6 1. INTRODUCTION
for a quasispecies to emerge and we recover the finite population counterpart of
the error threshold. Moreover, in the regime of very small mutations, we obtain
a lower bound on the population size allowing the emergence of a quasispecies: if
α ln 4/ ln σ then the equilibrium population is totally random, and a quasispecies
can be formed only when α ln 4/ ln σ. Finally, in the limit of very large popu-
lations, we recover an error catastrophe reminiscent of Eigen’s model: if σe−a 1
then the equilibrium population is totally random, and a quasispecies can be formed
only when σe−a 1. These results are supported by computer simulations. The
good news is that, already for small values of , the simulations are very conclusive.
Master
sequence
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Figure 3. Simulation of the equilibrium density of the Master sequence
It is certainly well known that the population dynamics depends on the population
size (see the discussion of Wilke [41]). In a theoretical study [30], Van Nimwe-
gen, Crutchfield and Huynen developed a model for the evolution of populations
on neutral networks and they show that an important parameter is the product of
the population size and the mutation rate. The nature of the dynamics changes
radically depending on whether this product is small or large. Sumedha, Martin
and Peliti [37] analyze further the influence of this parameter. In [39], Van Nimwe-
gen and Crutchfield derived analytical expressions for the waiting times needed to
increase the fitness, starting from a local optimum. Their scaling relations involve
the population size and show the existence of two different barriers, a fitness barrier
and an entropy barrier. Although they pursue a different goal than ours, most of
the heuristic ingredients explained previously are present in their work, and much
more; they observe and discuss also the transition from the quasispecies regime for
large populations to the disordered regime for small populations. The dependence
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