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.

Fri Nov 12 21:13:40 MET 1999