In our model, we determine the trait value of an individual using two components. The first component is associated with the direct additive genetic value for the trait, which we term as “the baseline value.” The second component involves the IGEs through the interaction with the trait value of the social partner, mediated through the interaction coefficient, ψ. We call this second component “the indirect value.” Furthermore, although recent theoretical studies have investigated the coevolutionary dynamics of phenotypes that are involved in social interactions between related individuals (Kölliker et al. 2010; McGlothlin et al. 2010), we follow the original framework of Moore et al. (1997) and focus our analyses on the case where social partners are not related to one another.
We use our theoretical framework to focus on three biological scenarios that involve interacting phenotypes, following Moore et al. (1997).
CASE 1—INTERACTIONS WITH NONRECIPROCAL EFFECTS BETWEEN TWO TRAITS
First, we consider the interaction between two different traits in two individuals, where one trait is partly determined by the value of the other trait (Fig. 1A). For example, in many species, a female’s investment into the offspring, such as the allocation of resources into eggs or the amount of parental care, depends on the quality or attractiveness of the male the female has mated with (e.g., Burley 1986; De Lope and Møller 1993; Kolm 2001). The male trait, here the quality or attractiveness, however, is not influenced by the female investment into the offspring. Consequently, there is a nonreciprocal interaction between the female trait and the male trait.
We model this biological scenario by considering two traits with values z1 and z2, where z1 is partly determined by z2 through the interaction coefficient ψ12. Accordingly, ψ12 is a trait of the focal individual that determines its phenotypic response to the trait of its social partner. In all equations, we use a prime (′) to denote the trait expressed in a social partner. We write the trait value z1 as:
In equation (1), we separate the value of trait z1 into two components. The first component, which we call the “baseline value”z1b, gives the part of the trait value that is independent of the phenotype of the social partner. Accordingly, the baseline value corresponds to the trait value of the focal individual in the absence of any information about the trait value of its social partner. The second component (i.e., ), which we call the “indirect value”, gives the part of the trait value that is determined by the phenotype of the social partner (). We further assume that these two components of the phenotype are under separate selection gradients, and , respectively. Considering two separate selection gradients allows us to examine biological cases where the two components of the phenotype are under differential, and potentially opposing, forces of selection. For example, in species where individuals adjust their parental effort according to the effort of their social partners (Burley 1986; Gowaty 2008), changes in biological conditions such as the degree of information about the offspring condition (Johnstone and Hinde 2006; also see “Discussion”) or relative costs and benefits of phenotypic plasticity (e.g., DeWitt et al. 1998) could favor a shift in the degree or direction of an individual’s plastic response to its social partner, but not in its baseline effort.
The baseline value for trait 1 (z1b) is composed of an additive genetic value, a1b, and a general environmental effect, e1b:
Because we are interested in modeling the interaction coefficient ψ12 as an evolving trait, we decompose the value of ψ12 into its additive genetic and environmental components:
On the other hand, in this first case, the value for trait 2 is not influenced by the value of trait 1 (i.e., ψ21= 0), and is only determined by its baseline value, z2b, which is composed of the additive genetic value, a2b, and the environmental effect, e2b:
Using equations (2 through 4), mean values for z1b, z2, and ψ12 are found as:
(5) (6) (7)
where environmental effects are assumed to be random with a mean of zero (Moore et al. 1997). Using equation (1), the mean of z1 is found as:
where we assume that social partners are not related and therefore .
Following Moore et al. (1997) and making standard quantitative genetic assumptions (Falconer and Mackay 1996), the breeding values for the baseline value of trait 1, z1b, the interaction coeefficient, ψ12, and value of trait 2, z2, are:
(9) (10) (11)
According to Price’s theorem, the change in mean value of a trait z is equal to the covariance between the breeding value and the fitness:
Following Wolf et al. (1999), we give the relative fitness, w, as a regression equation of the phenotype of an individual:
where α is the intercept, ε is the error term, and β1b, , and β2 are the selection gradients for z1b, ψ12, and z2, respectively.
Using equations (12) and (13), the change in the mean baseline value of trait 1, , per generation is:
where G11, G12, and G1ψ are respective genetic variances and covariances.
Similarly, the change in the mean values of traits z2 and ψ12 is found as:
where G22, G12, G1ψ, G2ψ, and Gψψ are respective genetic variances and covariances.
Using equation (8), the mean phenotypic value of z1 in the next generation, denoted as , can be written as:
It is important to note that equation (17) has two new terms (i.e., and ) in addition to the corresponding equation in Moore et al. (1997), which reflect the effect of the evolution of the interaction coefficient ψ on evolutionary dynamics. Subtracting (8) from (17), we find the per-generation change in the mean phenotypic value of the trait z1, ,
which can be written in terms of variances, covariances, and selection gradients using equations (14) through (16).
