Methods of small parameter in mathematical biology / Jacek Banasiak, Mirosław Lachowicz.
This monograph presents new tools for modeling multiscale biological processes. Natural processes are usually driven by mechanisms widely differing from each other in the time or space scale at which they operate and thus should be described by appropriate multiscale models. However, looking at all...
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Full Text (via Springer) |
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| Main Authors: | , |
| Format: | eBook |
| Language: | English |
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Cham :
Birkhäuser,
2014.
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| Series: | Modeling and simulation in science, engineering & technology.
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| Subjects: |
MARC
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| 100 | 1 | |a Banasiak, J., |e author. |0 http://id.loc.gov/authorities/names/n95057749 |1 http://isni.org/isni/0000000116423508. | |
| 245 | 1 | 0 | |a Methods of small parameter in mathematical biology / |c Jacek Banasiak, Mirosław Lachowicz. |
| 264 | 1 | |a Cham : |b Birkhäuser, |c 2014. | |
| 300 | |a 1 online resource (xi, 285 pages) : |b illustrations (some color) | ||
| 336 | |a text |b txt |2 rdacontent. | ||
| 337 | |a computer |b c |2 rdamedia. | ||
| 338 | |a online resource |b cr |2 rdacarrier. | ||
| 347 | |b PDF. | ||
| 347 | |a text file. | ||
| 490 | 1 | |a Modeling and Simulation in Science, Engineering and Technology, |x 2164-3679. | |
| 504 | |a Includes bibliographical references and index. | ||
| 505 | 0 | |6 880-01 |a 1 Small parameter methods -- basic ideas -- 2 Introduction to the Chapman-Enskog method -- linear models with migrations -- 3 Tikhonov-Vasilyeva theory -- 4 The Tikhonov theorem in some models of mathematical biosciences -- 5 Asymptotic expansion method in a singularly perturbed McKendrick problem -- 6 Diffusion limit of the telegraph equation -- 7 Kinetic model of alignment -- 8 From microscopic to macroscopic descriptions. -- 9 Conclusion. | |
| 520 | |a This monograph presents new tools for modeling multiscale biological processes. Natural processes are usually driven by mechanisms widely differing from each other in the time or space scale at which they operate and thus should be described by appropriate multiscale models. However, looking at all such scales simultaneously is often infeasible, costly, and provides information that is redundant for a particular application. Hence, there has been a growing interest in providing a more focused description of multiscale processes by aggregating variables in a way that is relevant and preserves the salient features of the dynamics. The aim of this book is to present a systematic way of deriving the so-called limit equations for such aggregated variables and ensuring that the coefficients of these equations encapsulate the relevant information from the discarded levels of description. Since any approximation is only valid if an estimate of the incurred error is available, the tools described allow for proving that the solutions to the original multiscale family of equations converge to the solution of the limit equation if the relevant parameter converges to its critical value. The chapters are arranged according to the mathematical complexity of the analysis, from systems of ordinary linear differential equations, through nonlinear ordinary differential equations, to linear and nonlinear partial differential equations. Many chapters begin with a survey of mathematical techniques needed for the analysis. All problems discussed in this book belong to the class of singularly perturbed problems; that is, problems in which the structure of the limit equation is significantly different from that of the multiscale model. Such problems appear in all areas of science and can be attacked using many techniques. Methods of Small Parameter in Mathematical Biology will appeal to senior undergraduate and graduate students in appled and biomathematics, as well as researchers specializing in differential equations and asymptotic analysis. | ||
| 588 | 0 | |a Online resource; title from PDF title page (SpringerLink, viewed April 25, 2014) | |
| 650 | 0 | |a Biomathematics. |0 http://id.loc.gov/authorities/subjects/sh85014235. | |
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| 650 | 7 | |a Biomathematics. |2 fast |0 (OCoLC)fst00832555. | |
| 700 | 1 | |a Lachowicz, Mirosław, |e author. |0 http://id.loc.gov/authorities/names/n2008183788 |1 http://isni.org/isni/0000000059602146. | |
| 776 | 0 | 8 | |i Printed edition: |z 9783319051390. |
| 830 | 0 | |a Modeling and simulation in science, engineering & technology. |0 http://id.loc.gov/authorities/names/n96087523. | |
| 856 | 4 | 0 | |u https://colorado.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-319-05140-6 |z Full Text (via Springer) |
