Table Of ContentLecture Notes 1n Biomathematics
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of Ep1dem1cs. V, 101 pages. 1974. Chemotherapy. IX, 431 pages. 1979.
Vol. 2: Mathematical Problems in Biology, Victoria Conference 1973. Vol. 31: E. Akin, The Geometry of Population Genetics. IV, 205 pages.
Ed1ted by P van den Driessche. VI, 280 pages. 1974. 1979.
Vol. 3: D. Ludw1g, Siochas!lc Population Theories. VI, 108 pages. Vol. 32: Systems Theory 111 Immunology. Proceedings, 1978. Edited
1974. by G. Bruni et al. XI, 273 pages. 1g79.
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M. Conrad, W Gutllflger, and M. Dal C1n. XI, 584 pages. 1974. ings, 1979. Edited by W. M. Getz. VIII, 355 pages. 1980.
Vol. 5 Mathemat1cal Analysts of Decision ProblemS in Ecology. Vol. 34: R. Collins, T.J. van der Werff, Mathematical Models of the
Proc1wd1ngs 1973. Ed1Jed by A. Charnes and W. R. Lynn. VIII, 421 Dynamics of the Human Eye. VII, 99 pages. 1980.
pages. 1970. Vol. 35: U. an der Heiden, Analysis of Neural Networks. X, 159 pages.
Vol. 6: H. T. Banks, Modeltng and Control in the Biomedical Sciences. 1980.
V 114 pages. 1975. Vol. 36: A. Wiirz-Busekros, Algebras in Genetics. VI, 237 pages.
Vol. 7: M.C. Mackey, ion Transport through Biological Membranes, 1980.
An Integrated Theoretical Approach. IX, 240 pages. 1975.
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Vol. 9: N. Dubm, A Stochast1c Model for Immunological Feedback Vol. 38: Biolog1cal Growth and Spread: Mathematical Theories and
in Carcinogenesis: Analysis and Approximations. XIII, 163 pages. Applications. Proceedings, 1979. Edited by W. Jager, H. Rost and
1976. P. Tautu. XI, 511 pages. 1980.
Vol. 10: J.J. Tyson, The Belousov-Zhabotinskii Reaktion. IX, 128 Vol. 39: Vito Volterra Symposium on Mathematical Models 1n Biology.
pages. 1976. Proceedings, 1979. Edited by C. Barigozzi. VI, 417 pages. 1980.
Vol. II. Mathematical Models in Medicine. Workshop 1976. Edited Vol. 40: Renewable Resource Management. Proceedings, 1980.
by J. Berger, W. Buhler, R. Repges, and P. Tautu. XII, 281 pages. 1976. Edited by T. Vincent and J. Skowronski. XII, 236 pages. 1981.
Vol. 12: A. V. Holden, Models of the Stochastic Act1vity of Neurones. Vol. 41: Modeles Mathematiques en Biologie. Proceedings, lg78.
VII, 368 pages. 1976. Edited by C. Chevalet and A. Micali. XIV, 219 pages. 1981.
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D. L. Solomon and C. Walter. VI, 240 pages. 1977. VIII, 282 pages. 1981.
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Biology. VI, 200 pages. 1977. ings, 1980. Edited by M. H. Holmes and L.A. Rubenfeld. V, 104 pages.
Vol. 15: Th. Nagylaki, Selection 1n One-and Two-Locus Systems. 1981.
VIII, 208 pages. 1977. Vol. 44: Recogn1tlon of Pattern and Form. Proceedings, 1979. Edited
Vol. 16: G. Sampath, S.K. Srinivasan, Stochastic Models for Spike by D. G. Albrecht. Ill, 226 pages. 1982.
Trains of S1ngle Neurons. VIII, 188 pages. 1977. Vol. 45: Competition and Cooperation in Neutral Nets. Proceedings,
VoL 17: T. Maruyama, Stochastic Problems in Population Genetics. 1982. Edited by S. Amari and M.A. Arbib. XIV, 441 pages.1982.
VIII, 245 pages. 1977. Vol. 46: E. Walter, Identifiability of State Space Models with applica
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Ed1ted by D. E. Matthews. VII, 385 pages. 1977. Vol. 47: E. Frehland, Stochastic Transport Processes in Discrete
Vol. 19: Measuring Select1on in Natural Populations. Edited by Biolog1cal Systems. VIII, 169 pages. 1982.
