Stochastic Modelling For Systems Biology Second Edition

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Stochastic Modelling for Systems Biology, Second Edition

Author : Darren J. Wilkinson
Publisher : CRC Press
ISBN 13 : 1439837724
Total Pages : 365 pages
Book Rating : 4.4/5 (398 download)

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Book Synopsis Stochastic Modelling for Systems Biology, Second Edition by : Darren J. Wilkinson

Download or read book Stochastic Modelling for Systems Biology, Second Edition written by Darren J. Wilkinson and published by CRC Press. This book was released on 2011-11-09 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Stochastic Modelling for Systems Biology, Third Edition

Author : Darren J. Wilkinson
Publisher : CRC Press
ISBN 13 : 1351000896
Total Pages : 366 pages
Book Rating : 4.3/5 (51 download)

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Book Synopsis Stochastic Modelling for Systems Biology, Third Edition by : Darren J. Wilkinson

Download or read book Stochastic Modelling for Systems Biology, Third Edition written by Darren J. Wilkinson and published by CRC Press. This book was released on 2018-12-07 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Having been thoroughly updated to reflect this, this third edition covers everything necessary for a good appreciation of stochastic kinetic modelling of biological networks in the systems biology context. New methods and applications are included in the book, and the use of R for practical illustration of the algorithms has been greatly extended. There is a brand new chapter on spatially extended systems, and the statistical inference chapter has also been extended with new methods, including approximate Bayesian computation (ABC). Stochastic Modelling for Systems Biology, Third Edition is now supplemented by an additional software library, written in Scala, described in a new appendix to the book. New in the Third Edition New chapter on spatially extended systems, covering the spatial Gillespie algorithm for reaction diffusion master equation models in 1- and 2-d, along with fast approximations based on the spatial chemical Langevin equation Significantly expanded chapter on inference for stochastic kinetic models from data, covering ABC, including ABC-SMC Updated R package, including code relating to all of the new material New R package for parsing SBML models into simulatable stochastic Petri net models New open-source software library, written in Scala, replicating most of the functionality of the R packages in a fast, compiled, strongly typed, functional language Keeping with the spirit of earlier editions, 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. An effective introduction to the area of stochastic modelling in computational systems biology, this new edition adds additional detail and computational methods that will provide a stronger foundation for the development of more advanced courses in stochastic biological modelling.

Stochastic Modelling for Systems Biology, Third Edition

Author : Darren J. Wilkinson
Publisher : CRC Press
ISBN 13 : 135100090X
Total Pages : 405 pages
Book Rating : 4.3/5 (51 download)

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Book Synopsis Stochastic Modelling for Systems Biology, Third Edition by : Darren J. Wilkinson

Download or read book Stochastic Modelling for Systems Biology, Third Edition written by Darren J. Wilkinson and published by CRC Press. This book was released on 2018-12-07 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Having been thoroughly updated to reflect this, this third edition covers everything necessary for a good appreciation of stochastic kinetic modelling of biological networks in the systems biology context. New methods and applications are included in the book, and the use of R for practical illustration of the algorithms has been greatly extended. There is a brand new chapter on spatially extended systems, and the statistical inference chapter has also been extended with new methods, including approximate Bayesian computation (ABC). Stochastic Modelling for Systems Biology, Third Edition is now supplemented by an additional software library, written in Scala, described in a new appendix to the book. New in the Third Edition New chapter on spatially extended systems, covering the spatial Gillespie algorithm for reaction diffusion master equation models in 1- and 2-d, along with fast approximations based on the spatial chemical Langevin equation Significantly expanded chapter on inference for stochastic kinetic models from data, covering ABC, including ABC-SMC Updated R package, including code relating to all of the new material New R package for parsing SBML models into simulatable stochastic Petri net models New open-source software library, written in Scala, replicating most of the functionality of the R packages in a fast, compiled, strongly typed, functional language Keeping with the spirit of earlier editions, 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. An effective introduction to the area of stochastic modelling in computational systems biology, this new edition adds additional detail and computational methods that will provide a stronger foundation for the development of more advanced courses in stochastic biological modelling.

