reported by Hans Metz.

This minisymposium was on the occasion of Sido Mylius' Ph.D. defense, May 26, 1999, at the van der Klaauw Laboratory in Leiden, organised in cooperation with the Netherlands Society for Theoretical Biology (NVTB), and the Institute for Evolutionary and Ecological Sciences (EEW) in Leiden.

The first speaker, **Regis Ferriere** (Ecole Normale Superieure de
Paris) presented some results obtained by him and Ulf Dieckmann on the
simultaneous evolution of cooperativity and dispersal, using the pair
approximation methods developed by David Rand and Minus van Baalen
(see e.g. the chapter by David Rand in J. MacGlade (1999) *
Advanced ecological Theory*, Blackwell Science, and the chapter by
Minus van Baalen in U. Dieckmann, R. Law & J.A.J. Metz (1999) *
The Geometry of Ecological Interactions: Simplifying Spatial
Complexity*, Cambridge University Press). The main outcome was that
if only altruism evolves, it moves to some positive level which
decreases (eventually to zero) with increasing dispersal rate. If both
parameters are allowed to evolve altruism initially increases (or
decreases depending on the initial condition), to the value set by the
current dispersal rate, whereupon the dispersal rate starts to
increase and the the altruism to decrease on a much slower time scale.
The difference in time scales reflects the differences on selective
pressure which in the case of the dispersal rate is mainly indirect.
For other values of the parameters there also was a possibility for
evolutionary branching (characterized by an inequality reminiscent of
Hamilton's rule!) which still needs further exploration.

The second speaker, **Sido Mylius** (sectie Populatiebiologie, UvA),
explained how the invasion by a mutant may result in a jump between
population-dynamical attractors resulting in the invader being ousted
again, for which he coined the phrase: The Resident Strikes Back. The
most intriguing case occurred in a model originally devised to study
the evolution of the return to the spawning grounds of Pacific salmon.
Here a population of individuals reproducing in their third year may
be invaded by individuals reproducing in their fourth year even when
the product of the incurred mortality over the years and the basic
fertility of the latter is smaller, and the competition, assumed to
occur only upstream, is fully symmetrical. The point is that the
3-year salmon may oscillate in density in such a manner that 4-year
salmon invading at the right phase of the oscillation sample a
majority of low-density years, where the 3-year salmon always sample
equal proportions of good and bad years. However, the introduction of
the 4-year salmon produces a phase shift in the oscillation such that
from then on the 4-year salmon actually sample a majority of bad
years. In other words, the 4-year invader seems to dig its own grave
and the 3-year resident strategy is invasible, yet invincible. He
further analysed this problem using a toy-model of 1-year and 2-year
salmon that he could analyse in great detail. This way he could
analyse the transition to the resident-strikes-back scenario by way of
a heteroclinic bifurcation.

The third speaker, **Karl Sigmund** (Institut
für Mathematik, Universität Wien), started by classifying two
strategy games on the basis of the relative sizes of the pay-offs, in
a small number of qualitative equivalence classes. Within this
framework he then considered repeated games and strategies with
particular simple sorts of memories, for which the strategy for the
next move is set by the outcomes of the previous move relative to some
aspiration level, plus some updating rule for the aspiration level,
also depending on the outcome of the moves and the present aspiration
level. First he showed results for the case of fixed aspiration
levels, with an emphasis on when such aspiration levels can lead to
good outcomes when once in a while, but only rarely, a player
mistakenly makes a wrong move. As a next step he considered the class
of convex update rules, where the new aspiration level is set by a
convex combination of the present aspiration level and a perceived
pay-off. He only had full results for the two extreme cases, to wit a
full switch to the received pay-off (or of the pay-off received by the
opponent) or a small adaptation. In the latter case the time-evolution
of the system can be approximated by a differential equation (or
rather a differential inclusion) with a piecewise constant right hand
side. In the former case the problem can be analyzed by combinatorial
and Markov chain methods. One intriguing outcome was that envy, i.e.,
set your aspiration level equal to what your opponent got, can
actually lead to a rather good overall pay-off.

The last speaker, **Odo Diekmann** (Mathematisch Instituut, UU),
discussed the case of the missing year classes: when can in a
semelparous organism maturating at age exactly *k*, a population
dynamics be such that the dynamics with only one year class present is
uninvadable by other year classes. To this end he further analysed a
model first conceived by Michael Bulmer in order to analyse the
periodic cicada problem. The main feature of the model is that all
interactions take place through a single variable *I*, a weighted (and
gauged to include some basic, environment independent, survival
probabilities) sum of the population sizes in all year classes. In
good Ricker tradition this variable then feeds through in an
exponential manner to the survival of the year classes (and/or
multiplicatively to the reproduction), with a different sensitivity
for each year class, like , with *g* the, year-class specific,
sensitivity. For the simplest case of *k*=2 he could fully analyze this
model, provided the basic reproduction ratio is sufficiently small so
that no complicated dynamics occurs. It turns out that there was a
nice complete dichotomy where there exists either an interior
equilibrium with both year classes present, or the boundary
equilibrium with only one year class present is uninvadable. The
latter case i.a. occurred when the two weights decreased with age and
the sensitivities increased with age. The full necessary and
sufficient conditions were a little more complicated, but could be
written as explicit inequalities in the parameters. The weaker result
that the fulfilment of two ordering rules in opposite direction for
the weights and sensitivities precludes an invasion by the missing
year classes holds true in general for arbitrary *k*.

Tue Aug 10 12:25:42 MET DST 1999