12 Sep 2003; Mat-Nat Faculty (UITO), room U7, 10:15-12:00

Einar Mjølhus "Some rigorous results for a class of nonlinear growth models with age structure and semelparity, leading to synchronization".

This seminar presents rather complete results for a topic that was also discussed in the Lie seminar of 28/11/2002. The problem arose during investigations in the period 1992 - 1995, leading to the Ph. D. dissertation of Arild Wikan of Harstad College. We investigated models of growth of a biological species with discrete time, age structure, and self-inhibition.

As a particular feature, we pursued the effect on stability of delay in reproduction. As an extreme of this, we investigated the case when only the last generation reproduces, and then dies. This is called semelparity. An example is the so-called 17-years cicada. We found some surprising results:

1. The equilibrium (between age classes) is "very often" (to be further specified in the seminar) unstable.
2. A state with the whole population in one age class (Single Year Class ("SYC") dynamics, or "synchronization") is always attracting.

These phenomena were further studied in Tale Solberg's master thesis.

We now have some rather complete results on this. They will be formulated, and some of the proofs will be scetched. In particular, we have pursued cases where some but not all of the age classes are populated. This we have called Multiple Year Class (MYC) dynamics. Our results include all these states. The core problem was to prove that an equilibrium always exists for MYC states. A proof for the latter was found in July this year.