Stochastic Modelling for Systems Biology

Author : Darren James Wilkinson
Publisher :
ISBN 13 : 9780429152030
Total Pages : 360 pages
Book Rating : 4.1/5 (52 download)

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Book Synopsis Stochastic Modelling for Systems Biology by : Darren James Wilkinson

Download or read book Stochastic Modelling for Systems Biology written by Darren James Wilkinson and published by . This book was released on 2012 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 accessib.

Systems Biology

Author : Edda Klipp
Publisher : John Wiley & Sons
ISBN 13 : 3527675663
Total Pages : 504 pages
Book Rating : 4.5/5 (276 download)

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Book Synopsis Systems Biology by : Edda Klipp

Download or read book Systems Biology written by Edda Klipp and published by John Wiley & Sons. This book was released on 2016-03-28 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: This advanced textbook is tailored for an introductory course in Systems Biology and is well-suited for biologists as well as engineers and computer scientists. It comes with student-friendly reading lists and a companion website featuring a short exam prep version of the book and educational modeling programs. The text is written in an easily accessible style and includes numerous worked examples and study questions in each chapter. For this edition, a section on medical systems biology has been included.

Markov Processes for Stochastic Modeling

Author : Oliver Ibe
Publisher : Newnes
ISBN 13 : 0124078397
Total Pages : 515 pages
Book Rating : 4.1/5 (24 download)

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Book Synopsis Markov Processes for Stochastic Modeling by : Oliver Ibe

Download or read book Markov Processes for Stochastic Modeling written by Oliver Ibe and published by Newnes. This book was released on 2013-05-22 with total page 515 pages. Available in PDF, EPUB and Kindle. Book excerpt: Markov processes are processes that have limited memory. In particular, their dependence on the past is only through the previous state. They are used to model the behavior of many systems including communications systems, transportation networks, image segmentation and analysis, biological systems and DNA sequence analysis, random atomic motion and diffusion in physics, social mobility, population studies, epidemiology, animal and insect migration, queueing systems, resource management, dams, financial engineering, actuarial science, and decision systems. Covering a wide range of areas of application of Markov processes, this second edition is revised to highlight the most important aspects as well as the most recent trends and applications of Markov processes. The author spent over 16 years in the industry before returning to academia, and he has applied many of the principles covered in this book in multiple research projects. Therefore, this is an applications-oriented book that also includes enough theory to provide a solid ground in the subject for the reader. - Presents both the theory and applications of the different aspects of Markov processes - Includes numerous solved examples as well as detailed diagrams that make it easier to understand the principle being presented - Discusses different applications of hidden Markov models, such as DNA sequence analysis and speech analysis.

An Introduction to Stochastic Modeling

Author : Howard M. Taylor
Publisher : Academic Press
ISBN 13 : 1483269272
Total Pages : 410 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis An Introduction to Stochastic Modeling by : Howard M. Taylor

Download or read book An Introduction to Stochastic Modeling written by Howard M. Taylor and published by Academic Press. This book was released on 2014-05-10 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Introduction to Stochastic Modeling provides information pertinent to the standard concepts and methods of stochastic modeling. This book presents the rich diversity of applications of stochastic processes in the sciences. Organized into nine chapters, this book begins with an overview of diverse types of stochastic models, which predicts a set of possible outcomes weighed by their likelihoods or probabilities. This text then provides exercises in the applications of simple stochastic analysis to appropriate problems. Other chapters consider the study of general functions of independent, identically distributed, nonnegative random variables representing the successive intervals between renewals. This book discusses as well the numerous examples of Markov branching processes that arise naturally in various scientific disciplines. The final chapter deals with queueing models, which aid the design process by predicting system performance. This book is a valuable resource for students of engineering and management science. Engineers will also find this book useful.