F. B. Christiansen and T. M. Fenchel. XXXI, 564 pages. 1977. Vol. 48: Tracer Kinetics and Physiologic Modeling. Proceedings,
Vol. 20: J. M. Cushing, lntegrodifferential Equations and Delay 1983. Edited by R.M. Lambrecht and A. Rescigno. VIII, 509 pages.
Models in Population Dynam1cs. VI, 196 pages. 1977. 1983.
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1977 Edited by R. He1m and G. Palm. VI, 244 pages. 1978. ceedings, 1981. Edited by M. Cosnard, J. Demongeot and A. Le
Vol. 2 2: F. M. Scud a and J. R. Z1egler, The Golden Age of Theoretical Breton. VII, 400 pages. 1983.
Ecology: 1923-1940. XII, 490 pages. 1978. Vol. 50: D. H. Anderson, Compartmental Modeling and Tracer
Kinetics. VII, 302 pages. 1983.
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J. Serra. XII, 338 pages. 1978. Edited by J. P E. Hodgson. VI, 196 pages. 1983.
Vol. 24: F.L. Bookste1n, The Measurement of Biological Shape and Vol. 52: Population Biology. Proceedings, 1982. Edited by
Shape Change. VIII, 191 pages. 1978. H. I. Freedman and C. Strebeck. XVII, 440 pages. 1983.
Vol. 25: P Yodzis, Competition for Space and the Structure of Vol. 53: Evolutionary Dynamics of Genetic Diversity. Proceedings,
Ecological Communities. VI, 191 pages. 1978. 1983. Edited by G.S. Mani. VII, 312 pages. 1984.
Vol. 26: M. B Katz, Questions of Uniqueness and Resolution in Vol. 54: Mathematical Ecology. Proceedings, 1982. Edited by
Reconstruction from Projections. IX, 175 pages. 1978. S.A. Levin and T.G. Hallam. XII, 513 pages. 1984.
Vol. 27: N. MacDonald, T1me Lags in Biolog1cal Models. VII, 112 Vol. 55: Modelling of Patterns in Space and Time. Proceedings,
pages. 1978. 1983. Edited by W. Jager and J.D. Murray. VIII, 405 pages. 1984.
Vol. 28: P.C. Fife, Mathematical Aspects of Reactmg and D1ffus1ng Vol. 56: H. W. Hethcote, J. A. Yorke, Gonorrhea Transmission Dynam
Systems. IV, 185 pages. 1979. ics and Control. IX, 105 pages. 1g94.
Vol. 29: Kinetic Logic - A Boolean Approach to the Analys1s of Vol. 57: Mathematics in Biology and Medicine. Proceedings, 1983.
Complex Regulatory Systems. Proceedings, 1977. Edited by R. Tho Edited by V. Capasso, E. Grosso and S. L. Paveri-Fontana. XVIII,
mas. XIII, 507 pages. 1979. 524 pages. 1985.
ctd. on inside back cover
Lecture Notes in
Biomathematics
Managing Editor: S. Levin
68
The Dynamics of
Physiologically
Structured Populations
Edited by J.A.J. Metz and 0. Diekmann
Springer-Verlag Berlin Heidelberg GmbH
Editorial Board
M. Arbib H.J. Bremermann J.D. Cowan W. Hirsch S. Karlin
J.B. Keller M. Kimura S. Levin (Managing Editor) R.C. Lewontin R. May J. Murray
G.F. Oster A.S. Perelson T. Poggio L.A. Segel
Editors
Johan A.J. Metz
Odo Diekmann
Institute of Theoretical Biology
Groenhovenstraat 5, 2311 BT Leiden, The Netherlands
and
Centre for Mathematics and Computer Science
Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
Mathematics Subject Classification (1980): 92A 15, 92A 17; 45K05, 47D05
ISBN 978-3-540-16786-0 ISBN 978-3-662-13159-6 (eBook)
DOI 10.1007/978-3-662-13159-6
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986
Preface
Physiological processes within individuals and behaviour patterns displayed by individuals rank high among the sub
ject matters of biology. The catch, handling and digestion of a prey by a predator, the uptake of nutrients by and the
growth and fission of a unicellular organism, and the succession of larval stages of an insect, all are biological
phenomena attracting great interest.
