Author: D.S. Jones,Michael Plank,B.D. Sleeman

Publisher: CRC Press

ISBN: 9781420083583

Category: Mathematics

Page: 462

View: 3575

Deepen students’ understanding of biological phenomena Suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, Differential Equations and Mathematical Biology, Second Edition introduces students in the physical, mathematical, and biological sciences to fundamental modeling and analytical techniques used to understand biological phenomena. In this edition, many of the chapters have been expanded to include new and topical material. New to the Second Edition A section on spiral waves Recent developments in tumor biology More on the numerical solution of differential equations and numerical bifurcation analysis MATLAB® files available for download online Many additional examples and exercises This textbook shows how first-order ordinary differential equations (ODEs) are used to model the growth of a population, the administration of drugs, and the mechanism by which living cells divide. The authors present linear ODEs with constant coefficients, extend the theory to systems of equations, model biological phenomena, and offer solutions to first-order autonomous systems of nonlinear differential equations using the Poincaré phase plane. They also analyze the heartbeat, nerve impulse transmission, chemical reactions, and predator–prey problems. After covering partial differential equations and evolutionary equations, the book discusses diffusion processes, the theory of bifurcation, and chaotic behavior. It concludes with problems of tumor growth and the spread of infectious diseases.
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Author: Darren J. Wilkinson

Publisher: CRC Press

ISBN: 1439837724

Category: Mathematics

Page: 363

View: 8034

Since the first edition of Stochastic Modelling for Systems Biology, there have been many interesting developments in the use of "likelihood-free" methods of Bayesian inference for complex stochastic models. Re-written to reflect this modern perspective, this second edition covers everything necessary for a good appreciation of stochastic kinetic modelling of biological networks in the systems biology context. Keeping with the spirit of the first edition, all of the new theory is presented in a very informal and intuitive manner, keeping the text as accessible as possible to the widest possible readership. New in the Second Edition All examples have been updated to Systems Biology Markup Language Level 3 All code relating to simulation, analysis, and inference for stochastic kinetic models has been re-written and re-structured in a more modular way An ancillary website provides links, resources, errata, and up-to-date information on installation and use of the associated R package More background material on the theory of Markov processes and stochastic differential equations, providing more substance for mathematically inclined readers Discussion of some of the more advanced concepts relating to stochastic kinetic models, such as random time change representations, Kolmogorov equations, Fokker-Planck equations and the linear noise approximation Simple modelling of "extrinsic" and "intrinsic" noise An effective introduction to the area of stochastic modelling in computational systems biology, this new edition adds additional mathematical detail and computational methods that will provide a stronger foundation for the development of more advanced courses in stochastic biological modelling.
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Design Principles of Biological Circuits

Author: Uri Alon

Publisher: CRC Press

ISBN: 1584886420

Category: Mathematics

Page: 320

View: 8432

Thorough and accessible, this book presents the design principles of biological systems, and highlights the recurring circuit elements that make up biological networks. It provides a simple mathematical framework which can be used to understand and even design biological circuits. The textavoids specialist terms, focusing instead on several well-studied biological systems that concisely demonstrate key principles. An Introduction to Systems Biology: Design Principles of Biological Circuits builds a solid foundation for the intuitive understanding of general principles. It encourages the reader to ask why a system is designed in a particular way and then proceeds to answer with simplified models.
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Mathematical Modeling and Model Analysis

Author: Andreas Kremling

Publisher: CRC Press

ISBN: 1466567902

Category: Mathematics

Page: 379

View: 9698

Drawing on the latest research in the field, Systems Biology: Mathematical Modeling and Model Analysis presents many methods for modeling and analyzing biological systems, in particular cellular systems. It shows how to use predictive mathematical models to acquire and analyze knowledge about cellular systems. It also explores how the models are systematically applied in biotechnology. The first part of the book introduces biological basics, such as metabolism, signaling, gene expression, and control as well as mathematical modeling fundamentals, including deterministic models and thermodynamics. The text also discusses linear regression methods, explains the differences between linear and nonlinear regression, and illustrates how to determine input variables to improve estimation accuracy during experimental design. The second part covers intracellular processes, including enzymatic reactions, polymerization processes, and signal transduction. The author highlights the process–function–behavior sequence in cells and shows how modeling and analysis of signal transduction units play a mediating role between process and function. The third part presents theoretical methods that address the dynamics of subsystems and the behavior near a steady state. It covers techniques for determining different time scales, sensitivity analysis, structural kinetic modeling, and theoretical control engineering aspects, including a method for robust control. It also explores frequent patterns (motifs) in biochemical networks, such as the feed-forward loop in the transcriptional network of E. coli. Moving on to models that describe a large number of individual reactions, the last part looks at how these cellular models are used in biotechnology. The book also explains how graphs can illustrate the link between two components in large networks with several interactions.
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Author: Suzanne Lenhart,John T. Workman

