Infinite-dimensional characters are those in which the phenotype of an individual is described by a function, rather than by a finite set of measurements. Examples include growth trajectories, morphological shapes, and norms of reaction. Methods are presented here that allow individual phenotypes, population means, and patterns of variance and covariance to be quantified for infinite-dimensional characters. A quantitative-genetic model is developed, and the recursion equation for the evolution of the population mean phenotype of an infinite-dimensional character is derived. The infinite-dimensional method offers three advantages over conventional finite-dimensional methods when applied to this kind of trait: (1) it describes the trait at all points rather than at a finite number of landmarks, (2) it eliminates errors in predicting the evolutionary response to selection made by conventional methods because they neglect the effects of selection on some parts of the trait, and (3) it estimates parameters of interest more efficiently.

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SUMMARY Suppose that Yi = X'iβ + f(ti) + εi, 1 ≤ i ≤ n, where β, f, and εi are unknown, but the m‐th derivative of f is square integrable. Estimates of β and f are given which minimize the sum of the residual sum of squares and a roughness penalty. It is shown that these estimates are Bayes under a diffuse prior on β and f, and that, under mild conditions on Xi, ti, εi, and the roughness penalty, the estimate of β is consistent and asymptotically normal.

Why do females of many species mate with males on the basis of traits apparently detrimental to male survival? The answer may lie in the fact that these male traits are correlated with male condition. We consider the argument that high male condition directly benefits female fecundity and/or viability (e.g. through lower transmission of parasites, improved control of resources, or better paternal care). Using a quantitative genetic model we show how female preferences for male traits that indicate condition can evolve, even if the male traits themselves have deleterious effects on both the male and the female's fecundity. So-called 'arbitrary preferences' can spread in this way because male traits subject to sexual selection are often under additional selection to become correlated with condition. At equilibrium the positive effects of male condition on a female's fecundity and the negative effects of the male trait on her fecundity are balanced and the female preference is under stabilizing selection. The male trait will often be correlated with viability, but not with fecundity, even though the preference evolved as a rrsult of differences in male fecundity. The mean fecundity of females is not maximized, and can steadily decline as the male trait and female preference evolve. If the male trait has no direct deleterious effects on female fecundity, as may happen in species with no paternal care, female preferences are under continuous directional selection to increase. ADDITIONAL KEY WORDS:-Fecundity selection --handicapsmean fitnessfemale preferencesconditionpaternal carequantitative geneticstheoretical modelsbirds.

Vision is a sensory modality of fundamental importance for many animals, aiding in foraging, detection of predators and mate choice. Adaptation to local ambient light conditions is thought to be commonplace, and a match between spectral sensitivity and light spectrum is predicted. We use opsin gene expression to test for local adaptation and matching of spectral sensitivity in multiple independent lake populations of threespine stickleback populations derived since the last ice age from an ancestral marine form. We show that sensitivity across the visual spectrum is shifted repeatedly towards longer wavelengths in freshwater compared with the ancestral marine form. Laboratory rearing suggests that this shift is largely genetically based. Using a new metric, we found that the magnitude of shift in spectral sensitivity in each population corresponds strongly to the transition in the availability of different wavelengths of light between the marine and lake environments. We also found evidence of local adaptation by sympatric benthic and limnetic ecotypes to different light environments within lakes. Our findings indicate rapid parallel evolution of the visual system to altered light conditions. The changes have not, however, yielded a close matching of spectrum-wide sensitivity to wavelength availability, for reasons we discuss.