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Mathematical Models in the Biosciences I [Pehme köide]

4.00/5 (2 hinnangut Goodreads-ist)
  • Formaat: Paperback / softback, 544 pages, kõrgus x laius x paksus: 235x156x38 mm, 250 b-w illus.
  • Ilmumisaeg: 10-Aug-2021
  • Kirjastus: Yale University Press
  • ISBN-10: 0300228317
  • ISBN-13: 9780300228311
Teised raamatud teemal:
Mathematical Models in the Biosciences ISuurem pilt
  • Formaat: Paperback / softback, 544 pages, kõrgus x laius x paksus: 235x156x38 mm, 250 b-w illus.
  • Ilmumisaeg: 10-Aug-2021
  • Kirjastus: Yale University Press
  • ISBN-10: 0300228317
  • ISBN-13: 9780300228311
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780300228311.html
Märksõnad:
An award-winning professor&;s introduction to essential concepts of calculus and mathematical modeling for students in the biosciences

This is the first of a two-part series exploring essential concepts of calculus in the context of biological systems. Michael Frame covers essential ideas and theories of basic calculus and probability while providing examples of how they apply to subjects like chemotherapy and tumor growth, chemical diffusion, allometric scaling, predator-prey relations, and nerve impulses. Based on the author&;s calculus class at Yale University, the book makes concepts of calculus more relatable for science majors and premedical students.

An award-winning professor&;s introduction to essential concepts of calculus and mathematical modeling for students in the biosciences

Arvustused

The choice of material is interesting and refreshing, and finds concrete applications for mathematical topics that might not be standard fare in the physics or chemistry curricula. The applications of the Poincaré-Bendixson theorem to locating limit cycles are a remarkable highlight.Chay Paterson, zbMATH Open





"This is a wonderful book, wise and witty. It would have taught me most of the math I needed for my career in research if I did all the problems."Stephen Stearns, author of The Evolution of Life Histories and Evolutionary Medicine  

This enlightening book covers not only the essential parts of calculus and dynamical system, but also how one can apply these tools in biological sciences. In addition, the last chapter of this book is a concise introduction to probability theory. Michael Frame motivates students with very well-selected examples.Hongyu He, Professor of Mathematics, Louisiana State University





This work is an important step toward a new curriculum model for the nascent field of mathematical biology: different content and authentic applications, geared toward a truly interdisciplinary audience.Rebecca Gasper, Assistant Professor of Mathematics, Creighton University

