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El. knyga: Dynamical Systems for Biological Modeling: An Introduction

(University of British Columbia, Vancouver, Canada), (University of Texas at Arlington, USA)
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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.
Preface xi
Acknowledgments xiii
I Elementary Topics
1(280)
1 Introduction to Biological Modeling
3(20)
1.1 The nature and purposes of biological modeling
3(2)
1.2 The modeling process
5(6)
1.3 Types of mathematical models
11(3)
1.4 Assumptions, simplifications, and compromises
14(3)
1.5 Scale, and choosing units
17(6)
2 Difference Equations (Discrete Dynamical Systems)
23(76)
2.1 Introduction to discrete dynamical systems
23(10)
2.1.1 Linear difference equations
24(2)
2.1.2 Solution of linear difference equations
26(2)
2.1.3 Nonlinear difference equations
28(5)
2.2 Graphical analysis
33(4)
2.3 Qualitative analysis and population genetics
37(13)
2.3.1 Linearization and local stability
37(5)
2.3.2 A problem in population genetics
42(8)
2.4 Intraspecies competition
50(8)
2.4.1 Two metered fish models
54(2)
2.4.2 Between contest and scramble
56(2)
2.5 Harvesting
58(11)
2.5.1 Fishery harvesting and graphical equilibrium analysis
60(9)
2.6 Period doubling and chaos
69(13)
2.6.1 Dispersal
75(3)
2.6.2 Dynamical diseases and physiological control systems
78(4)
2.7 Structured populations
82(11)
2.7.1 Spatial dispersal
86(4)
2.7.2 Two-stage populations
90(3)
2.8 Predator-prey systems
93(6)
2.8.1 A plant-herbivore model
93(1)
2.8.2 A host-parasitoid model
94(3)
Miscellaneous exercises
97(2)
3 First-Order Differential Equations (Continuous Dynamical Systems)
99(86)
3.1 Continuous-time models and exponential growth
99(9)
3.1.1 Exponential growth
100(5)
3.1.2 Radioactive decay
105(3)
3.2 Logistic population models
108(15)
3.2.1 All creatures great and small?
110(4)
3.2.2 Competition among plants
114(3)
3.2.3 The spread of infectious diseases
117(6)
3.3 Graphical analysis
123(7)
3.3.1 Solutions
123(3)
3.3.2 Direction fields
126(4)
3.4 Equations and models with variables separable
130(20)
3.4.1 A linear model for the cardiac pacemaker
131(4)
3.4.2 General procedure
135(4)
3.4.3 Solution of logistic equations
139(2)
3.4.4 Discrete-time metered population models
141(3)
3.4.5 Allometry
144(6)
3.5 Mixing processes and linear models
150(15)
3.5.1 Chemostats
152(3)
3.5.2 Drug dosage
155(3)
3.5.3 Newton's law of cooling
158(2)
3.5.4 Migration
160(5)
3.6 First-order models with time dependence
165(20)
3.6.1 Superposition
165(2)
3.6.2 Integrating factors
167(2)
3.6.3 Substitution and integration
169(2)
3.6.4 Mixing processes with variable coefficients
171(3)
3.6.5 Bernouilli equation
174(8)
Miscellaneous exercises
182(3)
4 Nonlinear Differential Equations
185(96)
4.1 Qualitative analysis tools
185(18)
4.1.1 Possible end behaviors
187(4)
4.1.2 Stability
191(5)
4.1.3 Phase portraits
196(3)
4.1.4 Foraging ants and phase transitions
199(4)
4.2 Harvesting
203(25)
4.2.1 Constant-yield harvesting
203(11)
4.2.2 Constant-effort harvesting
214(8)
4.2.3 Migration and dispersal as harvesting
222(2)
4.2.4 Conclusions
224(4)
4.3 Mass-action models
228(14)
4.3.1 A simple chemical reaction
228(3)
4.3.2 The spread of infectious diseases, revisited
231(6)
4.3.3 Contact rate saturation and the "Pay It Forward" model
237(5)
4.4 Parameter changes, thresholds, and bifurcations
242(16)
4.4.1 Hysteresis
249(2)
4.4.2 The spruce budworm
251(7)
4.5 Numerical analysis of differential equations
258(23)
4.5.1 Approximation error
259(1)
4.5.2 Euler's method
260(4)
4.5.3 Other numerical methods
264(6)
4.5.4 Eutrophication
270(7)
Miscellaneous exercises
277(4)
II More Advanced Topics
281(132)
5 Systems of Differential Equations
283(38)
5.1 Graphical analysis: The phase plane
283(8)
5.2 Linearization of a system at an equilibrium
291(6)
5.3 Linear systems with constant coefficients
297(14)
5.3.1 A liver chemistry example
305(6)
5.4 Qualitative analysis of systems
311(10)
Miscellaneous exercises
319(2)
6 Topics in Modeling Systems of Populations
321(38)
6.1 Epidemiology: Compartmental models
321(10)
6.1.1 An epidemic model
321(6)
6.1.2 A model for endemic situations
327(4)
6.2 Population biology: Interacting species
331(20)
6.2.1 Species in competition
331(6)
6.2.2 Predator-prey systems
337(8)
6.2.3 Symbiosis
345(6)
6.3 Numerical approximation to solutions of systems
351(8)
6.3.1 Example: A two-sex model
352(7)
7 Systems with Sustained Oscillations and Singularities
359(54)
7.1 Oscillations in neural activity
359(7)
7.1.1 The Fitzhugh-Nagumo equations
360(3)
7.1.2 A model for cat neurons
363(3)
7.2 Singular perturbations and enzyme kinetics
366(13)
7.2.1 Bursting
372(2)
7.2.2 An example from enzyme kinetics
374(5)
7.3 HIV: An example from immunology
379(13)
7.3.1 A basic model
381(6)
7.3.2 Including infected cells
387(5)
7.4 Slow selection in population genetics
392(10)
7.4.1 Equally fit genotypes
393(3)
7.4.2 Slow genetic selection
396(6)
7.5 Second-order differential equations: Acceleration
402(11)
7.5.1 The harmonic oscillator
402(3)
7.5.2 The van der Pol oscillator
405(3)
7.5.3 A model of oxygen diffusion in muscle fibers
408(5)
III Appendices
413(20)
A An Introduction to the Use of Maple™
415(10)
A.1 Plotting graphs of functions
416(1)
A.2 Graphical solution of first-order differential equations
417(2)
A.3 Graphical solution of systems of differential equations
419(1)
A.4 The cobwebbing method for graphical solution of first-order difference equations
420(2)
A.5 Solution of difference equations and systems of difference equations
422(2)
A.6 A bifurcation program
424(1)
B Taylor's Theorem and Linearization
425(2)
C Location of Roots of Polynomial Equations
427(2)
D Stability of Equilibrium of Difference Equations
429(4)
Answers to Selected Exercises 433(26)
Bibliography 459(8)
Index 467
Fred Brauer, PhD, University of British Columbia, Vancouver, Canada

Christopher Kribs, PhD, University of Texas at Arlington, USA
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