The Evolutionary Response to Frequency- and Density-Dependent Interactions

For a group of species interacting in a community both demographic and evolutionary changes must be taken into account to understand the structure of the community. One goal of evolutionary theory is the characterization of the effects of natural selection on each species in a community. While there is no guarantee that such a characterization in terms of elementary principles is always possible, it is important o find and examine those cases where it is. Otherwise each community and each species and each type of interaction must be considered as a special case. There are already several theoretical studies that show there are simple evolutionary principles that can be found under various restrictions on the kinds of selection acting in a population. Fisher (1930) and Wright (1931) showed that frequencyand density-independent selection at a single locus would cause the mean fitness of a randomly mating population to approach a local maximum. Subsequent workers (e.g., Kimura 1958 and Crow and Nagylaki 1976) have extended Fisher's "fundamental theorem" of natural selection to more complex cases with multiple alleles. Leon and Charlesworth (1976) have shown a similar result for each of two competing species. In studies of purely density-dependent selection, several workers (MacArthur 1962; Charlesworth 1971, 1973; Anderson 1971; Roughgarden 1971, 1976; Ginzburg 1977) have shown that total population size will be maximized as a function of allele frequency at an equilibrium. Roughgarden (1976) has extended these results to models of interacting species for which there is density dependence within each species. None of these studies considers frequency dependence within a species, and all of the principles obtained are in terms of "extremal properties" of the models. That is, the derivative with respect to allele frequency of some function (mean fitness, total population size, or a constrained population size) is zero at the equilibrium. It is well known and obvious from the algebraic structure of these models that any withinspecies frequency dependence would invalidate the general principles that are obtained. Consequently, studies of frequency-dependent selection (e.g., Lewontin