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Project Progress

As noted below, there have been two mathematical publications produced during the reporting period, and these two papers are in different directions.

The first paper sought to address the generic behavior of niche formulation in genotypic models —- in these models, the idea is that the genomes compete on a fitness landscape using competitive exclusion. It has been observed in many such models that niche formation and speciation can occur, even though these features are not built into the model. We were able to show that the formation of such structures is deeply dependent on the symmetries of the effective landscape.

The second paper is a study of a general game-theoretic model for competing species where it is assumed that there are multiple distinct evolutionarily stable strategies (ESS); these are strategies where a population completely consisting of a wild type of organism is robust with respect to a small invasion of a mutant. Since we are considering a finite population, it is always possible that random fluctuations can move a population away from an ESS due to a sequence of “lucky” events —- analogously to a thermally-activated system moving out of a stable potential well. We studied the general case where multiple ESS exist, and were able to give formulas for the switching times between various ESS and the relative probabilities of observing each ESS in equilibrium, all in terms of the parameters governing the abstract strategies of each of the various populations.

These works seem a bit distinct, but in fact the next step is to marry these two perspectives in a more general model. In the first paper, the model lacked a genotype-phenotype map, so that selection pressures worked directly on the genotype. Of course, in life, the pressures are on the phenotype instead; adding this component then gives a model which is a higher-dimensional version of the class of systems studied in the second paper. So in fact the next step is to consider the analysis of this more complex model, where the techniques developed in the second paper should prove to be useful.