Finally, because the value of trait 2, z2, does not depend on the phenotype of the social partner, the per-generation change in its mean value is given simply by .
CASE 2—INTERACTIONS BETWEEN TWO DIFFERENT TRAITS WITH RECIPROCAL EFFECTS
Second, we consider the case with two different traits in two interacting individuals, where the value of a trait of an individual is affected by the value of the other trait in its social partner (Fig. 1B). For example, in some species with male parental care, such as sand gobies, females prefer to mate with males who exhibit greater care for previous clutches of eggs (Lindström et al. 2006). In such a system, the initial investment in eggs might depend on whether a female mates with a more or less-preferred male (e.g., Burley 1986; De Lope and Møller 1993; Kolm 2001), in this case with a male that provides more or less care. In turn, males might adjust the amount of care that they provide depending on the level of female investment, using cues such as clutch size (e.g., Sargent 1988; Ridgway 1989; Lindström 1998; Karino and Arai 2006) or egg size (e.g., Nussbaum and Schultz 1989). Consequently, there would be a reciprocal interaction between the amount of care provided by males and the amount of resources invested into eggs by females.
Adapting our theoretical framework to this biological scenario, the value of the first trait, z1, is given as:
Similarly, the value of the second trait, z2, is:
Incorporating (20) into (19), the value for the first trait is found as:
Similarly, the value of the second trait is found as:
Because we consider the interaction coefficients ψ12 and ψ21 as evolvable traits, it is mathematically complex to derive an analytical solution for the means of z1 and z2. Therefore, we approximate the values of and by simulating a population with a large number of individuals, calculating the trait values for each individual, and finding the mean values of z1 and z2 in this population. In these analyses, we use given values of , , , , and respective standard deviations to simulate values of z1b, z2b, ψ12, and ψ21 for each individual.
Similarly, we calculate the mean trait values in the next generation, and , by simulating a large population of individuals, but using new values for z1b, z2b, ψ12, and ψ21, such that:
(23) (24) (25) (26)
where , , , are found as in equations (14–16).
CASE 3—INTERACTIONS WITH RECIPROCAL EFFECTS ON A SINGLE TRAIT
Here, we consider a single trait whose value is partly determined by the value of the same trait in the social partner (Fig. 1C). For example, in many species that exhibit biparental care, the amount of care provided by each parent is affected by the the amount of care provided by the other parent (reviewed in Johnstone and Hinde 2006), resulting in a reciprocal interaction between the values of the same trait (i.e., parental care) in two social partners.
Under this biological scenario, the value of the trait, z1, in the focal individual is written as:
Using the procedure we described above in Case 2, we approximate numerically.
ANALYSES OF THE MODEL
In all analyses, we assumed that baseline trait values are under positive directional selection (i.e., and ) and that starting values for interaction coefficients are positive (i.e., , , ), such that trait values are expected to increase when interaction coefficients are evolutionarily fixed. We implemented these conditions into our model by setting parameter values and . Assigning different positive starting values for interaction coefficients or changing the strength of the positive directional selection on baseline values did not alter the main conclusions that arise from our model. We used this modeling framework to investigate whether the evolution of ψ influences predictions about the rate and direction of evolutionary change in traits. To this end, we first analyzed the models under the assumption that interaction coefficients do not evolve. Accordingly, we set corresponding parameters for the per-generation change in mean interaction coefficients to 0 (i.e., ), and analyzed the coevolutionary dynamics of mean values of traits, and . We then considered interaction coefficients as evolving traits. Here, we assumed that the interaction coefficient is under directional selection to increase (, , ) or to decrease (, , ). We further assumed that the strength of selection on baseline trait values and interaction coefficients does not change over time (but see “Discussion” for cases where this is likely not to be the case). To implement these conditions, we set corresponding parameters for the per-generation change in mean interaction coefficients or , or , and or . Even though the magnitude of values of these parameters relative to the per-generation change in baseline trait values affected the degree of change in the coevolutionary dynamics, main conclusions from our analyses were not affected. Finally, in Cases 2 and 3, we estimated mean trait values by simulating 10,000 individuals, calculating trait values, and repeating this procedure 10,000 times. In all simulations, we used mean baseline values for traits (i.e., and ), mean interaction coefficients (i.e., , and ) and corresponding standard deviations of 1.0 and 0.05 for baseline values and interaction coefficients, respectively. Simulating more individuals, performing a greater number of iterations, or using a different standard deviation for mean baseline values and interaction coefficients did not affect significantly the results of our analyses.