An explicit genetic model for ecological character displacement

Ecology, March, 1996 by Michael Doebeli

INTRODUCTION

If the strength of competition between individuals is determined by a quantitative character, selection could lead to a permanent difference in the distributions of the character in two competing species. Such ecological character displacement is an intuitively appealing explanation for phenotypic differences among related sympatric species. However, it is controversial how often observed patterns are due to ecological character displacement, and much theoretical work has been devoted to the study of the conditions under which it can occur (e.g., Bulmer 1974, Crozier 1974, Lawlor and Maynard Smith 1976, Roughgarden 1976, Slatkin 1980, Matessi and Jayakar 1981, Case 1982, Lundberg and Stenseth 1985, Milligan 1985, Rummel and Roughgarden 1985, Taper and Case 1985, Abrams 1986, 1987a, b, 1990, Brown and Vincent 1987, Gotelli and Bossert 1991, Taper and Case 1992a, Vincent et al. 1993). These efforts have been reviewed by Taper and Case (1992a). They argued that, among the models they compared, the most realistic is the quantitative genetic model of Slatkin (1980), because it incorporates population genetics and contains the fewest constraining assumptions. It can be considered as a null-model: the competing species are assumed to have the same carrying capacity curves, competition in and between the species is symmetrical, and there are no constraints on the utilization of the resources. However, this null-model does not yield significant character displacement (Slatkin 1980, Taper and Case 1985). Substantial displacement only results when ecological asymmetries in the carrying capacity curves of the species or in the competition functions are introduced (Slatkin 1980, Milligan 1985, Taper and Case 1992a), or when resource use is constrained, e.g., by constraining the phenotypic variance in the species (Slatkin 1980, Taper and Case 1992a), or by introducing explicit resource dynamics (Taper and Case 1985). Although asymmetries and constraints are common in natural systems (Taper and Case 1992b), analyzing character displacement under symmetrical and unconstrained conditions is also important, for example when studying adaptive radiation from ancestral lineages into otherwise empty phenotype space (e.g., Schluter and McPhail 1993, Schluter 1994). Moreover, studying this scenario with its few assumptions can reveal mechanisms preventing or enhancing character displacement that are otherwise masked by additional assumptions, and may thus give new insights for other scenarios as well. In this paper I show that the null-result for Slatkin's null-model derives from the way the quantitative genetics are modeled, rather than from the lack of asymmetries or constraints. Thus, while character displacement is more likely under such conditions, they are not necessary for displacement to occur.

Slatkin (1980) modeled the genetics by assuming that random mating results in normal character distributions in each generation. This is very common in quantitative genetic theory, but it seems to be too rigid an assumption for studying ecological dynamics (Doebeli 1995b). It obscures important details of the dynamics of single phenotypes. I propose a more flexible quantitative genetic model that keeps track of the frequencies of single phenotypes. This explicit modeling of the genetics leads to character displacement even in the null-model, i.e., without ecological asymmetries and without constraints on the phenotype distributions. Therefore, conclusions from models for character displacement depend not only on the ecological features of the models, but also on the genetic assumptions. In particular, more detailed genetic models make displacement more likely. The reason seems to be that these models describe more subtly how competition shapes character distributions, so that inherent tendencies for divergence manifest themselves more clearly. Thus ecological character displacement might be more common than previously believed.

THE QUANTITATIVE GENETIC MODEL

Following Slatkin (1980), my starting point was the models for the dynamics of populations with discrete generations:

[N.sub.t 1] = [N.sub.t][multiplied by]f([N.sub.t]). (1)

Here [N.sub.t] is the density of the population at time t, and f(N) is the fitness function. Exploitative competition for resources is modeled implicitly by assuming that the fitness depends on the density: the higher N, the lower f(N). Slatkin (1980) used the logistic fitness function

f(N) = 1 r - r/KN. (2)

Here 1 r is the intrinsic growth rate of the population, and K is its equilibrium density, i.e., its carrying capacity. Although often used, the logistic function has the drawback of negative fitness values for high densities. This is not a problem in the homogenous and deterministic setting given by Eq. 2, because such high densities are never attained if the parameter r is chosen appropriately. However, in phenotypically variable populations like those considered below, negative fitness values can occur even for biologically reasonable choices of r if the system does not have stable equilibrium dynamics. This happens because the densities of the single phenotypes in the population add up to the total density that determines the fitness, and this total density can fluctuate to values that are too high. Therefore, for some of the results I used the following alternative for the fitness function: