Starting July 5 a small one-week workshop was held at the Lorentz Center in Leiden, bringing together a number of people working on the general theory of Physiologically Structured Populations. The roots of this theory go back to the early 1980s, when a colloquium on this topic was held at the then Mathematical Centre in Amsterdam. The early developments were mainly concerned with the modelling part. A PDE formalism was used as a vehicle for communication. This led naturally to mathematical approaches centering around classical and not-so-classical semigroup theory. For a special subclass of models in addition an approach based on Volterra integral equations was available. The results of the early days are reported in J.A.J. Metz & O. Diekmann (1986) The dynamics of physiologically structured populations, Springer Verlag, Lecture Notes in Biomathematics 68, XII+511 p. That early work has given rise to a considerable number of applications as well as extensions. One of the most notable offspring at the mathematical end was the encompassing functional analytic approach to delay-differential equations reported in O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, H.-O. Walther (1995) Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag XI+534 p. However elegant the latter approach may be, within population dynamics it is only of use for a relatively small subclass of physiologically structured models, essentially equivalent to age based ones. This in stark contrast to the needs brought to the fore by the by now florishing industry dealing with biological applications. Since it looked as if the usual semigroup approaches, in which the semigroup is constructed from a differential generator, were leading to a dead end, a small collective of researchers have tried their hands on extending the integral equation approach, concentrating first on a proper bookkeeping of births over time from which then the population semigroup is constructed in a direct manner thus obviating the necessity to construct the semigroup from a differential generator. At the biological end all the ingredients are the same, but the mathematical end is reached through a different, less classical, route. An integral equation formalism clearly can do the job in full generality in the linear case, although the required formalism is a little daunting; see O. Diekmann, M. Gyllenberg, J.A.J. Metz & H.R. Thieme (1998) On the formulation and analysis of general deterministic structured population models I Linear theory. J. Math. Biol. 36: 349-388. The hunt is now on for an extension of this theory to the nonlinear case. The chosen approach is to concentrate on the environmental feedback-loop. Here the population is considered as an operator, transforming environmental inputs as functions of time into outputs to the environment, again as functions of time. The closed loop population process then is considered as a fixed point of this operator. Simple though this may sound, there are considerable technical difficulties that have to be overcome to make this scheme work, not the least of which is the impenetrability of the usual formalisms.
The Leiden workshop was the third in a series of workshops devoted to overcoming these difficulties. It was preceded by workshops in Turku ( August 13-20, 1997) and Diepenveen (January 6-10, 1999) an NWO sponsored visit of M. Gyllenberg to Utrecht (one month, February 1999), and and will be followed by a next workshop in Turku (November 11-15, 1999). The attendees were Odo Diekmann (Utrecht), Mats Gyllenberg (Turku), Hiayang Huang (Beijing), Markus Kirkilionis (Heidelberg) and Hans Metz (Leiden). The main progress made this time was in the development of an efficient formalism for handling input-output semigroups in a manner compatible with both the later goal of using a contraction argument for determining the fixed point, as well as the various semigroup constructions needed to prove, for example, the principle of linearized stability. Given the results already available from previous occasions the workshop ended with a rather elated feeling: perhaps, perhaps, the end of a long road is in sight. If indeed all promises are fulfilled we shall have a framework that can deal with many of the mathematical intricacies of physiologically structured populations in one go, although necessarily on a very abstract level. Yet the framework is constructed in such a manner that in the longer run it should be possible to deal also with, for example, numerical techniques along exactly the same lines. In the end physiologically structured population models should be understood mathematically on the same level as we understand differential equation models. Of course, the developments in differential equations show that this isn't the end of the story. But at least we can then immediately get down to the nitty gritty of dealing with concrete models unhampered by the present inconclusiveness of an incomplete mathematical framework, and helped on by the then available strong tools of dynamical systems theory and bifurcation theory such as, for example, the principle of exchange of stability.