| 880 | 0 | |6 505-01/(S |g Machine generated contents note: |g 1. |t Small Parameter Methods: Basic Ideas -- |g 1.1. |t Introduction -- |g 1.2. |t Small Parameter in Physical Models -- |g 1.2.1. |t Classical Mechanics and Relativistic Mechanics -- |g 1.2.2. |t Classical Mechanics and Quantum Mechanics -- |g 1.2.3. |t Theory of Inviscid Fluids (TIF) and Theory of Viscous Fluids (TVF) -- |g 1.2.4. |t Macroscopic and Mesoscopic Description in the Framework of Kinetic Theory -- |g 1.3. |t Small Parameter in Biological Models -- |g 1.3.1. |t Allee-Type Model -- |g 1.3.2. |t Epidemiological Model -- |g 1.3.3. |t Structured Population Dynamics with Fast Migrations -- |g 1.3.4. |t Equations of Random Walks -- |g 1.3.5. |t Alignment -- |g 1.3.6. |t Michaelis--Menten Kinetics -- |g 1.4. |t Basics of Asymptotic Analysis -- |g 1.4.1. |t General Framework of Asymptotic Procedures -- |g 1.4.2. |t Introductory Problems -- |g 2. |t Introduction to the Chapman--Enskog Method: Linear Models with Migrations -- |g 2.1. |t Basics of Linear Dynamical Systems -- |g 2.1.1. |t Fundamental Solution Matrix -- |g 2.1.2. |t Eigenvalues, Eigenvectors and Associated Eigenvectors -- |g 2.1.3. |t Exponential of a Matrix -- |g 2.1.4. |t Spectral Decomposition -- |g 2.1.5. |t Transition Matrices -- |g 2.2. |t Asymptotic Procedure -- |g 2.2.1. |t Bulk Approximation -- |g 2.2.2. |t Initial Layer -- |g 2.3. |t Interacting Populations with Space Structure -- |g 2.3.1. |t Emerging Properties -- |g 2.3.2. |t Asymptotics on [0, [∞][-- |g 3. |t Tikhonov--Vasilyeva Theory -- |g 3.1. |t Regular Perturbation -- |g 3.2. |t Singular Perturbation -- |g 3.2.1. |t Simple Case -- |g 3.3. |t Tikhonov Theorem -- |g 3.4. |t Initial Layer -- |g 3.5. |t Error Estimates -- |g 3.6. |t Vasilyeva Theorem -- |g 4. |t Tikhonov Theorem in Some Models of Mathematical Biosciences -- |g 4.1. |t Allee Model -- |g 4.1.1. |t Case of Short Satiation Time: σ/μ ̃ 1/epsilon; -- |g 4.1.2. |t Case with Short Satiation and High Searching Efficiency: σ/μ ̃ 1/ε -- |g 4.1.3. |t Case with Short Satiation, High Searching Efficiency and High Searching Death Rate: σ/μ ̃ 1/ε, ξ/μ ̃ 1/ε -- |g 4.2. |t SIS Model with Basic Age Structure -- |g 4.2.1. |t SIS Model -- |g 4.2.2. |t SIS Model with an Age Structure and Its Basic Properties -- |g 4.2.3. |t Application of the Tikhonov Theorem -- |g 4.2.4. |t Comments on the S̀table--Unstable' Case -- |g 4.2.5. |t Numerical Illustration -- |g 4.2.6. |t Conclusions -- |g 4.3. |t Population Problems with Fast Migrations -- |g 4.3.1. |t Model -- |g 4.3.2. |t Basic Quasi Steady State Approximation -- |g 4.3.3. |t Approximation Preserving Long Time Dynamics -- |g 5. |t Asymptotic Expansion Method in a Singularly Perturbed McKendrick Problem -- |g 5.1. |t Rudiments of the Semigroup Theory -- |g 5.1.1. |t Generation of Semigroups -- |g 5.1.2. |t Nonhomogeneous Problems -- |g 5.1.3. |t Bounded Perturbation Theorem -- |g 5.2. |t Singularly Perturbed McKendrick Model with Geographic Structure -- |g 5.3. |t Preliminary Properties of (5.14)--(5.16) -- |g 5.3.1. |t Spectral Properties of C -- |g 5.3.2. |t Lifting Theorem -- |g 5.4. |t Formal Asymptotic Expansion -- |g 5.4.1. |t Projections of Operators -- |g 5.4.2. |t Bulk Approximation -- |g 5.4.3. |t Initial Layer -- |g 5.4.4. |t Boundary Layer -- |g 5.4.5. |t Corner Layer -- |g 5.5. |t Numerical Illustration -- |g 5.5.1. |t Computational Example -- |g 6. |t Diffusion Limit of the Telegraph Equation -- |g 6.1. |t Further Semigroups -- |g 6.1.1. |t Hilbert Spaces -- |g 6.1.2. |t Dissipative and Coercive Operators in Hilbert Spaces -- |g 6.1.3. |t Analytic Semigroups -- |g 6.1.4. |t Mathematical Setting for (6.1) -- |g 6.2. |t Singularly Perturbed System: Case (1.47) -- |g 6.3. |t Singularly Perturbed Systems: Case (1.49) -- |g 6.4. |t Application to the Random Walk Theory -- |g 7. |t Kinetic Model of Alignment -- |g 7.1. |t Space Homogeneous Case -- |g 7.2. |t Space Inhomogeneous Case -- |g 7.3. |t Entropy -- |g 7.4. |t Formal Macroscopic Limits -- |g 7.5. |t Macroscopic Limit: Aligned Picture -- |g 8. |t From Microscopic to Macroscopic Descriptions -- |g 8.1. |t Multiscale Descriptions -- |g 8.2. |t Microscopic Scale: Individually Based Models -- |g 8.3. |t Mesoscopic Scale: Kinetic Models -- |g 8.3.1. |t Nonlinear Kinetic Equations: The Mesoscopic Level -- |g 8.3.2. |t Binary Interactions -- |g 8.3.3. |t Bilinear Kinetic Equations: Examples -- |g 8.3.4. |t Bilinear Kinetic Equations: Existence and Uniqueness Results -- |g 8.3.5. |t Bilinear Kinetic Equations: The Logistic Growth -- |g 8.3.6. |t Bilinear Kinetic Equations: A Mesoscopic Model of DNA Denaturation -- |g 8.3.7. |t Lotka--Volterra Mesoscopic Model -- |g 8.3.8. |t Bilinear Kinetic Equations: Equilibrium Solutions -- |g 8.3.9. |t Bilinear Kinetic Equations: Diffusive Limit -- |g 8.4. |t Microscopic Systems: Equilibrium Solutions -- |g 9. |t Conclusion. | |
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