r 6 Lecture Notes in Biomathematics Managing Editor: S. Levin 68 The Dynamics of Physiologically Structured Populations Edited by J.A.J. Metz arid O. Diekmann Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
TABLE OF CONTENTS Part A. Mathematical Models for Physiologically Structured Populations: a Systematic Exposition 1 I. A Gentle Introduction to Structured Population Models: Three Worked Examples 3 J.A.J. Metz & O. Diekmann 1. Introduction 3 2. The invertebrate functional response 5 2.1. Introduction 5 2.2. Holling's disk equation 1: the underlying time scale argument, 6 as exemplified by the finite state predator 2.3. Holling's disk equation 2: general handling times 8 2.4. The general invertebrate predator 1: the basic biology 12 2.5. The general invertebrate predator 2: the population equation 14 2.6. The general invertebrate predator 3: calculating the functional response 16 2.7. The general invertebrate predator 4: the Rashevsky limit 18 2.8. Concluding remarks and summary 20 3. Size dependent reproduction in ectothermie animals 21 3.1. The dynamics of individuals 21 3.2. Formulation of the population equations 23 3.3. Constant environments 24 3.4. Constant environments: reduction to an age-dependent problem 26 3.5. Variable environments 29 3.6. Summary 31 4. The cell size distribution 31 4.1. The dynamics of individuals 32 4.2. Formulation of the^ equations 34 4.3. The stable cell size distribution, 35 4.4. The inverse problem \ 38 4.5. Limits to growth 39 4.6. Summary 42 5. On semigroups and generators 43 II. The Cell Size Distribution and Semigroups of Linear Operators 46 0. Diekmann. 1. Formulation of the problem 46 2. Strongly continuous semigroups of bounded linear operators 49 3. Do growth, death and division generate a semigroup? 53 4. The spectrum of A 56 5. The characteristic equation 58
VIII 6. Decomposition of the population state space X 60 7. Relations between the spectra of A and of T(t) 61 8. Exponential estimates 62 9. The stable size distribution 65 10. Interlude: integration along characteristics 68 11. The merry-go-round 71 12. The merry-go-round with an absorbing exit 74 13. Remarks about positivity 76 14. A somewhat special nonlinear problem 76 III. Formulating Models for Structured Populations. 78 J.A.J. Metz & O. Diekmann. 1. Introduction: six examples for later use 78 2. Some modelling philosophy 83 2.1. The state concept 83 2.2. Obtaining an /-state representation - 86 2.3. From the individual to the population level 88 3. Mass balance 90 3.1. Mass transport due to continuous i-state movement 92 3.1.1. A biologist's shortcut 92 3.1.2. The mathematician's derivation 94 3.2. The (local) loss of/>-mass, 96 3.2.1. The local disappearance of />-mass from the interior of Q \ 96 3.2.2. The disappearance of ^-mass across the boundary of U 98 3.3. The (reappearance of /Mnass 99 3.3.1. The (re)appearance of p-mass in the interior of U 100 3.3.2. The (reappearance of ^-mass across the boundary of fi 101 3.4. Boundaries and side conditions: picking up the strands 101 3.5. Summary and concluding remarks 103 4. Integration along characteristics, transformation of variables, and the following of 104 cohorts through time 4.1. Integration along characteristics 104 4.2. Transformation of variables 107 4.3. Following cohorts through time 111 5. About delta-functions and related topics 112 5.1. The delta-function formalism 113 5.2. Delta-functions and transition conditions 114 5.3. Delta-functions in initial conditions 115 5.4. Delta-functions in more dimenions 116
5.1. Models with a separable death rate 176 IX 6. Limiting processes and model simplification 117 6.1. Introduction: the role of limit arguments 117 6.2. Time scale arguments 118 6.2A. An explicit expression for F from the nursery competition model from example 6.2.4 122 6.3. Laws of large numbers on the individual level: the step from paniculate to nonparticulate 123 * 6A. A justification for the limit arguments: the Trotter-Kato Theorem 124 A. Calculus in W, a short refresher 125 Al. Differentiation 125 A2. Integration 127 A3. Some useful relations from linear algebra: the differentiation of determinants 129 B. Stochastic continuous i-state movements 130 C. The />-equations for the examples from section 1 133 IV. Age Dependence 136 J.A.J. Metz & O. Diekmann 1. Age as a substitute for comprehension " 136 1.1. Why this special attention 136 1.2. Which problems allow an age representation? 