Stochastic Modeling

Author : Nicolas Lanchier
Publisher : Springer
ISBN 13 : 3319500384
Total Pages : 305 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Stochastic Modeling by : Nicolas Lanchier

Download or read book Stochastic Modeling written by Nicolas Lanchier and published by Springer. This book was released on 2017-01-27 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: Three coherent parts form the material covered in this text, portions of which have not been widely covered in traditional textbooks. In this coverage the reader is quickly introduced to several different topics enriched with 175 exercises which focus on real-world problems. Exercises range from the classics of probability theory to more exotic research-oriented problems based on numerical simulations. Intended for graduate students in mathematics and applied sciences, the text provides the tools and training needed to write and use programs for research purposes. The first part of the text begins with a brief review of measure theory and revisits the main concepts of probability theory, from random variables to the standard limit theorems. The second part covers traditional material on stochastic processes, including martingales, discrete-time Markov chains, Poisson processes, and continuous-time Markov chains. The theory developed is illustrated by a variety of examples surrounding applications such as the gambler’s ruin chain, branching processes, symmetric random walks, and queueing systems. The third, more research-oriented part of the text, discusses special stochastic processes of interest in physics, biology, and sociology. Additional emphasis is placed on minimal models that have been used historically to develop new mathematical techniques in the field of stochastic processes: the logistic growth process, the Wright –Fisher model, Kingman’s coalescent, percolation models, the contact process, and the voter model. Further treatment of the material explains how these special processes are connected to each other from a modeling perspective as well as their simulation capabilities in C and MatlabTM.

Stochastic Dynamics in Computational Biology

Author : Stefanie Winkelmann
Publisher : Springer Nature
ISBN 13 : 3030623874
Total Pages : 284 pages
Book Rating : 4.0/5 (36 download)

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Book Synopsis Stochastic Dynamics in Computational Biology by : Stefanie Winkelmann

Download or read book Stochastic Dynamics in Computational Biology written by Stefanie Winkelmann and published by Springer Nature. This book was released on 2021-01-04 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations. Another main objective is to illustrate and compare diverse numerical simulation schemes and their computational effort. Unlike related works, this book presents a broad scope in its applications, from offering a detailed introduction to hybrid approaches for the case of multiple population scales to discussing the setting of time-scale separation resulting from widely varying firing rates of reaction channels. Additionally, it also addresses modeling approaches for non well-mixed reaction-diffusion dynamics, including deterministic and stochastic PDEs and spatiotemporal master equations. Finally, by translating and incorporating complex theory to a level accessible to non-mathematicians, this book effectively bridges the gap between mathematical research in computational biology and its practical use in biological, biochemical, and biomedical systems.

Stochastic Dynamics for Systems Biology

Author : Christian Mazza
Publisher : CRC Press
ISBN 13 : 1466514949
Total Pages : 274 pages
Book Rating : 4.4/5 (665 download)

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Book Synopsis Stochastic Dynamics for Systems Biology by : Christian Mazza

Download or read book Stochastic Dynamics for Systems Biology written by Christian Mazza and published by CRC Press. This book was released on 2016-04-19 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Computational Systems Biology

Author : Andres Kriete
Publisher : Academic Press
ISBN 13 : 0124059384
Total Pages : 549 pages
Book Rating : 4.1/5 (24 download)

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Book Synopsis Computational Systems Biology by : Andres Kriete

Download or read book Computational Systems Biology written by Andres Kriete and published by Academic Press. This book was released on 2013-11-26 with total page 549 pages. Available in PDF, EPUB and Kindle. Book excerpt: This comprehensively revised second edition of Computational Systems Biology discusses the experimental and theoretical foundations of the function of biological systems at the molecular, cellular or organismal level over temporal and spatial scales, as systems biology advances to provide clinical solutions to complex medical problems. In particular the work focuses on the engineering of biological systems and network modeling. - Logical information flow aids understanding of basic building blocks of life through disease phenotypes - Evolved principles gives insight into underlying organizational principles of biological organizations, and systems processes, governing functions such as adaptation or response patterns - Coverage of technical tools and systems helps researchers to understand and resolve specific systems biology problems using advanced computation - Multi-scale modeling on disparate scales aids researchers understanding of dependencies and constraints of spatio-temporal relationships fundamental to biological organization and function.

