Combining the Price Equation with our previous approach, by substituting eqn 4 into eqn 1, gives an accounting of how genotype dynamics (Cov(m,z)), phenotype dynamics unrelated to genetic change (E[Δz]) and environmental dynamics k(t) interactively affect the response variable X:
We will refer to eqn 5 and eqn 6, which separate change in X into components caused by genotype, phenotype and environmental change, as the GPE equation. The first term on the right-hand side of both equations is the effect of gene frequency change on X; the last term on the right-hand side is the direct effect of k on X. The interpretation of the middle term, involving transmission bias, depends on the situation. Transmission bias that occurs because k affects trait expression would be an ‘effect of ecology’. If transmission bias is strictly the result of endogenous dynamics (e.g. changes in age structure), or imposed on individuals by an external factor (disease, partial predation, etc.), it might instead be regarded as a third distinct factor. Purely random transmission bias is possible in small populations.
Although eqn 5 and eqn 6 are mathematically equivalent, they differ in an important way. Equation 5 is prospective, predicting how X will change by combining the processes driving change in z;eqn 6 is retrospective, with the trait dynamics dz/dt a ‘given’ and the selection-driven component computed by subtracting off the transmission bias. As a result, eqn 6 can be used without modelling selection, and without data on how trait values correlated with realized fitness components (of the sort used by Coulson & Tuljapurkar 2008 and Ozgul et al. 2009). This is especially important when the response X is affected by several evolving traits (possibly in several different species). The Price Equation still applies to each trait separately, because it does not require any assumptions about the cause of fitness differentials. With multiple traits the selection and transmission bias terms in eqn 5 and eqn 6 are just replaced by sums of those terms over all traits, as in Hairston et al. (2005), eqn 3); the same is true if multiple environmental factors affect X. But prospective analysis with eqn 5 has the complication that calculation of Cov(w,z) has to include correlated selection, e.g. a trait may correlate with fitness only because it is genetically correlated with a functionally unrelated trait that is undergoing selection. For these reasons, eqn 6 will usually be preferable for analysing data on trait evolution and its effects.
However, eqn 5 can be used for forecasting if Cov(w,z) can be calculated from a mechanistic selection model, which may be useful for integrating evolutionary and ecological components of predicted responses to environmental change (e.g. prospective versions of the retrospective analyses in Collins & Gardner 2009). Equation 5 also provides one way of incorporating the dynamics of species- or genotype-specific trait values into trait-based approaches to community assembly and function (e.g. Shipley 2009, 2010; Webb et al. 2010).
The rest of this paper focuses on demonstrating how eqn 6 can be applied to real data (Table 1). We concentrate on estimating transmission bias, because we have previously demonstrated how the other terms can be estimated from empirical data (Hairston et al. 2005). We focus on cases where transmission bias occurs due to phenotypic plasticity or demographic processes. Under the PMA (Fig. 2), the transmission bias equals the rate of change in trait mean for a typical individual, whose trait value equals the population mean. We also discuss two situations in which eqn 6 must be modified: first, if data are only available on the net change over an extended period of time, rather than on short-term rates of change; second, if the trait’s effect on X is indirect and mediated through effects on the dynamics of other components of the community or ecosystem.
Example: rapid evolution of mixis in rotifers
We begin with a simple example to illustrate how the components of eqn 6 can be estimated from data when phenotypic plasticity causes transmission bias. Fussmann et al. (2003) studied how the dynamics of a laboratory population of the rotifer, Brachionus calyciflorus, were affected by rapid evolution of the propensity for sexual (‘mictic’) vs. asexual (‘amictic’) reproduction. Brachionus is cyclically parthenogenetic. Amictic females produce diploid daughters by parthenogenesis. Sexual reproduction arises when some of those daughters develop into mictic females, which produce haploid eggs. Unfertilized eggs develop into males, while fertilized diploid eggs develop into female ‘diapausing eggs’ that hatch to become amictic females, initiating another period of asexual reproduction.
The trait z is the fraction of mictic daughters produced by an amictic female, which is subject to selection and also environmentally cued, typically by high rotifer population density (Gilbert 1963). Fussmann et al. (2003) found an approximately linear response, z = aR, with total rotifer abundance R (individuals mL−1). The slope parameter a describes the strength of response to the environmental cue. Selection against mixis occurs because turbulence in the chemostat prevents mating, so mictic females only produce male offspring, which then leave no progeny. Fussmann et al. (2003) showed that the selection against mixis caused the mean value of a to decrease over time.
For the response variable X we choose the per-capita rate of amictic offspring production by amictic females; this is the birth rate that determines rotifer population growth, because mictic daughters produce only males which have zero fitness. The ecological variable k affecting X is C, the abundance of the rotifers’ algal prey. We have X(z, C) = (1 − z)F(C), where F is the per-capita fecundity of amictic females as a function of C.