Preface xi
Ways to use this book xv
Acknowledgments xix
1 Review
1(5)
1.1 Rules for differentiation
1(1)
1.2 Interpretations of the derivative
2(1)
1.3 Intermediate and mean value theorems
3(1)
1.4 The fundamental theorem of calculus
4(1)
1.5 Discrete and continuous models
4(2)
2 Discrete dynamics
6(46)
2.1 Logistic map dynamics
6(7)
2.2 Fixed points and their stability
13(5)
2.3 Cycles and their stability
18(6)
2.4 Chaos
24(12)
2.5 Basic cardiac dynamics
36(1)
2.6 A simple cardiac model
37(7)
2.7 Arnold's model
44(8)
3 Differential equations models
52(37)
3.1 Some types of models
53(2)
3.2 Examples: diffusion, Gompertz, SIR
55(12)
3.3 Lairds growth logic
67(8)
3.4 The Norton-Simon model
75(14)
4 Single-variable differential equations
89(36)
4.1 Simple integration
90(4)
4.2 Integration by substitution
94(3)
4.3 Integration by parts
97(6)
4.4 Integral tables and computer integration
103(4)
4.5 Fixed point analysis
107(6)
4.6 Separation of variables
113(5)
4.7 Integration by partial fractions
118(7)
5 Definite integrals and improper integrals
125(24)
5.1 Definite integrals by substitution
126(3)
5.2 Volume by integration
129(2)
5.3 Lengths of curves
131(2)
5.4 Surface area by integration
133(4)
5.5 Improper integrals
137(3)
5.6 Improper integral comparison tests
140(5)
5.7 Stress testing growth models
145(4)
6 Power laws
149(24)
6.1 The circumference of a circle
149(3)
6.2 Scaling of coastlines, log-log plots
152(4)
6.3 Allometric scaling
156(4)
6.4 Power laws and dimensions
160(7)
6.5 Some biological examples
167(6)
7 Differential equations in the plane
173(30)
7.1 Vector fields and trajectories
174(7)
7.2 Differential equations software
181(1)
7.3 Predator-prey equations
182(5)
7.4 Competing populations
187(4)
7.5 Nullcline analysis
191(4)
7.6 The Fitzhugh-Nagumo equations
195(8)
8 Linear systems and stability
203(29)
8.1 Superposition of solutions
204(1)
8.2 Types of fixed points
205(6)
8.3 The matrix formalism
211(2)
8.4 Eigenvalues and eigenvectors
213(6)
8.5 Eigenvalues at fixed points
219(8)
8.6 The trace-determinant plane
227(5)
9 Nonlinear systems and stability
232(60)
9.1 Partial derivatives
232(4)
9.2 The Hartman-Grobman theorem
236(11)
9.3 The pendulum as guide
247(3)
9.4 Liapunov functions
250(1)
9.5 Fixed points not at the origin
251(5)
9.6 Limit cycles
256(16)
9.7 The Poincare-Bendixson theorem
272(11)
9.8 Lienard's and Bendixson's theorems
283(9)
10 Infinite series and power series
292(41)
10.1 The Integral Test
295(2)
10.2 The Comparison Test
297(3)
10.3 Alternating series
300(7)
10.4 The Root and Ratio Tests
307(3)
10.5 Numerical series practice
310(2)
10.6 Power series
312(3)
10.7 Radius and interval of convergence
315(5)
10.8 Taylor's theorem, series manipulation
320(7)
10.9 Power series solutions
327(6)
11 Some probability
333(102)
11.1 Discrete variables
334(9)
11.2 Continuous variables
343(1)
11.3 Some combinatorial rules
343(7)
11.4 Simpson's paradox
350(6)
11.5 Causality calculus
356(9)
11.6 Expected value and variance
365(12)
11.7 The binomial distribution
377(9)
11.8 The Poisson distribution
386(13)
11.9 The normal distribution
399(9)
11.10 Infinite moments
408(10)
11.11 Classical hypothesis tests
418(2)
11.12 Bayesian Inference
420(15)
12 Why this matters
435(9)
Appendix A Technical notes
444(41)
A.1 Integrating factors
444(3)
A.2 Existence and uniqueness
447(5)
A.3 The threshold theorem
452(2)
A.4 Volterra's trick
454(5)
A.5 Euler's formula
459(1)
A.6 Proof of the second-derivative test
460(2)
A.7 Some linear algebra
462(9)
A.8 A proof of Liapunov's theorem
471(2)
A.9 A proof of the Hartman-Grobman theorem
473(4)
A.10 A proof of the Poincare-Bendixson theorem
477(5)
A.11 The Law of Large Numbers
482(3)
Appendix B Some Mathematica code
485(9)
B.1 Cycles and their stability
486(1)
B.2 Chaos
487(1)
B.3 A simple cardiac model
488(1)
B.4 The Norton-Simon model
488(1)
B.5 Log-log plots
489(1)
B.6 Vector fields and trajectories
490(1)
B.7 Differential equations software
490(1)
B.8 Predator-prey models
491(1)
B.9 The Fitzhugh-Nagumo equations
492(1)
B.10 Eigenvalues and eigenvectors
493(1)
Appendix C Some useful integrals and hints
494(4)
Figure credits 498(1)
References 499(9)
Index 508
Michael Frame retired in 2016 as adjunct professor of mathematics at Yale University. For more than twenty years Frame taught courses on fractal geometry and calculus based on applications in biology and medicine. Amelia Urry and he are the coauthors of Fractal Worlds: Grown, Built, and Imagined.