Publisher: CRC Press

ISBN: 1584886404

Category: Mathematics

Page: 280

View: 7048

From economics and business to the biological sciences to physics and engineering, professionals successfully use the powerful mathematical tool of optimal control to make management and strategy decisions. Optimal Control Applied to Biological Models thoroughly develops the mathematical aspects of optimal control theory and provides insight into the application of this theory to biological models. Focusing on mathematical concepts, the book first examines the most basic problem for continuous time ordinary differential equations (ODEs) before discussing more complicated problems, such as variations of the initial conditions, imposed bounds on the control, multiple states and controls, linear dependence on the control, and free terminal time. In addition, the authors introduce the optimal control of discrete systems and of partial differential equations (PDEs). Featuring a user-friendly interface, the book contains fourteen interactive sections of various applications, including immunology and epidemic disease models, management decisions in harvesting, and resource allocation models. It also develops the underlying numerical methods of the applications and includes the MATLAB® codes on which the applications are based. Requiring only basic knowledge of multivariable calculus, simple ODEs, and mathematical models, this text shows how to adjust controls in biological systems in order to achieve proper outcomes.
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Author: Michael Small

Publisher: CRC Press

ISBN: 1439853363

Category: Mathematics

Page: 284

View: 1217

From the spontaneous rapid firing of cortical neurons to the spatial diffusion of disease epidemics, biological systems exhibit rich dynamic behaviour over a vast range of time and space scales. Unifying many of these diverse phenomena, Dynamics of Biological Systems provides the computational and mathematical platform from which to understand the underlying processes of the phenomena. Through an extensive tour of various biological systems, the text introduces computational methods for simulating spatial diffusion processes in excitable media, such as the human heart, as well as mathematical tools for dealing with systems of nonlinear ordinary and partial differential equations, such as neuronal activation and disease diffusion. The mathematical models and computer simulations offer insight into the dynamics of temporal and spatial biological systems, including cardiac pacemakers, artificial electrical defibrillation, pandemics, pattern formation, flocking behaviour, the interaction of autonomous agents, and hierarchical and structured network topologies. Tools from complex systems and complex networks are also presented for dealing with real phenomenological systems. With exercises and projects in each chapter, this classroom-tested text shows students how to apply a variety of mathematical and computational techniques to model and analyze the temporal and spatial phenomena of biological systems. MATLAB® implementations of algorithms and case studies are available on the author’s website.
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Author: Fred Brauer,Dawn Bies

Publisher: Springer Science & Business Media

ISBN: 1475735162

Category: Science

Page: 417

View: 8049

The goal of this book is to search for a balance between simple and analyzable models and unsolvable models which are capable of addressing important questions on population biology. Part I focusses on single species simple models including those which have been used to predict the growth of human and animal population in the past. Single population models are, in some sense, the building blocks of more realistic models -- the subject of Part II. Their role is fundamental to the study of ecological and demographic processes including the role of population structure and spatial heterogeneity -- the subject of Part III. This book, which will include both examples and exercises, is of use to practitioners, graduate students, and scientists working in the field.
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Author: Christian Mazza,Michel Benaim

