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0 Executive summary
In 1993, a joint research project ``Population dynamics and epidemiology'' was granted by the Priority Program Nonlinear Systems (NLS), which involved some 20 participants. This initiated frontline research in the field of individual-based population dynamics and epidemics. It stimulated intensive exchange of information, thanks to a financial contribution to the costs of travelling, of workshops, summer and winterschools. A newsletter has been issued to coordinate the activities.

Initially the emphasis was on stimulating interaction between mathematicians and population biologists through a chain of people with mutual interests and partially overlapping mathematical and biological expertise. This was very successful and has resulted in a number of close collaborations between mathematicians, theoretical biologists and experimental biologists that would not as easily have flourished without the stimulus of the NLS program. Having established these links, recently the emphasis shifted to stimulating these research collaborations individually through short term visits by experts from abroad.

After four years several of the established research links have gained a momentum of their own. There is, however, a substantial community of population biologists outside the group of original applicants with currently weak links with nonlinear mathematicians and intermediaries. Due to the weakness - or even absence - of these links, the scientists involved were not easily reached by the activities of the program in the first period. We think that by organising frequent one-day discussion meetings on various challenging themes of broad mutual interest - fuelled by a number of lectures and opinions of experts from biology as well as from mathematics - these `missing links' can be forged and strengthened. It is our aim therefore in the next period of the program to enlarge the number of chains in the Dutch population biology community along which mathematical results can flow in one direction and mathematically challenging problems can flow in the opposite direction. The original community that has been formed as a result of the NLS program is now in a position to widen its reach and connect to other individuals and groups and broaden the range of topics where both population dynamics and nonlinear mathematics can fruitfully interact.

This proposal asks for financial support for: (i) short term visits by various international experts as part of the series of discussion meetings; (ii) short term collaborating visitors from abroad; (iii) support for the organization of two summer/winter schools (with the same format as the 1996 winterschool); (iv) support for a mini-symposium on Nonlinear Methods in Population Biology and Epidemiology at the 4th European Conference for Mathematics in Biology and Medicine, in Amsterdam, 1999.

A list of topics for the meetings and the experts to be invited for them will be drawn up by the coordinators (see 3 below) soon in discussion with all interested parties inside but predominantly also outside the original group of applicants. A provisional list of prospective short term visitors to strengthen or initiate collaboration with a Dutch group is part of the proposal. The coordinators aim to `force' a combination of visits for the purpose (i) and (ii) above as much as possible.

1 Title
A continuation research proposal for NLS Population dynamics and epidemics.

2 Applicants
List of the participants of the population dynamics platform:

dr.R.J. de Boer UU Utrecht dr.F. van den Bosch LUW Wageningen
prof.dr.O. Diekmann UU Utrecht Grasman LUW Wageningen
dr.P. Haccou RUL Leiden dr.J.A.P. Heesterbeek GLW-DLO Wageningen
dr.A. Hemerik LUW Wageningen prof.dr.P. Hogeweg UU Utrecht Jong UU Utrecht dr.M.C.M. de Jong ID-DLO Lelystad
dr.M. Kirkilionis CWI Amsterdam dr.B.W. Kooi VUA Amsterdam
prof.dr.S. Kooijman VUA Amsterdam prof.dr.J.A.J. Metz RUL Leiden
dr.P. Opdam IBN-DLO Wageningen dr.A.M. de Roos UvA Amsterdam
dr.P.C. de Ruiter AB-DLO Haren prof.dr.M.W. Sabelis UvA Amsterdam
dr.M. Scheffer RIZA Lelystad dr.J. Verboom IBN-DLO Wageningen

3 Institute
prof.dr. S.A.L.M. Kooijman, Dept. Theoretical Biology
Vrije Universiteit, de Boelelaan 1087, 1081 HV Amsterdam
fax 020-4447123, tel 020-4447130, email (contact person).

dr. F. van den Bosch, Dept. of Mathematics, Agricultural University, Wageningen.

dr. J.A.P. Heesterbeek, Agricultural Mathematics Group (GLW-DLO), Wageningen

4 Abstract
Since the start of the project ``Population dynamics and epidemiology'' in 1993 the following activities took place:

We refer to the newsletters for a detailed account of all activities; the list of publications in sections 8 and 11 below illustrate the scientific productivity.

The personal interaction with Yuri Kuznetsov (as well as his lectures and subsequent book) provides a good example of a particularly beneficial activity for a substantial number of theoretical biologists and mathematicians, fuelled to a large extent by the NLS program. This interaction has resulted directly in a more advanced and fruitful use of bifurcation theory in their work and papers (for example work by Kooi, Boer and co-workers, Hantke, de Roos). In time this more advanced use of nonlinear methods will be passed on by them to less theoretical biologists further along the chain, and one can therefore conclude that the interaction as stimulated by NLS is certainly paying off in the Dutch population dynamical community.

