  We study how correlations in the random fitness assignment may affect the
structure of fitness landscapes. We consider three classes of fitness models.
The first is a continuous phenotype space in which individuals are
characterized by a large number of continuously varying traits such as size,
weight, color, or concentrations of gene products which directly affect
fitness. The second is a simple model that explicitly describes
genotype-to-phenotype and phenotype-to-fitness maps allowing for neutrality at
both phenotype and fitness levels and resulting in a fitness landscape with
tunable correlation length. The third is a class of models in which particular
combinations of alleles or values of phenotypic characters are "incompatible"
in the sense that the resulting genotypes or phenotypes have reduced (or zero)
fitness. This class of models can be viewed as a generalization of the
canonical Bateson-Dobzhansky-Muller model of speciation. We also demonstrate
that the discrete NK model shares some signature properties of models with high
correlations. Throughout the paper, our focus is on the percolation threshold,
on the number, size and structure of connected clusters, and on the number of
viable genotypes.