When trying to describe and predict the development of real world or laboratory populations by means of
mathematical models one usually concentrates on numbers of individuals only, while ignoring those aspects of the
life-cycle which would necessitate a further classification of individuals.
Structured population models bridge the gap between the individual and the population level. They allow us to
derive information about the dynamics of populations from information about the dynamical properties of individuals
or vice versa.
These lecture notes arose out of a colloquium on the Dynamics of Structured Populations, held at the Centre for
Mathematics and Computer Science in Amsterdam during the first half of 1983. The colloquium consisted of seven
one-day meetings; the number of participants, mainly biologists, ranged from 57 to 21 with an average of 34. At the
start of the colloquium the following ideals and aims were formulated:
Wishful thinking
I. To use knowledge about the behaviour and the physiology of individual animals or plants to describe, under
stand and predict the dynamics of real world and laboratory populations.
2. To build a qualitative theory of (possibly nonlinear) first order partial differential equations (balance laws) in
which non-local terms or side conditions occur.
Practical objectives
I. To take stock of existing knowledge, literature, models, analytical techuiques, ....
2. To collect examples of biological mechanisms which force us to distinguish individuals from each other on the
basis of certain traits.
3. To train the participants in formulating structured models as functional-partial differential equations. (One
has to get accustomed to the mathematical formalism before it becomes a convenient language for expressing
ones thoughts.)
4. To explore the connections between structured models and models formulated as ordinary differential equa
tions in Rk, through limit arguments, the taking of appropriate projections, etc.
5. To practice using both biological relevance and mathematical tractability as criteria for choosing problems and
making simplifications.
6. To train the participants in interpreting mathematical results in biological terms.
7. To learn analytical (and also numerical) techniques which are useful when studying functional partial
differential equations.
8. To elaborate simple prototype problems in detail in order to enhance the intuitive understanding of the rela
tions between the dynamics of individuals and the dynamics of populations.
9. To show how qualitative information obtained from simplified models complements quantitative information
obtained from simulation studies.
10. To let the participants become conversant with each other and to provide them with a common basis, and by
doing so, to stimulate cooperation.
IV
The present notes have grown out of the seeds sown at that colloquium. The text has grown and ripened during a
period of intense interdisciplinary interaction of the core participants after the colloquium itself had finished. Part A
consists of a systematic exposition of the present state of the mathematical theory. Part B is a collection of papers on
special topics. The connections with the main theory are manifest from the many forward references in part A. Each
of the topical sections of part B carries a short preface placing the contributions in perspective. The remaining part of
this preface refers mainly to part A.
The first step in building structured population models consists of finding a suitable explicit parametmation of the
state of the individuals (satiation, size, age, ... ). The state of the population is then given by a density function
describing their distribution in the individual state space. In the course of time the state of each individual changes
smoothly (owing to digestion, growth, aging. ... ). Moreover individuals are born, die or may make jumps in the indivi
dual state space (for instance, the satiation of a predator will change abruptly upon the swallowing of a prey). In the
second step one draws up the balance of these processes. At this point two roads depart.
The first road leads to the computer. The biological description of the various smooth and jump transitions is
translated directly into a computer program and subsequently biological reality is simulated by calculating the result of
all transitions during a certain period of time. This approach is touched upon in the contributions by Sabelis and
coworkers and by Goudriaan in part B, but it is certainly not the main theme of these notes. Whenever the individual
state space is high-dimensional and/ or the constants and functions specifying the model can be determined quantita
tively in advance, this is an appropriate road. The reward at the end of it is a quantitative understanding of the rela
tion between the behaviour of the individuals and the dynamics of the population.
The second road leads to (first order functional partial) differential equations describing the infinitesimal changes
in population state. The coefficients in the equation describe the functioning and the behaviour of individuals, but
their solution pertains to the population as a whole. Subsequently we look for typical properties of solutions, such as
convergence towards relatively simple time independent or periodic solutions. (Finding such properties can be
difficult, and often a mixture of numerical and analytical techniques will be used.) Whenever we use modelling as a
strategic tool for understanding biological reality, this second road is the appropriate one. The reward at the end is
insight into qualitative aspects of the relation between the behaviour of individuals and the dynamics of the popula
tion.
The aim of part A is to map out this second road. We want to describe its present state, and to stimulate our
readers to use it, to improve it, to strengthen its foundations, and to enlarge its number of lanes. The required
mathematical tl\eory has only just started and a wealth of biological phenomena is awaiting a mathematical descrip
tion which fits into this framework.
