Network Models in Population Biology

This book is an outgrowth of one phase of an upper-division course on quantitative ecology, given each year for the past eight at Berkeley. I am most grateful to the students in that course and to many graduate students in the Berkeley Department of Zoology and Colleges of Engineering and Natural Resources whose spirited discussions inspired much of the book's content. I also am deeply grateful to those faculty colleagues with whom, at one time or another, I have shared courses or seminars in ecology or population biology, D.M. Auslander, L. Demetrius, G. Oster, O.H. Paris, F.A. Pitelka, A.M. Schultz, Y. Takahashi, D.B. Tyler, and P. Vogelhut, all of whom contributed substantially to the development of my thinking in those fields, to my Depart­ mental colleagues E. Polak and A.J. Thomasian, who guided me into the litera­ ture on numerical methods and stochastic processes, and to the graduate students who at one time or another have worked with me on population-biology projects, L.M. Brodnax, S-P. Chan, A. Elterman, G.C. Ferrell, D. Green, C. Hayashi, K-L. Lee, W.F. Martin Jr., D. May, J. Stamnes, G.E. Swanson, and I. Weeks, who, together, undoubtedly provided me with the greatest inspiration. I am indebted to the copy-editing and production staff of Springer-Verlag, especially to Ms. M. Muzeniek, for their diligence and skill, and to Mrs. Alice Peters, biomathematics editor, for her patience.

Why Model?.- 1. Foundations of Modeling Dynamic Systems.- 1.1. Time.- 1.2. Dynamics.- 1.3. State.- 1.4. Discrete and Continuous Representations of Time.- 1.5. The Discrete Nature of Observed Time and Observed States.- 1.6. State Spaces.- 1.7. Progress Through State Space.- 1.8. The Conditional Probability of Transition from State to State.- 1.9. Network Representations of Primitive Markovian State Spaces.- 1.10. Conservation.- 1.11. State Variables Associated with Individual Organisms.- 1.12. Basic Analysis of Markov Chains.- 1.13. Vector Notation, State Projection Matrices.- 1.14. Elementary Dynamics of Homogeneous Markov Chains.- 1.15. Observation of Transition Probabilities.- 1.16. The Primitive State Space for an Entire Population of Identical Objects.- 1.17. Dynamics of Populations Comprising Indistinguishable Members.- 1.18. Deduction of Population Dynamics Directly from the Member State Space.- 1.19. A Situation in Which Member State Space Cannot be Used to Deduce Population Dynamics.- 1.20. The Law of Large Numbers.- 1.21. Summary.- 1.22. Some References for Chapter 1.- 2. General Concepts of Population Modeling.- 2.1. Lumped Markovian States from Irreducible Primitive Markovian State Spaces.- 2.2. Shannon’s Measure: Uncertainty in State Spaces and Lumped States.- 2.3. Lumped Markovian States from Reducible Primitive Markovian State Spaces.- 2.4. Frequency Aliasing: The Artifact of Lumped Time.- 2.5. Idealizations: Thought Experiments and Hypothesis Testing.- 2.6. Conservation: Defining Membership in a Given Population.- 2.7. Conservation and Constitutive Relationships for a Single State.- 2.8. Reproduction, Death and Life as Flow Processes.- 2.9. Further Lumping: Combining Age Classes for Simplified Situations and Hypotheses.- 2.10. The Use of Network Diagrams to Construct Models.- 2.11. Basic Principles of Network Construction.- 2.12. Some Alternative Representations of Common Network Configurations.- 2.13. Some References for Chapter 2.- 3. A Network Approach to Population Modeling.- 3.1. Introduction to Network Modeling of Populations.- 3.2. Network Models for Some Basic, Idealized Life Cycles.- 3.2.1. Simple Life Cycles.- 3.2.2. Models with Overlapping Time Classes.- 3.3. Scalor Parameters and Multiplier Functions.