We would like to continue these types of activity but maximize the range of people who benefit by making careful choices of the topics for seminars and discussion meetings (see section 9).

5 Duration of the project 4 years.

6 Personnel No personnel.

7 Cost estimates
The total costs for the four year period are estimated to be 110kf. This amount consists of travel and subsistence contributions for 5-10 visiting scientists per year (15kf per year), support for two summer/winter schools (15kf per school), and support for a mini-symposium on Nonlinear Methods in Population Biology and Epidemiology at the 4th European Conference on Mathematics in Biology and Medicine (20kf).

8 Publications
Although the publications listed are all the result of interaction between (nonlinear) mathematics and biology, the publications marked with an asterisk are examples where particularly the two-way interaction was fuelled either directly or indirectly by activities of the NLS program.
R.J. De Boer & A.S. Perelson. T-cell repertoires and competitive exclusion. J. theor. Biol., 169:375-390, 1994.

M.A. Boerlijst & P. Hogeweg. Spatial gradients enhance persistence of hypercycles. Physica D, 88:29-39, 1995.

M.A. Boerlijst & P. Hogeweg. Attractors and spatial patterns in hypercycles with negative interactions. J. theor. Biol, 176:199-210, 1995.

*F. van den Bosch & A.M. de Roos. Roguing and replanting in orchards J. Biol. Systems, 3:495-503, 1995.

F. van den Bosch & J.A.J. Metz. The velocity of spatial population expansion: an overview of the individual based approach. Aspects in Applied Biology, Modelling in applied biology: Spatial aspects, 46:231-238, 1996.

F. van den Bosch & J.A.J. Metz. The continental spread of plant disease. Aspects in Applied Biology, Modelling in applied biology: Spatial aspects, 46:249-251, 1996.

C.J. Briggs, R.M. Nisbet, W.W. Murdoch, T.R. Collier & J.A.J. Metz. Dynamical effects of host-feeding in parasitoids. Journal of Animal Ecology, 64:403-416, 1995.

O. Diekmann. An invitation to structured (meta) population models. In: S.A. Levin, T.M. Powell and J. H.Steele, eds. Patch Dynamics Springer Lect. Notes in Biomath. 96 pp. 162-175, 1993.

O. Diekmann, M. Gyllenberg & H.R. Thieme. Perturbing semigroups by solving Stieltjes renewal equations, Diff. Int. Equ., 6:155-181, 1993.

*O. Diekmann, M. Gyllenberg, J.A.J. Metz & H.R. Thieme. The 'cumulative' formulation of (physiologically) structured population models. In: Ph. Clement and G. Lumer, eds. Evolution equations, control theory and biomathematics, Lect. Notes in Pure Applied Math. 155: 145-154, Marcel Dekker, 1994.

*O. Diekmann, M.C.M. de Jong & J.A.P. Heesterbeek. The computation of R for discrete-time epidemic models with dynamic heterogeneity, Math. Biosc., 119: 94-114, 1994.

O. Diekmann & J.A.J. Metz. On the reciprocal relationship between life histories and population dynamics. In: S.A. Levin, ed. Frontiers in Theoretical Biology, Springer Lect. Notes in Biomath. vol. 100: 263-279, 1994.

*O. Diekmann, M.C.M. de Jong, A.A. de Koeijer & P. Reijnders. The force of infection in populations of varying size: a modelling problem. J. Biol. Syst., 3:519-529, 1995.

O. Diekmann, J.A.J. Metz & J.A.P. Heesterbeek. The legacy of Kermack and McKendrick. In: D. Mollison, ed. Epidemic Models: Their Structure and Relation to Data. Cambridge Univ. Press, pp. 95-115, 1995.

*O. Diekmann, A.A. de Koeijer & J.A.J. Metz. On the final size of epidemics within herds. Canadian Applied Mathematics Quarterly, 4:21-30, 1996.

J.A.P. Heesterbeek & J.A.J. Metz. The saturating contact rate in epidemic models. In: V. Isham & G. Medley, eds. Models for Infectious Human Diseases. Their Structure and Relation to Data. Cambridge Univ. Press, pp. 308-310, 1996.

*J.A.P. Heesterbeek & M.G. Roberts. Threshold quantities for helminth infections. J. Math. Biol., 33, 415-434, 1995.