Publisher: CRC Press

ISBN: 1466514949

Category: Mathematics

Page: 274

View: 9782

Stochastic Dynamics for Systems Biology is one of the first books to provide a systematic study of the many stochastic models used in systems biology. The book shows how the mathematical models are used as technical tools for simulating biological processes and how the models lead to conceptual insights on the functioning of the cellular processing system. Most of the text should be accessible to scientists with basic knowledge in calculus and probability theory. The authors illustrate the relevant Markov chain theory using realistic models from systems biology, including signaling and metabolic pathways, phosphorylation processes, genetic switches, and transcription. A central part of the book presents an original and up-to-date treatment of cooperativity. The book defines classical indexes, such as the Hill coefficient, using notions from statistical mechanics. It explains why binding curves often have S-shapes and why cooperative behaviors can lead to ultrasensitive genetic switches. These notions are then used to model transcription rates. Examples cover the phage lambda genetic switch and eukaryotic gene expression. The book then presents a short course on dynamical systems and describes stochastic aspects of linear noise approximation. This mathematical framework enables the simplification of complex stochastic dynamics using Gaussian processes and nonlinear ODEs. Simple examples illustrate the technique in noise propagation in gene networks and the effects of network structures on multistability and gene expression noise levels. The last chapter provides up-to-date results on stochastic and deterministic mass action kinetics with applications to enzymatic biochemical reactions and metabolic pathways.
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Author: H.T. Banks,H.T. Tran

Publisher: CRC Press

ISBN: 9781420073386

Category: Mathematics

Page: 298

View: 8646

Through several case study problems from industrial and scientific research laboratory applications, Mathematical and Experimental Modeling of Physical and Biological Processes provides students with a fundamental understanding of how mathematics is applied to problems in science and engineering. For each case study problem, the authors discuss why a model is needed and what goals can be achieved with the model. Exploring what mathematics can reveal about applications, the book focuses on the design of appropriate experiments to validate the development of mathematical models. It guides students through the modeling process, from empirical observations and formalization of properties to model analysis and interpretation of results. The authors also describe the hardware and software tools used to design the experiments so faculty/students can duplicate them. Integrating real-world applications into the traditional mathematics curriculum, this textbook deals with the formulation and analysis of mathematical models in science and engineering. It gives students an appreciation of the use of mathematics and encourages them to further study the applied topics. Real experimental data for projects can be downloaded from CRC Press Online.
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Theory, Methods and Practice

Author: Azmy S. Ackleh,Edward James Allen,R. Baker Kearfott,Padmanabhan Seshaiyer

Publisher: CRC Press

ISBN: 9781420091588

Category: Mathematics

Page: 628

View: 3215

Classical and Modern Numerical Analysis: Theory, Methods and Practice provides a sound foundation in numerical analysis for more specialized topics, such as finite element theory, advanced numerical linear algebra, and optimization. It prepares graduate students for taking doctoral examinations in numerical analysis. The text covers the main areas of introductory numerical analysis, including the solution of nonlinear equations, numerical linear algebra, ordinary differential equations, approximation theory, numerical integration, and boundary value problems. Focusing on interval computing in numerical analysis, it explains interval arithmetic, interval computation, and interval algorithms. The authors illustrate the concepts with many examples as well as analytical and computational exercises at the end of each chapter. This advanced, graduate-level introduction to the theory and methods of numerical analysis supplies the necessary background in numerical methods so that students can apply the techniques and understand the mathematical literature in this area. Although the book is independent of a specific computer program, MATLAB® code is available on the authors' website to illustrate various concepts.
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Proceedings of the International Symposium on Mathematical and Computational Biology

