students and researchers in
mathematical biology who have a solid background in linear algebra,
differential equations and dynamical systems. Readers can find gems
of unexpected beauty within these pages, and those who knew K.P.
(as he was often called) well will likely feel his presence and
hear him speaking to them as they read.
This book provides an overview of the main approaches used to
analyze the dynamics of cellular automata. Cellular automata are an
indispensable tool in mathematical modeling. In contrast to
classical modeling approaches like partial differential equations,
cellular automata are relatively easy to simulate but difficult to
analyze. In this book we present a review of approaches and
theories that allow the reader to understand the behavior of
cellular automata beyond simulations. The first part consists of an
introduction to cellular automata on Cayley graphs, and their
characterization via the fundamental Cutis-Hedlund-Lyndon theorems
in the context of various topological concepts (Cantor, Besicovitch
and Weyl topology). The second part focuses on classification
results: What classification follows from topological concepts
(Hurley classification), Lyapunov stability (Gilman
classification), and the theory of formal languages and grammars
(Kurka classification)? These classifications suggest that cellular
automata be clustered, similar to the classification of partial
differential equations into hyperbolic, parabolic and elliptic
equations. This part of the book culminates in the question of
whether the properties of cellular automata are decidable.
Surjectivity and injectivity are examined, and the seminal Garden
of Eden theorems are discussed. In turn, the third part focuses on
the analysis of cellular automata that inherit distinct properties,
often based on mathematical modeling of biological, physical or
chemical systems. Linearity is a concept that allows us to define
self-similar limit sets. Models for particle motion show how to
bridge the gap between cellular automata and partial differential
equations (HPP model and ultradiscrete limit). Pattern formation is
related to linear cellular automata, to the Bar-Yam model for the
Turing pattern, and Greenberg-Hastings automata for excitable
media. In addition, models for sand piles, the dynamics of
infectious d
This book provides an overview of the main approaches used to
analyze the dynamics of cellular automata. Cellular automata are an
indispensable tool in mathematical modeling. In contrast to
classical modeling approaches like partial differential equations,
cellular automata are relatively easy to simulate but difficult to
analyze. In this book we present a review of approaches and
theories that allow the reader to understand the behavior of
cellular automata beyond simulations. The first part consists of an
introduction to cellular automata on Cayley graphs, and their
characterization via the fundamental Cutis-Hedlund-Lyndon theorems
in the context of various topological concepts (Cantor, Besicovitch
and Weyl topology). The second part focuses on classification
results: What classification follows from topological concepts
(Hurley classification), Lyapunov stability (Gilman
classification), and the theory of formal languages and grammars
(Kurka classification)? These classifications suggest that cellular
automata be clustered, similar to the classification of partial
differential equations into hyperbolic, parabolic and elliptic
equations. This part of the book culminates in the question of
whether the properties of cellular automata are decidable.
Surjectivity and injectivity are examined, and the seminal Garden
of Eden theorems are discussed. In turn, the third part focuses on
the analysis of cellular automata that inherit distinct properties,
often based on mathematical modeling of biological, physical or
chemical systems. Linearity is a concept that allows us to define
self-similar limit sets. Models for particle motion show how to
bridge the gap between cellular automata and partial differential
equations (HPP model and ultradiscrete limit). Pattern formation is
related to linear cellular automata, to the Bar-Yam model for the
Turing pattern, and Greenberg-Hastings automata for excitable
media. In addition, models for sand piles, the dynamics of
infectious d
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