Stochastic Processes in Cell Biology

Author : Paul C. Bressloff
Publisher : Springer Nature
ISBN 13 : 3030725154
Total Pages : 773 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Stochastic Processes in Cell Biology by : Paul C. Bressloff

Download or read book Stochastic Processes in Cell Biology written by Paul C. Bressloff and published by Springer Nature. This book was released on 2022-01-04 with total page 773 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops the theory of continuous and discrete stochastic processes within the context of cell biology. In the second edition the material has been significantly expanded, particularly within the context of nonequilibrium and self-organizing systems. Given the amount of additional material, the book has been divided into two volumes, with volume I mainly covering molecular processes and volume II focusing on cellular processes. A wide range of biological topics are covered in the new edition, including stochastic ion channels and excitable systems, molecular motors, stochastic gene networks, genetic switches and oscillators, epigenetics, normal and anomalous diffusion in complex cellular environments, stochastically-gated diffusion, active intracellular transport, signal transduction, cell sensing, bacterial chemotaxis, intracellular pattern formation, cell polarization, cell mechanics, biological polymers and membranes, nuclear structure and dynamics, biological condensates, molecular aggregation and nucleation, cellular length control, cell mitosis, cell motility, cell adhesion, cytoneme-based morphogenesis, bacterial growth, and quorum sensing. The book also provides a pedagogical introduction to the theory of stochastic and nonequilibrium processes – Fokker Planck equations, stochastic differential equations, stochastic calculus, master equations and jump Markov processes, birth-death processes, Poisson processes, first passage time problems, stochastic hybrid systems, queuing and renewal theory, narrow capture and escape, extreme statistics, search processes and stochastic resetting, exclusion processes, WKB methods, large deviation theory, path integrals, martingales and branching processes, numerical methods, linear response theory, phase separation, fluctuation-dissipation theorems, age-structured models, and statistical field theory. This text is primarily aimed at graduate students and researchers working in mathematical biology, statistical and biological physicists, and applied mathematicians interested in stochastic modeling. Applied probabilists should also find it of interest. It provides significant background material in applied mathematics and statistical physics, and introduces concepts in stochastic and nonequilibrium processes via motivating biological applications. The book is highly illustrated and contains a large number of examples and exercises that further develop the models and ideas in the body of the text. It is based on a course that the author has taught at the University of Utah for many years.

Encyclopedia of Bioinformatics and Computational Biology

Author :
Publisher : Elsevier
ISBN 13 : 0128114320
Total Pages : 3421 pages
Book Rating : 4.1/5 (281 download)

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Book Synopsis Encyclopedia of Bioinformatics and Computational Biology by :

Download or read book Encyclopedia of Bioinformatics and Computational Biology written by and published by Elsevier. This book was released on 2018-08-21 with total page 3421 pages. Available in PDF, EPUB and Kindle. Book excerpt: Encyclopedia of Bioinformatics and Computational Biology: ABC of Bioinformatics, Three Volume Set combines elements of computer science, information technology, mathematics, statistics and biotechnology, providing the methodology and in silico solutions to mine biological data and processes. The book covers Theory, Topics and Applications, with a special focus on Integrative –omics and Systems Biology. The theoretical, methodological underpinnings of BCB, including phylogeny are covered, as are more current areas of focus, such as translational bioinformatics, cheminformatics, and environmental informatics. Finally, Applications provide guidance for commonly asked questions. This major reference work spans basic and cutting-edge methodologies authored by leaders in the field, providing an invaluable resource for students, scientists, professionals in research institutes, and a broad swath of researchers in biotechnology and the biomedical and pharmaceutical industries. Brings together information from computer science, information technology, mathematics, statistics and biotechnology Written and reviewed by leading experts in the field, providing a unique and authoritative resource Focuses on the main theoretical and methodological concepts before expanding on specific topics and applications Includes interactive images, multimedia tools and crosslinking to further resources and databases

Systems Biology

Author : Andreas Kremling
Publisher : CRC Press
ISBN 13 : 1466567899
Total Pages : 382 pages
Book Rating : 4.4/5 (665 download)

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Book Synopsis Systems Biology by : Andreas Kremling

Download or read book Systems Biology written by Andreas Kremling and published by CRC Press. This book was released on 2013-11-12 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Physical Models of Living Systems

Author : Philip Nelson
Publisher : Macmillan Higher Education
ISBN 13 : 1319036902
Total Pages : 365 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Physical Models of Living Systems by : Philip Nelson

Download or read book Physical Models of Living Systems written by Philip Nelson and published by Macmillan Higher Education. This book was released on 2014-12-20 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written for intermediate-level undergraduates pursuing any science or engineering major, Physical Models of Living Systems helps students develop many of the competencies that form the basis of the new MCAT2015. The only prerequisite is first-year physics. With the more advanced "Track-2" sections at the end of each chapter, the book can be used in graduate-level courses as well.

