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.