species to maintain their motility as they increase in size during development (Solari, 2005; Solari et al., 2006a).
Tradeoffs among the contributions of cells to the fitness components of the group leads to the “covariance effect,” whereby the fitness of the group, W, is greater than the average fitness of its members, by the magnitude of the covariance among fitness components (Michod, 2006; Michod et al., 2006) as given in Eq. 1.
In Eq. 1, Cov[v, b] < 0 expresses a tradeoff, and The viability and fecundity of cell i (or its contribution to group viability and fecundity) are vi and bi, respectively, and i = 1,… N, where N is group size. We take fitness as the product of viability and fecundity, as is appropriate for organisms with discrete generations such as the volvocines. For groups to obtain the benefit of the covariance effect, cells must vary in their reproductive effort. As already mentioned, under a convex curvature of the tradeoff function, there is an advantage of cells specializing in different fitness components (Fig. 7.3).
Convexity or concavity of tradeoffs between fitness components is a basic issue in life-history theory (Levins, 1968; Schaffer, 1974; Michod, 1978; Reznick, 1985; Stearns, 1992; Benkman, 1993; Carriere and Roff, 1995; Takada and Nakajima, 1996; Benson and Stephens, 1996; Strohm and Linsenmair, 2000; Kisdi, 2001; Sato, 2002; Roff, 2002; Blows et al., 2004; Rueffler et al., 2004). For a convex function v(b) the second derivative is positive, and for a concave function v(b) the second derivative is negative, so if we take a particular point b* and two points equidistant below and above b*, b− and b+, then v(b−) + v(b+) > [<] 2 v(b*). If b is reproduction and v(b) is viability, then convexity of v implies there is an advantage to specializing in the two fitness components. Despite the central relevance of this issue to life-history theory, a recent review (Rueffler et al., 2004) states, “Unfortunately, there is no study known to us which has revealed the details of this curvature for any life-history tradeoff in a specific organism. However, these curvatures are central in life-history theory which indicates a major gap between theory and empirical knowledge.” We have addressed this difficult empirical problem by viewing a convex curve in a piece-wise linear fashion (Fig. 7.3) and quantifying the initial cost of reproduction to motility shown in Fig. 7.3C and as discussed in Cost of Reproduction.
The particular mathematical representation of the covariance effect given in Eq. 1 depends on additivity of fitness effects as described in