Systems Biology

Author : Edda Klipp
Publisher : John Wiley & Sons
ISBN 13 : 3527336362
Total Pages : 504 pages
Book Rating : 4.5/5 (273 download)

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Book Synopsis Systems Biology by : Edda Klipp

Download or read book Systems Biology written by Edda Klipp and published by John Wiley & Sons. This book was released on 2016-06-27 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: This advanced textbook is tailored for an introductory course in Systems Biology and is well-suited for biologists as well as engineers and computer scientists. It comes with student-friendly reading lists and a companion website featuring a short exam prep version of the book and educational modeling programs. The text is written in an easily accessible style and includes numerous worked examples and study questions in each chapter. For this edition, a section on medical systems biology has been included.

Modeling Biological Systems:

Author : James W. Haefner
Publisher : Springer Science & Business Media
ISBN 13 : 9780387250113
Total Pages : 500 pages
Book Rating : 4.2/5 (51 download)

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Book Synopsis Modeling Biological Systems: by : James W. Haefner

Download or read book Modeling Biological Systems: written by James W. Haefner and published by Springer Science & Business Media. This book was released on 2005-05-06 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: I Principles 1 1 Models of Systems 3 1. 1 Systems. Models. and Modeling . . . . . . . . . . . . . . . . . . . . 3 1. 2 Uses of Scientific Models . . . . . . . . . . . . . . . . . . . . . . . . 4 1. 3 Example: Island Biogeography . . . . . . . . . . . . . . . . . . . . . 6 1. 4 Classifications of Models . . . . . . . . . . . . . . . . . . . . . . . . 10 1. 5 Constraints on Model Structure . . . . . . . . . . . . . . . . . . . . . 12 1. 6 Some Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1. 7 Misuses of Models: The Dark Side . . . . . . . . . . . . . . . . . . . 13 1. 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 The Modeling Process 17 2. 1 Models Are Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2. 2 Two Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . 18 2. 3 An Example: Population Doubling Time . . . . . . . . . . . . . . . . 24 2. 4 Model Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2. 5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Qualitative Model Formulation 32 3. 1 How to Eat an Elephant . . . . . . . . . . . . . . . . . . . . . . . . . 32 3. 2 Forrester Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3. 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3. 4 Errors in Forrester Diagrams . . . . . . . . . . . . . . . . . . . . . . 44 3. 5 Advantages and Disadvantages of Forrester Diagrams . . . . . . . . . 44 3. 6 Principles of Qualitative Formulation . . . . . . . . . . . . . . . . . . 45 3. 7 Model Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3. 8 Other Modeling Problems . . . . . . . . . . . . . . . . . . . . . . . . 49 viii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. 9 Exercises 53 4 Quantitative Model Formulation: I 4. 1 From Qualitative to Quantitative . . . . . . . . . . . . . . . . . Finite Difference Equations and Differential Equations 4. 2 . . . . . . . . . . . . . . . . 4. 3 Biological Feedback in Quantitative Models . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 4 Example Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 5 Exercises 5 Quantitative Model Formulation: I1 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 1 Physical Processes 81 . . . . . . . . . . . . . . . 5. 2 Using the Toolbox of Biological Processes 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 3 Useful Functions 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 4 Examples 102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 5 Exercises 104 6 Numerical Techniques 107 . . . . . . . . . . . . . . . . . . . . . . . 6. 1 Mistakes Computers Make 107 . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 2 Numerical Integration 110 . . . . . . . . . . . . . . . . 6. 3 Numerical Instability and Stiff Equations 115 . . . . . . . . . . . . . .