January 5-9, 2000, Woudschoten, the Netherlands

The Third Winterschool on Population Dynamics was organised by Fleur Kelpin (Department of Theoretical Biology, Vrije Universiteit, Amsterdam), Odo Diekmann (Department of Mathematics, Utrecht University), Hans Heesterbeek (Centre for Biometry, Wageningen) and Bob Kooi (Department of Theoretical Biology, Vrije Universiteit, Amsterdam). This Winterschool is part of the "Population Dynamics and Epidemiology" project of the NWO priority program "Nonlinear Systems".

About 66 young scientists from all over the World (Europe, China, USA, South Africa) and from all kind of backgrounds (population biology, mathematics, epidemiology) attended this Winterschool.

The general lines that made the success of the first two Winterschools were kept. Five lecturers gave each a series of five lectures:

Bas Kooijman | (Department of Theoretical Biology, Vrije Universiteit, Amsterdam) |
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Rob de Boer | (Department of Theoretical Biology, Utrecht University) |

Horst Thieme | (Department of Mathematics, Arizona State University, Tempe) |

Fred Adler | (Departments of Mathematics and Biology, University Of Utah, Salt Lake City) |

Roger Nisbet | (Department Ecology, Evolution & Marine Biology, University of California, Santa Barbara) |

Each lecture lasted around forty five minutes which was enough for the "lecturer" to tell the essential and not to long for the participants to keep concentrated. The readers provided a substantially good understanding of the way mathematics is applied to study problems in biology. They also mentioned the limits of modelling to fully represent the reality. The lectures were very lively, thanks to the questions of many participants or to the additional informations some of them could give.

Bas Kooijman presented the Dynamics Budget Theory, a theory with which most of the participants were not familiar. He first began with the basic model for individual organisms. The DEB model specifies the rules for uptake and use of food for maintenance, growth and reproduction. Three life stages are distinguished; the embryo (which does not eat), the juveniles (which eats, but does not reproduce) and the adult. He pointed out the importance of taking the reserves into account in order to understand biological phenomenon. Multivariate extensions of the basic DEB model were presented on the basis of Synthesizing Units, with several substrates, several reserves and two structures. This extensions can provide a good model for plants (the two structures are roots and leafs) and symbiosis phenomenon. In his last lecture, Bas Kooijman presented the applications of the theory in several fields of social interest (Ecotoxicology, tumor growth, production optimization, ). He insisted on the fact that Society asks for Biology, and that it is dangerous to let only non-biologists provide an answer.

Rob de Boer provided models to explain some apparently unexpected properties of the immune system. The way of modelling was different for each model, giving thus an idea of how extensive the contribution of mathematics can be. For instance, he applied bifurcation analysis, genetic algorithms and quasi-steady state approximations. The first lectures dealt with the diversity of lymphocyte repertoire. A probabilistic-based model suggests that this diversity principally reflects the diversity of self-antigens rather than that of foreign antigens. The competition between T-cells for MHC-peptide complexes can explain why the total T-cell number hardly depends upon the diversity of the T cell repertoire or the diversity of the set of presented peptides for a sufficient diversity and/or degree of cross-reactivity. A simple model was used to explain some properties of T-cell vaccination and MHC diversity was presented as a result of host-pathogen coevolution.

The lectures presented by Horst Thieme gave a solid mathematical foundation for the study of Population Dynamics. The first lecture presented the most rudimentary model with two stages (Larvae and Adult) and density dependence. Sufficient conditions were given to obtain a bounded solution. The possibility of stable periodic solution was discussed (Bendixon Negative Criterion, Hopf bifurcation).The cannibalism in Arizonan tiger salamanders was then used to illustrate the persistence and invasion theory. As for discrete structured metapopulation models, Horst Thieme provided the mathematical theory for the study of quasi-positives matrices. He ended his lectures with continuous structured population models, illustrated by the Pease/Inaba influenza model.

Fred Adler provided tools to study the evolution of spatial patterns for the use of resource with interactions between neighbours (He tries to model the STAR, that means Space Time And Resource). He first overviewed the classical theory for altruism and extended it for the use of resources in a simple resource dynamics. As storing the resource is good for everyone including the one who stores, this behaviour can be selected. For sedentary organisms, long-term local interactions with neighbours are unavoidable and provide a place where "cooperative" strategies for resource use might evolve. This evolution was studied with models for competition for space and applied to ant colonies. The importance of signals and informations shared by plants in plant-plant interactions were also discussed.

The lectures by Roger Nisbet gave a good understanding of structured and unstructured population models and of the way the results can be applied to analyse ecological data. He focused in his first lectures on the population cycles. For the delay-differential equations, the lower limit for the period of the cycles is twice the developmental delay. For an extended Nicholson-Bailey host-parasite model, this lower limit is twice the developmental delay of the host and four this of the parasite. These results are powerful diagnostic properties when determining mechanisms of population regulation from time series. Roger Nisbet dealt in his last lectures with biomass-based models to illustrate with different examples the predictive power and limitations of simple population models.

Participants were provided with two readers containing papers dealing with the presented topics.

For two afternoons (and sometimes evenings), small groups were formed to study anddiscuss papers related to the subjects developed in the lectures. A presentation of the work was done on Saturday afternoon. These working groups were a good opportunity to appreciate the different ways of understanding biology and mathematics by people from very different backgrounds.

The evenings, the meals and the tea and coffee breaks were not only moments to rest but although to discuss about all topics that can be of interest for young PhD students: What will I do after my PhD? How will I manage to reconcile a life devoted to research and family life? These discussion were of course followed by lighter discussions around the bar or the billiard table.