Foundations of Mathematical Biology

List of Contributors

Preface

Contents of Other Volumes

Chapter 1 Physiological Regulation and Control

I. Introduction

II. Mathematical Formulation of Systems

A. Some General Definitions

B. Assumptions

C. Description of Linear Systems

D. State Vector and State Equations

E. The Transfer Function and Its Properties

F. Vector Representation of Transfer Functions

G. Example: Electrical Phenomena across Excitable Membranes

III. Control Theory

A. Structure and Dynamics of Control Systems

B. Structure of Physiological Control

IV. Stability and Oscillations

A. Stability

B. Oscillations and Stability in Physiological Systems

V. Optimization

A. An Evolutionary Basis for Design and Operating Optimization

B. Hierarchical Aspects

C. Performance Indices, Cost Functions

D. Analytical Trade-off and Optimality Conditions

E. The Respiratory-Cardiovascular System

References

Chapter 2A Mathematical Aspects of Some Cardiovascular Phenomena

I. Introduction and Scope

II. Linear Case

III. Some Applications of the Linear Case: A Two-Chamber Theory

IV. Nonlinear Case

V. Applications of the Nonlinear Theory

VI. Volume Elasticity and the Elasticity of the Blood Vessel Wall:



Propagation of Pulse Waves

References

Chapter 2B The Principle of Adequate Design

I. Models and General Principles in Biology

II. Quantitative Description of a Form of an Organism

III. Form of Plants

IV. An Example of the Application to the External Shape of a Quadruped

V. Application to the Size of the Aorta

VI. Overall Design of the Circulatory System: Peripheral Resistance

VII. The Overall Design of the Cardiovascular System and the Evaluation



of Its Basic Parameters

References

Chapter 2C A Unified Approach to Physics, Biology, and Sociology

Chapter 3 Automata Theory in the Context of Theoretical Neurophysiology

I. The Concept of State

II. Finite-State Models of Neural Nets

III. Complexity Theory for Pattern Recognition Networks

IV. An Aside: Gödel's Incompleteness Theorem

V. From External to Internal Descriptions

VI. The Correction of Errors in Communication and Computation

A. Reliable Brains from Unreliable Neurons

B. Von Neumann's Multiplexing Scheme

C. Shannon's Communication Theory

D. Communication Theory and Automata

E. The Cowan-Winograd Theory of Reliable Automata

References

Chapter 4 The Deterministic Theory of Population Dynamics

I. Introduction

II. The Dynamics of an Isolated Species

A. Malthus' Equation

B. The Pearl-Verhulst Equation

C. A General Logistics Growth Rate Function

III. The Modes of Interaction between Two Species

A. Competition and Volterra's Competitive Exclusion Principle: Two



Species Competing for a Common Ecological Niche

B. Two Species Living in a Predator-Prey Relationship

C. Symbiosis

D. Parasitism

E. A General Qualitative Theory for the Interactions between Two



Species

IV. The Interactions between Three or More Species

A. Competition Involving Several Species

B. Predation Involving Several Species

V. Incorporation of "Historical Actions"

A. The System Equations for Predation

B. Law of Conservation of the Means and Law of Perturbation of the



Means for "Historical Actions"

VI. Epilogue

References

Chapter 5 Is There a Unified Mathematical Biology?

Author Index

Subject Index, Volume I

Subject Index, Volume II

Subject Index, Volume III