A continuous reaction norm or performance curve represents a phenotypic trait of an individual or genotype in which the trait value may vary with some continuous environmental variable. We explore patterns of genetic variation in thermal performance curves of short-term caterpillar growth rate in a population of Pieris rapae. We compare multivariate methods, which treat performance at each test temperature as a distinct trait, with function-valued methods that treat a performance curve as a continuous function. Mean growth rate increased with increasing temperatures from 8 to 35°C, was highest at 35°C, and declined at 40°C. There was substantial and significant variation among full-sib families in their thermal performance curves. Estimates of broad-sense genetic variances and covariances showed that genetic variance in growth rate increased more than 30-fold from low (8–11°C) to high (35–40°C) temperatures, even after differences in mean growth rate across temperatures were removed. Growth rate at 35 and 40°C was negatively correlated genetically, suggesting a genetic trade-off in growth rate at these temperatures; this trade-off may represent either a generalist-specialist trade-off and/or variation in the optimal temperature for growth. The estimated genetic variance-covariance function (G function), the function-valued analog of the variance-covariance matrix (G matrix), was quite bumpy compared with the estimated G matrix; and results of principal component analyses of the G function were difficult to interpret. The use of orthogonal polynomials as the basis functions in current function-valued estimation methods may generate artifacts when the true G function has prominent local features, such as strong negative covariances at nearby temperatures (e.g. at 35 and 40°C); this may be a particular issue for thermal performance curves and other highly nonlinear reaction norms.
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