Trait values were not directly observable because mictic and amictic females are morphologically indistinguishable (Fussmann et al. 2003). Instead, trait dynamics were inferred by fitting a differential equation model for the dynamics of a and of the algal and rotifer abundances to the available data: the total abundance of algae and rotifers, and the numbers of rotifers carrying mictic vs. amictic eggs. Combining the fitted a(t) and rotifer fecundity function F, with the experimental data on R and C (Fig. 1, left column), gives estimates for z(t) = a(t)R(t) and X(t) = (1 − z(t))F(C(t)).
Next we compute the transmission bias E[Δz], which is the rate at which the trait mean would change if genotype frequencies were held constant. With z = aR and genotype determining a, we have . The contribution of evolution to the change in X is therefore
In this case the transmission bias is plasticity driven by an ecological factor, the change in rotifer abundance. Therefore, eqn 7 is the only evolutionary contribution to the change in X, and the ecological contribution can be computed as dX/dt minus the value of eqn 7. The response and the transmission bias are both linear functions of z (for all else held constant), so the PMA holds.
Figure 3 shows the population data and resulting estimates of evolutionary and ecological contributions to change in X for the four experimental replicates, using the data on changes in C and R, the inferred changes in a, and the fitted fecundity F. Even though the experimental design created very rapid ecological dynamics following rotifer introduction, the contribution of evolution to changes in rotifer per-capita fecundity was roughly 25% as large as the effects of the ecological dynamics (EVO : ECO ≈ 0.25). The turbulence driving selection against mixis is always present, but the selection is weak when rotifer density is very low (because the trait is not expressed then), or when food is too scarce for much reproduction to occur. The contribution of evolution was therefore small at the start of each replicate because rotifer densities were low, larger while rotifers reproduced rapidly and reached high density, and small again when food and rotifer densities declined.
Figure 3. Comparison of evolutionary and ecological contributions in the experiments of Fussmann et al. (2003) on predator–prey dynamics in a chemostat system. Panels in the left column show the abundance of the rotifer, Brachionus calyciflorus (females mL−1, filled red circles), and its algal prey, Chlorella vulgaris (106 cells mL−1, open green circles); the dashed curve and symbols are the experimental data, and the solid curves show the data smoothed by local third-order polynomial regression (R package locpol, using plug-in bandwidth selection to choose the amount of smoothing). The solid purple curve without symbols is the computed value of the trait z = aR, the fraction of mictic daughters, and the solid black curve without symbols is the computed value of the response variable X, the per-capita production rate of amictic daughters (amictic females day−1) by amictic breeding rotifers. The z curve was computed from the smoothed population data and the fitted values of the mixis propensity a, and X was computed from these plus the fitted rotifer fecundity function (Fussmann et al. 2003). Panels in the right column show the estimated evolutionary (blue, solid) and ecological (black, heavy dashed) contributions to dX/dt as described in the text; the derivatives involved in these terms were computed by numerically differentiating the smoothed population data. The time range in these panels omits the start and end of the experiment, when the contribution of evolution is necessarily zero because the trait is not expressed at very low rotifer (Rot) or algal (Alg) densities. The figure captions on the right-column panels give the ratio between the integrated absolute values of the ecological and evolutionary components over the time period plotted in those panels. Values in parentheses are the ratio between ‘evolutionary’ and ‘ecological’ components evaluated using equation (1) instead of equation (6), so that the effects of ecological change mediated by phenotypic plasticity are misclassified, as discussed in the text.
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Plasticity is central to this example, because mixis is a plastic response and evolution results from genetic variation in the degree of plasticity. If the importance of evolution were estimated by our previous approach which ignored trait plasticity, the results would be very different. Using eqn 1 instead of eqn 6, the ‘evolutionary’ contribution (first term on the right-hand side of eqn 1) is much larger than the ‘ecological’ contribution (Fig. 3, right-hand column headings – EVO : ECO values in parentheses). The reason for this difference can be seen by writing X as a function of a, R, and C:
The first two terms in the last expression sum to , so eqn 1 puts both terms in the ‘evolutionary’ contribution. Using eqn 6, the evolutionary contribution consists (correctly) of only the first term, while the second, representing plasticity, goes into the ecological contribution. In all four experimental replicates, the second term in eqn 8 was the largest by far, so shifting it between categories has an enormous impact. In addition, the correct calculation using eqn 6 produces consistent results across experiments for the relative importance of evolution [EVO : ECO = 0.25 ± 0.04(SE)], as expected if evolutionary processes are similar across replicates, whereas the results using eqn 1 varied nearly threefold, suggesting (incorrectly) that replicates followed very different evolutionary paths.
The role of evolution in all replicates was to partially counteract the maladaptive plastic response to increased rotifer density, an increase in mixis. Although the plastic response was dominant (ECO > EVO), in all replicates the evolution of a reduced propensity to mixis was fast and large enough to temporarily reverse the decline in rotifer numbers as their food supply was exhausted.