Author: Rubem P Mondaini

Publisher: World Scientific

ISBN: 981460223X

Category: Mathematics

Page: 428

View: 7788

This is a book of a series on interdisciplinary topics on the Biological and Mathematical Sciences. The chapters correspond to selected papers on special research themes, which have been presented at BIOMAT 2013 International Symposium on Mathematical and Computational Biology which was held in the Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, Canada, on November 04 – 08, 2013. The treatment is both pedagogical and advanced in order to motivate research students as well as to fulfill the requirements of professional practitioners. There are comprehensive reviews written by prominent scientific leaders of famous research groups. Contents:Population Dynamics:The Princess and the Pea: The Unexpected Importance of Movement Algorithms (Rebecca Tyson)Plankton Nutrient Interaction Model with Harvesting under Constant Environment (Samares Pal and A Chatterjee)Traveling Wave Solutions for a Chemotaxis System (F Catrina and V M Reyes G)Dynamics of a General Stage Structured N Parallel Food Chains (Isam Al-Darabsah and Yuan Yuan)Pattern Recognition of Biological Phenomena:Complex Data Clustering: From Neural Network Architecture to Theory and Applications of Nonlinear Dynamics of Pattern Recognition (Guojun Gan, Jialun Yin, Yulia Wang and Jianhong Wu)Dynamic and Geometric Modelling of Biomolecular Structures:A Two-Step Kinetic Model of Insulin Aggregation with a Competitive Inhibitor (Mark Whidden, Allison Ho and Santiago Schnell)Optimal Control Techniques in Mathematical Modelling of Biological Phenomena:Optimal Control of Resource Coefficient in a Parabolic Population Model (J Bintz, H Finotti and S Lenhart)Optimization of Costs for Combating Aedes Aegypti in Optimal Time-Windows (W O Dias, G A Xavier, D A P Lima, E F Wanner and R T N Cardoso)Dynamics of a Varroa-Infested Honey Bee Colonies Model (K O Okosun)Computational Biology:Probability Distributions of GC Content Reflect the Evolution of Primate Species (Marco V José, Qi Lu and Juan R Bobadilla)Mining the Constraints of Protein Evolution (Fernando Encinas and Antonio Basilio de Miranda)Entropy Measures Based Methods for the Classification of Protein Domains into Families and Clans (Nicolas Carels, Cecilia F Mondaini and Rubem P Mondaini)Modelling Physiological Disorders:Modelling of Porous Elastic and Viscoelastic Media and Its Application to the Brain (R Begg, J Murley, M. Kohandel and S Sivaloganathan)The Mathematics of Liver Transplantation (F A B Coutinho, E Chaib, M Amaku, M M Burattini and E Massad)Complexity of Molecular Signaling Networks for Various Types of Cancer and Neurological Diseases Correlates with Patient Survivability (D Breitkreutz, E A Rietman, P Hinow, M Healey and J A Tuszynski)Mathematical Modelling of Infectious Diseases:Modelling Malaria Dynamics in Temperate Regions with Long Term Incubation Period (Kyeongah Nah, Gergely Röst and Yongkuk Kim)A Simulation of the U S Influenza Outbreak in 2009–2010 Using a Patch SIR Model based on Airport Transportation Data (D L Wallace and M Chen)Modelling Directly Transmitted Infections considering Age-structured Contact Rate and Vaccination (H M Yang and C H Dezotti)A General Framework for Agent-Based Modelling with Applications to Infectious Disease Dynamics (Marek Laskowski and Seyed M Moghadas)Analysis of the Basic Reproduction Number from the Initial Growth Phase of the Outbreak in Diseases Caused by Vectors (R P Sanches and E Massad)Parameter Estimation of a Tuberculosis Model in a Patchy Environment: Case of Cameroon (D P Moualeu, S Bowong and J Kürts)An Agent-Based Modelling Framework for Tuberculosis Infection with Drug-Resistance (Aquino L Espindola, A S Martinez and Seyed M Moghadas)Some Extensions of the Classical Epidemic Models (Fred Brauer) Readership: Undergraduates, graduates, researchers and all practitioners on the interdisciplinary fields of Mathematical Biology, Biological Physics and Mathematical Modelling of Biosystems. Keywords:Mathematical Biology;Biomathematics;Mathematical Modelling of Biosystems;Biological Physics;Biophysics;Computational Biology;Bioinformatics
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Author: James R. Kirkwood

Publisher: Academic Press

ISBN: 0123869110

Category: Mathematics

Page: 418

View: 9110

Suitable for advanced undergraduate and beginning graduate students taking a course on mathematical physics, this title presents some of the most important topics and methods of mathematical physics. It contains mathematical derivations and solutions - reinforcing the material through repetition of both the equations and the techniques.
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Theory, Applications and Advanced Topics, Third Edition