*O.A. van Herwaarden & J. Grasman Stochastic epidemics: major outbreaks and the duration of the endemic period. J. Math. Biol., 33, 581-601, 1995.

P. Hogeweg. Multilevel Evolution: replicators and the evolution of diversity. Physica D, 75 275-291, 1994.

P. Hogeweg. On the potential role of DNA in an RNA world: Pattern generation and information accumulation in replicator systems. Ber. Bunsengesel Phys. Chem., 98: 1135-1139, 1994.

*M.C.M. de Jong, O. Diekmann & J.A.P. Heesterbeek. How does transmission of infection depend on population size? In: D. Mollison, ed. Epidemic Models: Their Structure and Relation to Data. pp. 84-94. Cambridge Univ. Press, 1995.

*B.W. Kooi & M.P. Boer. Discrete and continuous time population models, a comparison concerning proliferation by fission. J. Biol. Systems, 3:543-558, 1995.

*B.W. Kooi & S.A.L.M. Kooijman. Many limiting behaviours in microbial food chain. In: O. Arino, M. Kimmel, & D. Axelrod, eds. Mathematical Population Dynamics: Analysis of Heterogeneity Vol 2, pp. 131-148, Wuerz, 1995.

S.A.L.M. Kooijman. Dynamic Energy Budgets in Biological Systems; Theory and Applications in Ecotoxicology Cambridge University Press, 1993.

S.A.L.M. Kooijman & J.J.M. Bedaux. The analysis of aquatic toxicity data. VU University Press, and software DEBtox, 1996.

*S.A.L.M. Kooijman & B.W. Kooi. Catastrophic behaviour of myxamoebae. Nonlin. World, 3:77-83, 1996.

*S.A.L.M. Kooijman, B.W. Kooi, & M.P. Boer. Rotifers do it with delay. the behaviour of reproducers vs dividers in chemostats. Nonlin. World, 3:107-128, 1996.

S.A.L.M. Kooijman. The stoichiometry of animal energetics. J. theor. Biol., 177:139-149, 1995.

J.D. van der Laan, L. Lhotka & P. Hogeweg. Sequential predation: a multi model study. J. theor Biol. 174:149-167, 1995.

*J.D. van der Laan & P. Hogeweg. Predator-prey coevolution: interactions among different time scales. Proc. R. Soc. Lond. B., 259:35-42, 1995.

*C.C. Lord, M.E.J. Woolhouse, J.A.P. Heesterbeek & P.S. Mellor. Vector-borne diseases and the basic reproduction number: a case study of African horse sickness. Med. Vet. Entomology, 10, 19-28, 1996.

*E.McCauley, W.G.Wilson and Roos. Dynamics of age- and spatially-structured predator-prey interactions: Individual based models and population level formulations. Amer. Natur. 142(3):412-442, 1993.

J.A.J. Metz & F. van den Bosch. Velocities of epidemic spread. In: D.Mollison ed. Epidemic models, their structure and relation to data. Cambridge University Press, 150-186, 1995.

*C.H. Ratsak, B.W. Kooi, & H.W. van Verseveld. Biomass reduction and mineralization increase due to the ciliate Terahymena pyriformis grazing on the bacterium Pseudomonas fluorescens. Wat. Sci. Tech., 29:119-128, 1994.

*C.H. Ratsak, B.W. Kooi, & S.A.L.M. Kooijman. Modelling the individual growth of Tetrahymena sp. and its population consequences. J. Eukaryotic Microbiol., 42:268-276, 1995.

*M.G. Roberts & J.A.P. Heesterbeek. The dynamics of nematode infections of farmed ruminants. Parasitology, 110, 493-502, 1995.

*A.M. de Roos & M.W. Sabelis. Why does space matter? OIKOS, 74(3):347.

*A.M. de Roos & F. van den Bosch. The dynamics of infectious diseases in orchards with roguing and replanting as control strategy. J. Math. Biol., 35, 129-157, 1996.

*M. Scheffer & R.J. De Boer. Implications of spatial heterogeneity for the paradox of enrichment. Ecology, 76:2270-2277, 1995.

*W.G. Wilson, A.M. de Roos & E.McCauley. Spatial instabilities within the diffusive Lotka-Volterra system: Individual-based simulation results. Theor. Pop. Biol., 43(1):91-127, 1993.

*W.G. Wilson, E. McCauley & A.M. de Roos. Effect of Dimensionality on Lotka-Volterra Predator-prey Dynamics: Individual Based Simulation Results, Bulletin of mathematical biology, 57(4):507-526, 1995.

9 Research Programme
The following projects are finished or will be finished soon:

Models for population dynamics evolved recently from a simple sets of ordinary differential equations, describing the dynamics of non-structured populations of prey and predators strategically, to partial differential equations or integral equations, describing physiologically structured systems of populations realistically.