- 3.3.1. Single-Species Models.- 3.3.2. Two-Species Models.- 3.4. Time-Delay Durations.- 3.5. Conversion to a Stochastic Model.- 3.5.1. Models without Interactions of Correlated Variables.- 3.5.2. Models with Interactions of Correlated Variables.- 3.5.3. Examples of Stochastic Network Models.- 3.6. Some References for Chapter 3.- 4. Analysis of Network Models.- 4.1. Introduction to Network Analysis.- 4.2. Interval by Interval Accounting on a Digital Computer.- 4.2.1. Digital Representation of Time Delays in Large-Numbers Models.- 4.2.2. Digital Representations for Stochastic Network Models.- 4.2.3. Examples of Digital Modeling.- 4.3. Graphical Analysis of One-Loop Networks with Lumpable Parameters.- 4.4. Large-Numbers Models with Constant Parameters.- 4.5. Inputs and Outputs of Network Models.- 4.6. Linearity, Cohorts, and Superposition-Convolution.- 4.7. The z-Transform: A Shorthand Notation for Discrete Functions.- 4.8. The Application of z-Transforms to Linear Network Functions.- 4.8.1. Straight-Chain Networks.- 4.8.2. Networks with Feed Forward Loops.- 4.8.3. Networks with Feedback Loops.- 4.9. Linear Flow-Graph Analysis.- 4.9.1. Graphical Reduction of Flow Graphs.- 4.9.2. Mason’s Rule.- 4.10. Interpretation of Unit-Cohort Response Functions: The Inverse z-Transform.- 4.10.1. Partial-Fraction Expansion.- 4.10.2. Finding the Coefficients of an Expansion.- 4.10.3. Dealing with Multiple Roots at the Origin.- 4.10.4. Exercises.- 4.11. Types of Common Ratios and Their Significances.- 4.11.1. The General Nature of Common Ratios.- 4.11.2. Real Common Ratios.- 4.11.3. Complex Common Ratios.- 4.11.4. Exercises.- 4.12. The Patterns of Linear Dynamics.- 4.12.1. Growth Patterns in Low-Level Populations and Populations with Constant Parameters.- 4.12.2. Time Required for Establishment of Dominance.- 4.12.3. Geometric Patterns of Growth and the Biotic Potential.- 4.12.4. Patterns of Dynamics Near Nonzero Critical Levels.- 4.12.5. Exercises.- 4.13. Constant-Parameter Models for Nonzero Critical Levels.- 4.13.1. Replacement for the Scalor.- 4.13.2. Modification of the Time Delay.- 4.13.3. Dynamics Close to a Nonzero Critical Level.- 4.14. Finding the Roots of Q(z).- 4.14.1. Euclid’s Algorithm.- 4.14.2. Descartes’ Rule and Sturm Sequences.- 4.14.3. Locating the Real Roots.- 4.14.4. Locating Imaginary and Complex Roots.- 4.14.5. Estimating the Magnitude of the Dominant Common Ratio.- 4.15. Network Responses to More Complicated Input Patterns.- 4.15.1. The Natural Frequencies of a Constant-Parameter Network Model.- 4.15.2. Exciting and Observing the Natural Frequencies of a Network.- 4.15.3. Responses to Category-1 Inputs.- 4.15.4. The Initial-and Final-Value Theorems, Steady-State Analysis.- 4.16. Elements of Dynamic Control of Networks.- 4.16.1. Linear Control with Category-2 Inputs.- 4.16.2. Comments on Nonlinear Control and Regulation.- 4.17. Dynamics of Constant-Parameter Models with Stochastic Time Delays.- 4.17.1. z-Transforms of Stochastic Time Delays.- 4.17.2. Moments of Stochastic Time Delay Distributions.- 4.17.3. Examples of Stochastic Time Delays and Their Transforms.- 4.17.4. Effects of Time Delay Distributions on Dynamics in a One-Loop Model.- 4.17.5. An Elementary Sensitivity Analysis.- 4.17.6. When the Minimum Latency is Not Finite.- 4.18. The Inverse Problem: Model Synthesis.- 4.19. Application of Constant-Parameter Network Analysis to More General Homogeneous Markov Chains.- 4.20. Some References for Chapter 4.- Appendix A. Probability Arrays, Array Manipulation.- A.l. Definitions.- A.2. Manipulation of Arrays.- A.3. Operations on Probability Arrays.- Appendix B. Bernoulli Trials and the Binomial Distribution.