Author: Ronald E. Mickens

Publisher: CRC Press

ISBN: 1482230798

Category: Mathematics

Page: 555

View: 5591

Difference Equations: Theory, Applications and Advanced Topics, Third Edition provides a broad introduction to the mathematics of difference equations and some of their applications. Many worked examples illustrate how to calculate both exact and approximate solutions to special classes of difference equations. Along with adding several advanced topics, this edition continues to cover general, linear, first-, second-, and n-th order difference equations; nonlinear equations that may be reduced to linear equations; and partial difference equations. New to the Third Edition New chapter on special topics, including discrete Cauchy–Euler equations; gamma, beta, and digamma functions; Lambert W-function; Euler polynomials; functional equations; and exact discretizations of differential equations New chapter on the application of difference equations to complex problems arising in the mathematical modeling of phenomena in engineering and the natural and social sciences Additional problems in all chapters Expanded bibliography to include recently published texts related to the subject of difference equations Suitable for self-study or as the main text for courses on difference equations, this book helps readers understand the fundamental concepts and procedures of difference equations. It uses an informal presentation style, avoiding the minutia of detailed proofs and formal explanations.
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Author: Linda J. S. Allen

Publisher: Chapman and Hall/CRC

ISBN: 9781439818824

Category: Mathematics

Page: 490

View: 2991

An Introduction to Stochastic Processes with Applications to Biology, Second Edition presents the basic theory of stochastic processes necessary in understanding and applying stochastic methods to biological problems in areas such as population growth and extinction, drug kinetics, two-species competition and predation, the spread of epidemics, and the genetics of inbreeding. Because of their rich structure, the text focuses on discrete and continuous time Markov chains and continuous time and state Markov processes. New to the Second Edition A new chapter on stochastic differential equations that extends the basic theory to multivariate processes, including multivariate forward and backward Kolmogorov differential equations and the multivariate Itô’s formula The inclusion of examples and exercises from cellular and molecular biology Double the number of exercises and MATLAB® programs at the end of each chapter Answers and hints to selected exercises in the appendix Additional references from the literature This edition continues to provide an excellent introduction to the fundamental theory of stochastic processes, along with a wide range of applications from the biological sciences. To better visualize the dynamics of stochastic processes, MATLAB programs are provided in the chapter appendices.
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Author: Andrei D. Polyanin,Vladimir E. Nazaikinskii

Publisher: CRC Press

ISBN: 1466581492

Category: Mathematics

Page: 1609

View: 1332

Includes nearly 4,000 linear partial differential equations (PDEs) with solutions Presents solutions of numerous problems relevant to heat and mass transfer, wave theory, hydrodynamics, aerodynamics, elasticity, acoustics, electrodynamics, diffraction theory, quantum mechanics, chemical engineering sciences, electrical engineering, and other fields Outlines basic methods for solving various problems in science and engineering Contains much more linear equations, problems, and solutions than any other book currently available Provides a database of test problems for numerical and approximate analytical methods for solving linear PDEs and systems of coupled PDEs New to the Second Edition More than 700 pages with 1,500+ new first-, second-, third-, fourth-, and higher-order linear equations with solutions Systems of coupled PDEs with solutions Some analytical methods, including decomposition methods and their applications Symbolic and numerical methods for solving linear PDEs with Maple, Mathematica, and MATLAB® Many new problems, illustrative examples, tables, and figures To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the material in increasing order of complexity.
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Author: Saber N. Elaydi

Publisher: Springer Science & Business Media

ISBN: 1475791682

Category: Mathematics

Page: 390

View: 7525

This book grew out of lecture notes I used in a course on difference equations that I taught at Trinity University for the past five years. The classes were largely pop ulated by juniors and seniors majoring in Mathematics, Engineering, Chemistry, Computer Science, and Physics. This book is intended to be used as a textbook for a course on difference equations at the level of both advanced undergraduate and beginning graduate. It may also be used as a supplement for engineering courses on discrete systems and control theory. The main prerequisites for most of the material in this book are calculus and linear algebra. However, some topics in later chapters may require some rudiments of advanced calculus. Since many of the chapters in the book are independent, the instructor has great flexibility in choosing topics for the first one-semester course. A diagram showing the interdependence of the chapters in the book appears following the preface. This book presents the current state of affairs in many areas such as stability, Z-transform, asymptoticity, oscillations and control theory. However, this book is by no means encyclopedic and does not contain many important topics, such as Numerical Analysis, Combinatorics, Special functions and orthogonal polyno mials, boundary value problems, partial difference equations, chaos theory, and fractals. The nonselection of these topics is dictated not only by the limitations imposed by the elementary nature of this book, but also by the research interest (or lack thereof) of the author.
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Author: Mark Broom,Jan Rychtar

