January 24, 1997, Leiden

The program explored the conceptual connections between economic theory and adaptive dynamics and evolutionary game theory as perceived by biologists, as well as some frontiers of the theory of adaptive dynamics as it is presently developing.

The opening talk by Peter Hammerstein (Berlin) discussed the problem of equilibrium selection in game theory. The classical Nash equilibrium concept has the problem that there are very many Nash equilibria, i.e. strategies such that if everybody plays that strategy it wouldn't pay a single individual to start playing an alternative strategy. Many of these are very counterintuitive. For instance, in a game which is played in a number of subsequent moves, there may exist Nash strategies of the overall game which are based on a threat. However, it may be that once the opponent has made the move that according to the threat is to be punished, their is no gain in effecting the threat. This would take away the threat. Strategies such that there reduction to a subgame are still Nash are called subgame-perfect. Removing the subgame-imperfect strategies reduces the set of allowed equilibrium strategies. Other idea's are the ``trembling hand'' in which only strategies are allowed which stay Nash when infrequent mistakes occur in the chosen action, or the idea that we should take into account the occurrence of mistakes in the perception of one's opponent moves. The problem is how all these ideas stand up against the context of the game. For instance effecting threats may actually be functional if a game is played repetitively or is observed by future opponents. Biology provides at least some context for the various possible assumptions. But we are a long way from a good overview yet.

Karl Sigmund (Vienna) gave a survey of the ways in which state-space representations can be constructed and employed to represent strategies with different sorts of memories for repeated games and explained the nature of the results obtained so far for various families of games with the same two-letter alphabet of possible pure strategies available per player and round, where the simultaneous moves of the two players were characterized by different inequalities among the pay-offs for the various combinations of moves.

Richard Law (York) tackled the interrelation of the social sciences and evolutionary biology from still another angle in a novel analysis of the ``tragedy of the commons''. Apart from the further mathematical elaboration his analysis of the problem differed in three ways from previous analyses. First of all he considered a renewable resource, secondly he kept the number of exploiters of that common resource finite, and thirdly he considered a scheme in which the exploiters changed their behaviour on the basis of the pay-off in the previous round in a manner which led to equations of the same form as the ``canonical equations of adaptive dynamics'' which also formed one of the main subjects of Ulf Dieckmann's talk. The outcome was that under mild conditions a much smaller level of overexploitation of the resource occurs than the tragical amount commonly predicted. The last two talks concentrated on the central concepts of adaptive dynamics per se.

Geza Meszena (Budapest) considered some extensions of the ideas of MacArthur and Tilman on resource based competition to an evolutionary context. In those cases where the environment can be characterized by a finite number of variables which monotonically influence fitness and which are monotonically influenced by the sizes of various evolving populations there is a duality between adaptive dynamics as a random walk in trait space and as a random walk in environment space. In environment space a new set of geometrical arguments come to the fore that can be exploited to derive new insights about the geometry of adaptive trait substitutions, thus forging a closer link between the ecological interpretation and the more abstracted viewpoint of adaptive dynamics. The final talk by Ulf Dieckmann (Laxenburg) concentrated on the derivation through formal limit arguments of adaptive dynamics as a continuous motion in trait space from individual-based models for the underlying population dynamics. This derivation went in two steps.

- The starting point is a population dynamics formalized as a number of coupled birth and death processes. Individuals in each population reproduce and die with birth and death rates which depend on the overall community composition. The individuals in each process are characterized by a trait that is inherited and that determines their role in the community interaction. These traits are inherited almost faithfully: most of the time a daughter is of the same type as her mother, but sometimes the trait value of the daughter is a little displaced.
- The assumptions of large population sizes plus mutation limitation give the population dynamics the possibility to deterministically equilibrate in between the arrivals of new mutants. Mutants then arrive as single individuals and therefore have to pass through a stochastic boundary layer where demographic noise sets the scene. This stochastic process results in either extinction or an entering of the deterministic realm. The extinction probability can be calculated from a branching process approximation. If we look on a large time scale we see a sequence of instantaneous mutant substitutions, inducing small jumps in the (equilibrium) population density.
- If the mutational steps are small the resulting random walk in trait space can be approximated by a differential equation known as ``the canonical equation of adaptive dynamics''. The resulting chain of models, related to each other through limit arguments, provides a nice connection between various previously disconnected approaches to phenotypic evolutionary change.

