生物数学引论

Contents

List of Figures

1. Single Species Population Dynamics

1.1 Introduction

1.2 Linear and Nonlinear First Order Discrete Time Models.

1.2.1 The Biology of Insect Population Dynamics

1.2.2 A Model for Insect Population Dynamics with Competition

1.3 Differential-Equation Models

1.4 EvolutionaryAspects

1.5 Harvesting and Fisheries

1.6 Metapopulations

1.7 Delay Effects

1.8Fibonaccis-Rabbits

1.9 Leslie Matrices： Age-structured Populations in Discrete Time

1 10 Euler-Lotka Equations

1.10.1 Discrete Time

1.10.2 Continuous Time

1.11 The McKendrick Approach to Age Structure

1 12 Conclusions

2. Population Dynamics of Interacting Species

2.1 Introduction

2 2 Host-parasitoid Interactions

2.3 The Lotka-Volterra Prey-predator Equations

2.4 Modelling the Predator Functional Response

2.5 Competition.

2.6 Ecosystems Modelling

2.7 Interacting Metapopulations

2.7.1 Competition

2.7.2 Predation

2.7.3 Predator-mediated Coexistence of Competitors

2.7.4 Effects of Habitat Destruction

2.8 Conclusions

3. Infectious Diseases

3.1 Introduction

3.2 The Simple Epidemic and SIS Diseases

3.3 SIR Epidemics

3.4 SIR Endemics

3.4.1 No Disease-related Death

3.4.2 Including Disease-related Death

3.5 Eradication and Control

3.6 Age-structured Populations

3.6.1 The Equations

3.6.2 Steady State

3.7 Vector-borne Diseases

3.8 Basic Model for Macroparasitic Diseases

3.9 Evolutionary Aspects

3.10 Conclusions

4. Population Genetics and Evolution

4.1 Introduction

4.2 Mendelian Genetics in Populations with Non-overlapping Generations

4.3 Selection Pressure

4.4 Selection in Some Special Cases

4.4.1 Selection for a Dominant Allele

4.4.2 Selection for a Recessive Allele

4.4.3 Selection against Dominant and Recessive Alleles

4.4.4 The Additive Case

4.5 Analytical Approach for Weak Selection

4.6 The Balance Between Selection and Mutation

4.7 Wrights Adaptive Topography

4.8 Evolution of the Genetic System

4.9 Game Theory

4.10 Replicator Dynamics

4.11 Conclusions

5. Biological Motion

5.1 Introduction

5.2 Macroscopic Theory of Motion; A Continuum Approach

5.2.1 General Derivation

5.2.2 Some Particular Cases

5.3 Directed Motion， or Taxis

5.4 Steady State Equations and Transit Times

5.4.1 Steady State Equations in One Spatial Variable

5.4.2 Transit Times

5.4.3 Macrophages vs Bacteria

5.5 Biological Invasions： A Model for Muskrat Dispersal

5.6 Travelling Wave Solutions of General Reaction-diffusion Equations

5.6.1 Node-saddle Orbits (the Monostable Equation)

5.6.2 Saddle-saddle Orbits (the Bistable Equation)

5.7 Travelling Wave Solutions of Systems of Reaction-diffusion

Equations： Spatial Spread of Epidemics

5.8 Conclusions

6. Molecular and Cellular Biology

6.1 Introduction

6.2 Biochemical Kinetics

6.3 Metabolic Pathways

6.3.1 Activation and Inhibition

6.3.2 Cooperative Phenomena

6.4 Neural Modelling

6.5 Immunology and AIDS

6.6 Conclusions

7. Pattern Formation

7.1 Introduction

7.2 Turing Instability

7.3 Turing Bifurcations

7.4 Activator-inhibitor Systems

7.4.1 Conditions for Turing Instability

7.4.2 Short-range Activation， Long-range Inhibition

7.4.3 Do Activator-inhibitor Systems Explain Biological Pattern Formation?

7.5 Bifurcations with Domain Size

7.6 Incorporating Biological Movement

7.7 Mechanochemical Models

7.8 Conclusions

8. Tumour Modelling

8.1 Introduction

8.2 Phenomenological Models

8.3 Nutrients： the Diffusion-limited Stage

8.4 Moving Boundary Problems

8.5 Growth Promoters and Inhibitors

8.6 Vascularisation

8.7 Metastasis

8.8 Immune System Response

8.9 Conclusions

Further Reading

A. Some Techniques for Difference Equations

A.I First-order Equations

A.I.1 Graphical Analysis

A.1.2 Linearisation

A.2 Bifurcations and Chaos for First-order Equations

A.2.1 Saddle-node Bifurcations

A.2.2 Transcritical Bifurcations

A.2.3 Pitchfork Bifurcations

A.2.4 Period-doubling or Flip Bifurcations

A.3 Systems of Linear Equations： Jury Conditions

A.4 Systems of Nonlinear Difference Equations

A.4.1 Linearisation of Systems

A.4.2 Bifurcation for Systems

B. Some Techniques for Ordinary Differential Equations

B.1 First-order Ordinary Differential Equations

B.I.1 Geometric Analysis

B.1.2 Integration

B.1.3 Linearisation

B.2 Second-order Ordinary Differential Equations

B.2.1 Geometric Analysis (Phase Plane)

B.2.2 Linearisation

B.2.3 Poincard-Bendixson Theory

B.3 Some Results and Techniques for ruth Order Systems

B.3.1 Linearisation

B.3.2 Lyapunov Functions

B.3.3 Some Miscellaneous Facts

B.4 Bifurcation Theory for Ordinary Differential Equations

B.4.1 Bifurcations with Eigenvalue Zero

B.4.2 Hopf Bifurcations

C. Some Techniques for Partial Differential Equations

C.1 First-order Partial Differential Equations and Characteristics

C.2 Some Results and Techniques for the Diffusion Equation

C.2.1 The Fundamental Solution

C.2.2 Connection with Probabilities

C.2.3 Other Coordinate Systems

C.3 Some Spectral Theory for Laplaces Equation

C.4 Separation of Variables in Partial Differential Equations

C.5 Systems of Diffusion Equations with Linear Kinetics

C.6 Separating the Spatial Variables from Each Other

D. Non-negative Matrices

D.1 Perron-Frobenius Theory

E. Hints for Exercises

Index

《生物数学引论》由浅入深讲述生物数学基础理论，从最经典的问题入手，最后走向学科前沿和近年的热点问题；内容先进，讲述方法科学，简洁明了，易读性好。生物数学在应用数学中占有日益重要的地位，数学系培养的学生至少一部分人应当对这个领域有所了解。随着生命科学的迅速发展，生物数学也发展很快。 　　《生物数学引论》自身具有完整体系，在“微积分”、“代数”等基础课知识之外，读者不需要其他预备知识。 　　《生物数学引论》适合用作数学及生命科学高年级本科生相关课程教材或参考书。