In these notes we deliberately address biologists and mathematicians alike. We try to present mathematical con
cepts, techniques and results together with their biological interpretation in the context of specific models in order to
introduce biological readers to the mathematics and we try to present biological phenomena and their verbal models
together with their mathematical counterparts (exotic equations with transformed arguments) in order to introduce
mathematical readers to the underlying biology, as painlesly as possible. Since one needs to practice a language to
learn it, we have included numerous exercises.
Perhaps at this point it should be made clear that we don't want to join issue with advocates of simulation
methods. Simulation ~odels undoubtedly can come closer to reality. This is a clear advantage. But it can be a
disadvantage as well, since the inherent complexity can make it difficult to interpret the results. Between present day
realistic simulation models and the usual idealized total population models formulated as ordinary differential equa
tions lies quite a gap. Models of the type discussed in these notes are specifically designed to fill this gap. A true
understanding of natural phenomena quite often requires a continuous spectrum of supplementary models and not just
one. Thus we strive for symbiosis, not competition.
Functional partial differential equations yield a new and interesting class of infinite-dimensional dynamical systems
which holds great appeal for mathematicians. In recent years many papers were written on this topic, mainly dealing
with age-structured populations. One of the basic objectives of structured population dynamics is the use of biological
knowledge to describe the interaction of a population and its environment, including other populations, by specifying
in detail how birth, death and growth processes depend on environmental quantities. In this respect "age" usually fails
to be a sufficient characteristic for distinguishing individuals, as it provides but scant information about their physio
logical condition, except in the special case of constant environmental conditions (and then only indirectly). Indeed,
in most nonlinear age-structured models density dependence is incorporated ad hoc, without any physiological founda
tion. Although this may lead to interesting mathematical problems, from a biological viewpoint it hardly improves
v
upon the usual unstructured models. We hope that these notes will stimulate applied mathematicians to spend more
time and energy on the equally (or rather more, we feel) interesting infinite-dimensional dynamical systems which
derive from choosing some physiological variable such as size as distinguishing characteristic. (Ibis also makes it
necessary to pay more attention to the modelling aspect. Chapters I and III provide some clues).
The mathematical theory of the class of linear functional partial differential equations corresponding to structured
population models is basically in good shape (see chapters II, IV and V), though 11 great deal of amendments and
extensions are certainly needed. This in sharp contrast with the nonlinear case where hardly any systematic theory
exists (except for the case of purely age-structured models, but even there the work is far from finished; in chapter VI
we present some general thoughts about the directions we feel that such a theory ought to take). Yet another aim we
have is to stimulate our mathematical readers to bring about drastic changes in this situation. So, in a sense, we hope
that these notes will soon be out of date!
Amsterdam and Leiden Odo Diekmann and Hans Metz
december 1985
Hints for reading part A
dependence between
the chapters:
linear theory nonlinear theory
primary subject:
"---y-----'
modelling
The stinTed parts of chapters I, III and IV are of a more mathematical nature than the rest of these chapters.
Acknowledgements
The editors wish to thank Henk Heijmans for all his help in the organization of the colloquium and the editing of
these notes, Henk Heijmans, Mats Gyllenberg, Bas Kooijman, Roger Nisbet, Mous Sabelis, Horst Thieme and John
Tyson for illuminating discussions on various topics from part A, Steve Blythe, Frank van den Bosch, Bas Kooijman,
Andre de Roos and especially Henk Heijmans and Sean Nee for constructive comments on various drafts of part A,
Hans Heesterbeek for preparing the table of contents and the index of examples, Frank van den Bosch, Kees Lankes
ter and Andre de Roos for drawing the figures, and Mrs G. Verloop for her skill in dealing with the text processor.
Hans Metz wishes to thank the Netherlands organization for the advancement of pure research ZWO for a stipen
dium and the L.:partment of Applied Physics of the University of Strathclyde, the School of Biological Sciences of the
University of Sussex, and the Edward Grey Institute of Field Ornithology, Oxford for their hospitality during the
aca1emic year 1984/85.
Odo Diekmann wishes to thank the University of Amsterdam for giving him the opportunity to lecture on struc
tured populations during the fall of 1983.