One further point that may be worth noting is that the meeting also led
to a number of somewhat heated discussions, in particular about the
scenarios underlying adaptive branching. The adaptive dynamics
community has a number of rather specific opinions about this topic
which at times give rise to considerable disagreement. As it happens
there also are a number of groups that work on these issues from a
more empirical perspective at the host institution, the Leiden EEW. In
the wake of the mini-symposium a series of discussions have been, and
are still, going on at the EEW. Through this link the mini-symposium
by now has contributed to at least one paper:

F. Galis & J.A.J. Metz (1998) *Why
are there so many cichlid species? On the interplay of speciation and
adaptive radiation*. TREE 13: 1-2.

Marek Kimmel was in Paris en route to Pau and this provided a good opportunity to suggest a small detour to the Netherlands for a visit to Odo Diekmann, Utrecht and Hans Heesterbeek, Centre for Biometry Wageningen. His area current interest is to extract information on historical changes in the size of populations from data on present-day DNA polymorphism in that population (e.g. looking at DNA repeat sequences such as microsatelite data). The theory is based on branching processes.

In the renewed spirit of our NLS-programme in its final phase, we tried to broaden the group of biologists that could benefit from the activities of the programme. On Monday morning we could arrange a meeting between Marek Kimmel and Pim van Hooft (Department of Terrestrial Ecology and Nature Management of the Agricultural University Wageningen), who represents a group that is interested in problems very similar to Marek's. The group studies microsatelites in African buffalo, to elucidate population changes during a major epidemic of rinderpest in the nineteenth century. Both Marek and Pim judged this discussion as valuable and illuminating for both parties.

In the afternoon there was a programme of four well-attended lectures: Marek Kimmel, Johannes Mueller, Sido Mylius and Hans Heesterbeek. The audience for Marek's lecture consisted of 25 people, of whom 15 were still left at the end of the afternoon. We had the feeling that the lectures were much appreciated. In summary, this extremely short visit was very worthwhile.

Department of Statistics Rice University Houston, TX, USA

Credits: Michael Bamshad, Ranajit Chakraborty, Leslea J.Davison,
Ranjan Deka, Lynn B.Jorde, J.Patrick King, Andrzej Polanski, W.Scott Watkins

It has been proposed that modern humans evolved from a small ancestral population, which appeared several hundred thousand years ago in Africa. Descendants of the founder group migrated to Europe and then to Asia, not mixing with the pre-existing local populations (e.g. the Neanderthal men in Europe), but replacing them. Two demographic elements are present in this "out of Africa" hypothesis: numerical growth of the modern humans and their migration into Eurasia.

Did these processes leave an imprint in our DNA? To address this
question, we use the classical Fisher-Wright-Moran model of population
genetics, assuming variable population size and two models of
mutation: the infinite-sites model and the stepwise-mutation model. We
use the coalescence theory, which amounts to tracing the common
ancestors of contemporary genes. We obtain mathematical formulae
expressing the distribution of alleles given the time changes of
population size *N*(*t*) (some of them analogous to the Lyapunov's
equation known from the control theory).

In the framework of the infinite-sites model, estimation of *N*(*t*)
requires using the reverse Laplace transform known to be unstable.
Nevertheless, simulations indicate that the pattern of past population
size change leaves its signature on the pattern of DNA polymorphism.
Application of the theory to the published mitochondrial DNA sequences
indicates that the current mitochondrial DNA sequence variation is not
inconsistent with the logistic growth of the modern human population.

In the framework of the stepwise-mutation model, we demonstrate that population bottleneck followed by growth in size causes an imbalance between allele-size variance and heterozygosity. We analyze a set of data on tetranucleotide repeats which reveals the existence of this imbalance. The pattern of imbalance is consistent with the bottleneck being most ancient in Africans, most recent in Asians and intermediate in Europeans.

These findings can be interpreted as consistent with the "out of Africa" hypothesis, although by no means do they constitute its proof.

We investigate the interplay of population- and adaptive dynamics in the context of the timing of reproduction of Pacific Salmon. Coexistence of different population-dynamical attractors and resonance of life-span relative to population dynamical fluctuations can have a profound effect on invasibility and the resulting evolutionarily stable life-history. Different local attractors of the resident population dynamics can have different invasion properties. Successful invasion in one attractor can be followed by extinction of the former invader, ultimately leading the resident to the same attractor, but phase-shifted, or to another attractor. So a strategy can be ``invasible, yet unbeatable''.

We show that a simple model for parasitic infection of farm animals, incorporating an environmental stage in the life-cycle of the parasite, acquired immunity and management practice can show complicated dynamics. The model incorporates the above aspects in a rudimentary but non-trivial way. The analysis shows that one has to be careful when using models to gauge effects of control measures for systems where these aspects play a major role.

Thu Dec 18 11:35:42 MET 1997