The increase in realism proved extremely valuable in a variety of contexts:

These important biological applications for realistic population dynamical models, combined with the recent progress in the development of a mathematical framework to address such problems, justify a continued interest for population dynamics within the NLS priority program.

The mathematical aspects of structured population dynamics involves the development of new mathematical frameworks to pose and analyse biological problems as well as the application of recently developed tools for the analysis of complex dynamical systems, especially to analyse local stability of attractors in infinite dimensional systems.

The groups formed in the first years of the program will continue their various collaborations. We mention only a few specifically to give a flavour of the questions studied and to show some chains running from nonlinear theory to biology. Van den Bosch and de Roos will analyse in more detail the dynamics of infectious diseases in orchards and the potential for control by management practices. Phytopathologists have expressed interest in this work and collaboration is hoped to start (prof. Jeger, Wageningen). Neutel (Haren), de Ruiter and Heesterbeek study causes of apparently inherent stability in food webs in soil combining experimental results, simulation and nonlinear analysis. De Roos will study effects of spatial heterogeneities. Kooi, Kooijman, de Boer and Kelpin will analyse food chains and food webs, for instance the effects of variation of parameter values among individuals, and the `large number assumption' at low reproduction rates. The latter problem will also be addressed by Kirkilionis and Diekmann within a more formal mathematical setting. They concentrate on the development of theory and numerical bifurcation analysis of models of physiologically structured populations. To that end BASE a modularly designed tool for 'Bifurcation Analysis of Structured Equations was written by Kirkilionis. Metz and others will study the problem of slow changes of parameters of individuals and its consequences for competition and evolutionary change. Heesterbeek, van den Bosch and Verboom will use and develop meta-population theory to answer practical questions in nature management that relate to measures taken for the preservation of wildlife natural diversity to counter effects of habitat destruction and fragmentation. Interaction with biologists working for various regional bodies (such as provinces) will be actively sought.

The fact that a joined program on population dynamics has been granted within the NWO priority program MPR (Massaal Parallel Rekenen) illustrates the rapid progress and interest in the algorithmic aspects of population dynamical models. (6 participants: Diekmann, Kirkilionis, Kooi, Kooijman, Metz and de Roos) of this proposal partake also in the MPR project. We believe that the analytical and the algorithmical aspects must be developed simultaneously and think that the NLS and MPR programs can fertilize each other in the population dynamical field.

Proposed activities
An initial list of possible visitors is given below. Appearing on this list does not necessarily imply approval. We propose that (subject to approval of the project of course) the coordinators send out a call for topics of discussion meeting and seminars (and accompanying names of visitors) once a year to a very broad representative slice of the Dutch population biological and biomathematical community. This does not mean that ideas arising later in the year will not be considered. The coordinators of the program will judge the relevance of the proposed activity with respect to the activities of the platform and will make a priority ordering of the proposals when necessary. The seminars, discussion meetings and visitors will be announced and later reported on in the Newsletter (editor Kooi) and will be spread by regular convocations by de Roos. As always co-financing will be stimulated, i.e. part of the costs should preferably be covered by the participant(s). In case of doubt, the coordinators will consult NWO (Dr. Maessen).

The next types of meetings are already planned:

10 Societal relevance
Population dynamics is a discipline applied to solve problems in among other areas Ecotoxicology, Water quality management, Production ecology, Landscape planning and Epidemiology; see the Research Programme and the attached letters of dr. M.Scheffer, dr. J.Verboom and dr. Jong.

11 References
In the reference list we give an overview of books and PhD-theses of which the subject is directly related to the project:

J. Grasman and G. van Straten, editors, Predictability and Nonlinear Modelling in Natural Sciences and Economics, Kluwer Academic Publishers, Dordrecht, 1994.

A.M. de Roos & M.W. Sabelis. Why does space matter? OIKOS, 74(3):347. This volume of OIKOS are the Proceedings of the summerschool held in Amsterdam, Aug 29-Sept 2, 1995.

Yu.A. Kuznetsov. Elements of applied bifurcation theory, Applied Mathematical Sciences Vol 112, Springer-Verlag Berlin, 1995.

J. Verboom. Modelling fragmented populations: between theory and application in landscape planning, PhD thesis, Rijksuniversiteit Leiden, 3 December 1996.

Ulf Dieckmann. The dynamical theory of coevolution, PhD thesis, Rijksuniversiteit Leiden, 23 January 1997.

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Bob Kooi
Thu Dec 11 13:19:33 MET 1997