Lecture Notes in Biomathematics

Transcription

1 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

2 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 Introduction 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 The general invertebrate predator 1: the basic biology The general invertebrate predator 2: the population equation The general invertebrate predator 3: calculating the functional response The general invertebrate predator 4: the Rashevsky limit Concluding remarks and summary Size dependent reproduction in ectothermie animals The dynamics of individuals Formulation of the population equations Constant environments Constant environments: reduction to an age-dependent problem Variable environments Summary The cell size distribution The dynamics of individuals Formulation of the^ equations The stable cell size distribution, The inverse problem \ Limits to growth Summary On semigroups and generators 43 II. The Cell Size Distribution and Semigroups of Linear Operators Diekmann. 1. Formulation of the problem Strongly continuous semigroups of bounded linear operators Do growth, death and division generate a semigroup? The spectrum of A The characteristic equation 58

3 VIII 6. Decomposition of the population state space X Relations between the spectra of A and of T(t) Exponential estimates The stable size distribution Interlude: integration along characteristics The merry-go-round The merry-go-round with an absorbing exit Remarks about positivity 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 Some modelling philosophy The state concept Obtaining an /-state representation From the individual to the population level Mass balance Mass transport due to continuous i-state movement A biologist's shortcut The mathematician's derivation The (local) loss of/>-mass, The local disappearance of />-mass from the interior of Q \ The disappearance of ^-mass across the boundary of U The (reappearance of /Mnass The (re)appearance of p-mass in the interior of U The (reappearance of ^-mass across the boundary of fi Boundaries and side conditions: picking up the strands Summary and concluding remarks Integration along characteristics, transformation of variables, and the following of 104 cohorts through time 4.1. Integration along characteristics Transformation of variables Following cohorts through time About delta-functions and related topics The delta-function formalism Delta-functions and transition conditions Delta-functions in initial conditions Delta-functions in more dimenions 116

4 5.1. Models with a separable death rate 176 IX 6. Limiting processes and model simplification Introduction: the role of limit arguments Time scale arguments A. An explicit expression for F from the nursery competition model from example 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 IV. Age Dependence 136 J.A.J. Metz & O. Diekmann 1. Age as a substitute for comprehension " Why this special attention Which problems allow an age representation? Integral equations as a natural modelling tool The calculation of some birth kernels Linear theory An explicit expression for the population birth rate Renewal theorems Semigroup approaches The age distribution Two semi-groups derived directly from the renewal equation itself. 149 * 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 Extensions of the linear theory Introduction ' Scar distributions in yeast Budding yeasts Fission yeasts Colony size in the diatom Asterionella Some nonlinear extensions of the linear theory Kermack's and McKendrick's (1927) general epidemic and the nonlinear renewal theorem Population decline in ectotherms Models allowing a reduction to a differential equation on U k 176

5 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 The model Reduction to an abstract renewal equation Existence and uniqueness of solutions Laplace transformation Positive operators Location of the singular points Computation of the residue in \ d The inverse Laplace transform Interpretation, conclusions and final remarks 199 A. Appendix 201 VI. Nonlinear Dynamical Systems: Worked Examples, Perspectives and Open Problems Diekmann & HJ.A.M. Heijmans (with contributions by F. van den Bosch). 1. Basic terminology and an outline of the program, Fundamental concepts of dynamical systems theory Linearized stability and bifurcation theory in the context of ordinary differential equations An impressionistic sketch of some global aspects An example of the construction of a dynamical system: an epidemic model with temporary immunity The model Existence and uniqueness The stability of the steady states Hopf bifurcation in scalar nonlinear renewal equations and nursery competition Introduction to the theory A first application Nursery competition Lyapunov functions and monotone methods: the G-M model in cell kinetics The Existence model and uniqueness

6 XI 4.3. Boundedness of solutions Extinction of the population Existence of a nontrivial equilibrium and monotonicity on an invariant subset Global stability of the nontrivial equilibrium Final remarks Reduction to an ODE-system: a chemostat model for a cell population reproducing by unequal fission The model The linear equation An ODE system related to the nonlinear problem The nonlinear problem 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) 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

7 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