And all of us wish to thank the Centre for Mathematics and Computer Science, Amsterdam, for providing facilities
for the 1983 Colloquium on Structured Populations and for the assistance in the preparation of these notes.
Ust of Contributors
T. ALDENBERG, RIVM (National Institute for Public Health and Environmental Hygiene), Postbus I, 3720 BA
Bilthoven, the Netherlands.
S.P. BLYTHE, Department of Applied Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 ONG,
Scotland, U.K.
F. VAN DEN BoscH, Center for Mathematics and Computer Science, Kruislaan 413, 1098 SJ Amsterdam, the Neth
erlands (Present address: Institute of Theoretical Biology, University of Leiden, Gtoenhovenstraat 5, 2311 BT
Leiden, the Netherlands).
N. DAAN, RIVO (Netherlands Institute for Fishery Investigations),Postbus 68, 1970 AB IJmuiden, the Netherlands
0. DIEKMANN, Center for Mathematics and Computer Science, Kruislaan 413, 1098 SJ Amsterdam, the Nether
lands & Institute of Theoretical Biology, University of Leiden, Groenhovenstraat 5, 2311 BT Leiden, the Nether
lands
J. GouDRIAAN, Department of Theoretical Production Ecology, Agricultural University, Bomsesteeg 65, 6708 PD
Wageningen, the Netherlands
W.S.C. GURNEY, Department of Applied Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 ONG,
Scotland, U.K.
H.J.A.M. HEJJMANS, Center for Mathematics and Computer Science, Kruislaan 413, 1098 SJ Amsterdam, the
Netherlands
A.L. KocH, Department of Biology, Indiana University, Bloomington, Indiana 47405, U.S.A.
S.A.L.M. KoOJJMAN, Biologi8ch Laboratorium, Vrije Universiteit, Postbus 7161, 1007 MC Amsterdam, tlte Nether
lands & Division of Technology for Society TNO, Postbus 217, 2600 AE Delft, the Netherlands
W.E.M. Laane, Department of Animal Ecology, Agricultural University, Postbus 61, 6700 AB Wageningen, the
Netherlands
J. VAN-DERMEER, Department of Population Biology, University of Leiden, Kaiserstraat 63, 2300 RA Leiden, the
Netherlands
J.A.J. METz, Institute of Theoretical Biology, University of Leiden, Groenhovenstraat 5, 2311 BT Leiden, the
N etlterlands
R.M. NISBET, Department of Applied Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 ONG,
Scotland, U.K.
P.A.C. RAATs, Institute for Soil Fertility, Postbus 30003, 9750 RA Haren (Gr.), the Netlterlands
M.W. SABELIS, Department of Population Biology, University of Leiden, Kaiserstraat 63, 2300 RA Leiden, the
Netherlands.
N.M. VAN STRAALEN, Biologisch Laboratorium, Vrije Universiteit, Postbus 7161, 1007 MC Amsterdam, the
Netherlands
H.R. THIEME, Sonderforschungsbereich 123, Universitlit Heidelberg, Im Neuenheimer Feld 294, 6900 Heidelberg,
Bundesrepublik Deutschland
W.J. VooRN, Department of Electron Microscopy and Molecular Cytology, University of Amsterdam (Present
address: Department of Medical Microbiology, University of Amsterdam, Meibergdreef 15, 1105 AZ. Amsterdam,
tlte Netherlands)
TABLE OF CON1ENTS
Part A. Mathemadcal Models for Physiologically Structured Populations: a Systematic Exposition
I. A Gentle Introduction to Structured Population Models: Three Worked Examples 3
J.A.J. Metz & 0. Diekmann
1. Introduction 3
2. The invertebrate functional response 5
2.1. Introduction 5
2.2. Holling's disk equation I: 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 ectothermic 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 p 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
49
2. Strongly continuous semigroups of bounded linear operators
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
II. 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 & 0. Diekmann.
I. Introduction: six examples for later use 78
2. Some modelling philosophy 83
2.1. The state concept 83
2.2. Obtaining an i-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 p-mass 96
3.2.1. The local disappearance of p-mass from the interior of 0 96
3.2.2. The disappearance of p-mass across the boundary of 0 98
3.3. The (re)appearance of p-mass 99
3.3.1. The (re)appearance of p-mass in the interior of 0 100
3.3.2. The (re)appearance of p-mass across the boundary of 0 101
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 III
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