136 1.3. Integral equations as a natural modelling tool 138 1.4. The calculation of some birth kernels 140 2. Linear theory 143 2.1. An explicit expression for the population birth rate 143 2.2. Renewal theorems 144 2.3. Semigroup approaches 148 2.3.1. The age distribution. 148 2.3.2. Two semi-groups derived directly from the renewal equation itself. 149 * 2.3.3. Finite representability 151 2A. Moments, cumulants and some approximations for r 153 2B. The dependence of the long run population size on the initial age distribution 155 3. Extensions of the linear theory 156 3.0. Introduction ' 156 3.1. Scar distributions in yeast 157 3.1.1. Budding yeasts 157 3.1.2. Fission yeasts 160 3.2. Colony size in the diatom Asterionella 163 4. Some nonlinear extensions of the linear theory 169 4.1. Kermack's and McKendrick's (1927) general epidemic and the nonlinear renewal theorem 169 4.2. Population decline in ectotherms 174 5. Models allowing a reduction to a differential equation on U k 176
5.2. Linear chain trickery 178 A. The Laplace transformation 182 V. The Dynamical Behaviour of the Age-Size-Distribution of a Cell Population 185 HJ.A.M. Heijmans 0. Introduction 185 1.. The model 186 2. Reduction to an abstract renewal equation 188 3. Existence and uniqueness of solutions 189 4. Laplace transformation 190 5. Positive operators 191 6. Location of the singular points 192 7. Computation of the residue in \ d 195 8. The inverse Laplace transform 196 9. Interpretation, conclusions and final remarks 199 A. Appendix 201 VI. Nonlinear Dynamical Systems: Worked Examples, Perspectives and Open Problems 203 0. Diekmann & HJ.A.M. Heijmans (with contributions by F. van den Bosch). 1. Basic terminology and an outline of the program, 203 1.1. Fundamental concepts of dynamical systems theory 203 1.2. Linearized stability and bifurcation theory in the context of ordinary differential equations 207 1.3. An impressionistic sketch of some global aspects 211 2. An example of the construction of a dynamical system: an epidemic model with temporary immunity 214 2.1. The model 214 2.2. Existence and uniqueness 216 2.3. The stability of the steady states 218 3. Hopf bifurcation in scalar nonlinear renewal equations and nursery competition 220 3.1. Introduction to the theory 220 3.2. A first application 222 3.3. Nursery competition 224 4. Lyapunov functions and monotone methods: the G-M model in cell kinetics 227 4.1. 4.2. The Existence model and uniqueness 227 229
XI 4.3. Boundedness of solutions 231 4.4. Extinction of the population - 231 4.5. Existence of a nontrivial equilibrium and monotonicity on an invariant subset 232 4.6. Global stability of the nontrivial equilibrium 235 4.7. Final remarks 236 5. Reduction to an ODE-system: a chemostat model for a cell population reproducing by unequal fission237 5.1. The model 237 5.2. The linear equation 239 5.3. An ODE system related to the nonlinear problem 240 5.4. The nonlinear problem 240 6. Interaction through the environment: some open problems 241 Bibliography 244 Index of examples 261 Part B. From Physiological Ecology to Population Dynamics: a Collection of Papers 263 Topic I. Individuals and laboratory populations. 265 S.A.L.M. Kooijman, Population dynamics on basis of budgets. 266 M.W. Sabelis, The functional response of predatory mites to the density of two-spotted spider rriites. 298 M.W. Sabelis & J. van der Meer, Local dynamics of the interaction between predatory mites and two-spotted spider mites 322 M.W. Sabelis & W.E.M. Laane, Regional dynamics of spider-mite populations that become extinct locally because of food source depletion and predation by phytoseiid mites (Acarina: Tetranychidae, Phytoseiidae).. 345 Topic II. Field populations. 376 N. Daan, Age structured models for exploited fish populations 377 N.M. van Straalen, The "inverse problem" in demographic analysis of stage-structured populations 393 T. Aldenberg, Structured population models and methods of calculating secondary production 409 Topic III. Cell populations. 429 W.J. Voorn & A.L. Koch, Characterization of the stable size distribution of cultured cells by moments 430 P.A.C. Raats, The kinematics of growing tissues 441 Topic IV. Numerical approaches. 452 J. Goudriaan, Boxcartrain methods for modelling of ageing, development, delays and dispersion. 453 W.S.C. Gurney, R.M. Nisbet & S.P. Blythe, The systematic formulation of models of stage-structured populations. 474
XII Topic V. Analytical approaches and a novel type of /-state. 495 H.R. Thieme, A differential-integral equation modelling the dynamics of populations with a rank structure 496