Abstract. Power-law relationships are among the most well-studied functional relationships in biology. Recently the common practice of fitting power laws using linear regression (LR) on log-transformed data has been criticized, calling into question the conclusions of hundreds of studies. It has been suggested that nonlinear regression (NLR) is preferable, but no rigorous comparison of these two methods has been conducted. Using Monte Carlo simulations, we demonstrate that the error distribution determines which method performs better, with NLR better characterizing data with additive, homoscedastic, normal error and LR better characterizing data with multiplicative, heteroscedastic, lognormal error. Analysis of 471 biological power laws shows that both forms of error occur in nature. While previous analyses based on log-transformation appear to be generally valid, future analyses should choose methods based on a combination of biological plausibility and analysis of the error distribution. We provide detailed guidelines and associated computer code for doing so, including a model averaging approach for cases where the error structure is uncertain.

Interference at the level of fine roots in the field was studied by detailed examination of fine root distribution in small soil patches. To capture roots as they occur in natural three-dimensional soil space, we used a freezing and slicing technique for microscale root mapping. The location of individual roots intersecting a sliced soil core surface was digitized and the identity of shrub and grass roots was established by a chemical technique. Soil patches were created midway between the shrub, Artemisia tridentata, and one of two tussock grasses, Pseudoroegneria spicata or Agropyron desertorum. Some soil patches were enriched with nutrients and others given only deionized water (control); in addition, patches were located between plants of different size combination (large shrubs with small tussock grasses and small shrubs with large tussock grasses). The abundance of shrub and grass roots sharing soil patches and the inter-root distances of individual fine roots were measured. Total average rooting density in patches varied among these different treatment combinations by only a factor of 2, but the proportion of shrub and grass roots in the patches varied sixfold. For the shrub, the species of grass roots sharing the patches had a pronounced influence on shrub root density; shrub roots were more abundant if the patch was shared with Pseudoroegneria roots than if shared with Agropyron roots. The relative size of plants whose roots shared the soil patches also influenced the proportion of shrub and grass roots; larger plants were able to place more roots in the patches than were the smaller plants. In the nutrient-enriched patches, these influences of grass species and size combination were amplified. At the millimeter- to centimeter-scale within patches, shrub and grass roots tended to segregate, i.e., avoid each other, based on nearest-neighbor distances. At this scale, there was no indication that the species-specific interactions were the result of resource competition, since there were no obvious patterns between the proportion of shrub and grass roots of the two species combinations with microsite nutrient concentrations. Other potential mechanisms are discussed. Interference at the fine-root level, and its species-specific character, is likely an influential component of competitive success, but one that is not easily assessed.

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We used ecotypic variation in big sagebrush (Artemisia tridentata) to examine potential trade-offs between inherent growth rate and tolerance or resistance to herbivory. Seeds were obtained from seven geographic populations, and 1,120 seedlings were established in a common garden. In one set of plots, plants were subjected to five treatments: control, regular insecticide spray, moderate browsing, severe browsing, or moderate browsing plus insecticide. Plants in a second set of plots were all untreated, and were used to estimate ambient growth, flower production, and susceptibility to herbivorous insects. In the first growing season, population differences in relative growth rate produced approximately seven-fold variation in mean biomass. Two populations of basin big sagebrush (A. tridentata tridentata) and one population of mountain big sagebrush (A. tridentata vaseyana) grew fastest; those of Wyoming big sagebrush (A. tridentata wyomingensis) showed the slowest growth. Bi-weekly application of insecticide for two growing seasons had no effect on the growth of either browsed or unbrowsed plants. All populations showed compensatory growth (but not overcompensation) in response to browsing, but the degree of compensation was unrelated to inherent growth rate. Similarly, there was no consistent relationship between plant growth rate and flower production in the second growing season. Some insects colonized fast-growing populations more frequently than slow-growing ones, but patterns of insect colonization were species-specific. At the level of geographic populations and subspecies, we found little evidence of a built-in trade-off between inherent growth rate and the ability to tolerate or resist herbivory. Because population ranks for growth rate changed substantially between seasons, attempts to correlate growth and defense characters need to account for differences in the growth trajectories of perennial plants.