Publisher: CRC Press

ISBN: 1439853215

Category: Mathematics

Page: 520

View: 4508

Covering the major topics of evolutionary game theory, Game-Theoretical Models in Biology presents both abstract and practical mathematical models of real biological situations. It discusses the static aspects of game theory in a mathematically rigorous way that is appealing to mathematicians. In addition, the authors explore many applications of game theory to biology, making the text useful to biologists as well. The book describes a wide range of topics in evolutionary games, including matrix games, replicator dynamics, the hawk-dove game, and the prisoner’s dilemma. It covers the evolutionarily stable strategy, a key concept in biological games, and offers in-depth details of the mathematical models. Most chapters illustrate how to use MATLAB® to solve various games. Important biological phenomena, such as the sex ratio of so many species being close to a half, the evolution of cooperative behavior, and the existence of adornments (for example, the peacock’s tail), have been explained using ideas underpinned by game theoretical modeling. Suitable for readers studying and working at the interface of mathematics and the life sciences, this book shows how evolutionary game theory is used in the modeling of these diverse biological phenomena.
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Using Modern Discrete Models

Author: Raina Robeva,Terrell Hodge

Publisher: Academic Press

ISBN: 0124157939

Category: Mathematics

Page: 372

View: 9440

Mathematical Concepts and Methods in Modern Biology offers a quantitative framework for analyzing, predicting, and modulating the behavior of complex biological systems. The book presents important mathematical concepts, methods and tools in the context of essential questions raised in modern biology. Designed around the principles of project-based learning and problem-solving, the book considers biological topics such as neuronal networks, plant population growth, metabolic pathways, and phylogenetic tree reconstruction. The mathematical modeling tools brought to bear on these topics include Boolean and ordinary differential equations, projection matrices, agent-based modeling and several algebraic approaches. Heavy computation in some of the examples is eased by the use of freely available open-source software. Features self-contained chapters with real biological research examples using freely available computational tools Spans several mathematical techniques at basic to advanced levels Offers broad perspective on the uses of algebraic geometry/polynomial algebra in molecular systems biology
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Linking Undergraduate Disciplines

Author: Lynn Arthur Steen

Publisher: MAA

ISBN: 9780883858189

Category: Mathematics

Page: 161

View: 6060

Math & Bio 2010: Linking Undergraduate Disciplines envisages a new educational paradigm in which the disciplines of mathematics and biology, currently quite separate, will be productively linked in the undergraduate science programs of the 21st century. As a science, biology depends increasingly on data, algorithms, and models; in virtually every respect, it is becoming more quantitative, more computational, and more mathematical. While these trends are related, they are not the same; they represent, rather, three different perspectives on what many are calling the "new biology." All three methods---quantitative, computational, mathematical---are spreading across the entire landscape of biological science from molecular to cellular, organismic and ecological. The aim of this volume is to alert members of both communities---biological and mathematical---to the expanding and exciting challenges of interdisciplinary work in these fields.
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An Introduction

Author: Fred Brauer,Christopher Kribs

Publisher: CRC Press

ISBN: 1498774040

Category: Mathematics

Page: 478

View: 3468

Dynamical Systems for Biological Modeling: An Introduction prepares both biology and mathematics students with the understanding and techniques necessary to undertake basic modeling of biological systems. It achieves this through the development and analysis of dynamical systems. The approach emphasizes qualitative ideas rather than explicit computations. Some technical details are necessary, but a qualitative approach emphasizing ideas is essential for understanding. The modeling approach helps students focus on essentials rather than extensive mathematical details, which is helpful for students whose primary interests are in sciences other than mathematics need or want. The book discusses a variety of biological modeling topics, including population biology, epidemiology, immunology, intraspecies competition, harvesting, predator-prey systems, structured populations, and more. The authors also include examples of problems with solutions and some exercises which follow the examples quite closely. In addition, problems are included which go beyond the examples, both in mathematical analysis and in the development of mathematical models for biological problems, in order to encourage deeper understanding and an eagerness to use mathematics in learning about biology.
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