The allometry between secondary sexual traits and body size is nonlinear among cervids

What is complex allometry?

Complex allometry describes a smooth, curvilinear relationship between logarithmic transformations of a biological variable and a corresponding measure for body size when the observations are displayed on a bivariate graph with linear scaling. The curvature in such a display is commonly captured by fitting a quadratic equation to the distribution; and the quadratic term is typically interpreted, in turn, to mean that the mathematically equivalent equation for describing the arithmetic distribution is a two-parameter power equation with an exponent that changes with body size. A power equation with an exponent that is itself a function of body size is virtually uninterpretable, yet numerous attempts have been made in recent years to incorporate such an exponent into theoretical models for the evolution of form and function in both plants and animals. However, the curvature that is described by a quadratic equation fitted to logarithms usually means that an explicit, non-zero intercept is required in the power equation describing the untransformed distribution - not that the exponent in the power equation varies with body size. Misperceptions that commonly accompany reports of complex allometry can be avoided by using nonlinear regression to examine untransformed data.

Signal detection shapes ornament allometry in functionally convergent Caribbean Anolis and Southeast Asian Draco lizards

Visual ornaments have long been assumed to evolve hyper-allometry as an outcome of sexual selection. Yet growing evidence suggests many sexually selected morphologies can exhibit other scaling patterns with body size, including hypo-allometry. The large conspicuous throat fan, or dewlap, of arboreal Caribbean Anolis lizards was one ornament previously thought to conform to the classical expectation of hyper-allometry. We re-evaluated this classic example alongside a second arboreal group of lizards that has also independently evolved a functionally equivalent dewlap, the Southeast Asian Draco lizards. Across multiple closely related species in both genera, the Anolis and Draco dewlaps were either isometric or had hypo-allometric scaling patterns. In the case of the Anolis dewlap, variation in dewlap allometry was predicted by the distance of conspecifics and the light environment in which the dewlap was typically viewed. Signal efficacy, therefore, appears to have driven the evolution of hypo-allometry in what was originally thought to be a sexually selected ornament with hyper-allometry. Our findings suggest that other elaborate morphological structures used in social communication might similarly exhibit isometric or hypo-allometric scaling patterns because of environmental constraints on signal detection.

A Revision of the Traditional Analysis Method of Allometry to Allow Extension of the Normality-Borne Complexity of Error Structure: Examining the Adequacy of a Normal-Mixture Distribution-Driven Error Term

Huxley’s model of simple allometry provides a parsimonious scheme for examining scaling relationships in scientific research, resource management, and species conservation endeavors. Factors including biological error, analysis method, sample size, and overall data quality can undermine the reliability of a fit of Huxley’s model. Customary amendments enhance the complexity of the power function-conveyed systematic term while keeping the usual normality-borne error structure. The resulting protocols bear multiple-parameter complex allometry forms that could pose interpretative shortcomings and parameter estimation difficulties, and even being empirically pertinent, they could potentially bear overfitting. A subsequent heavy-tailed Q-Q normal spread often remains undetected since the adequacy of a normally distributed error term remains unexplored. Previously, we promoted the advantages of keeping Huxley’s model-driven systematic part while switching to a logistically distributed error term to improve fit quality. Here, we analyzed eelgrass leaf biomass and area data exhibiting a marked size-related heterogeneity, perhaps explaining a lack of systematization at data gathering. Overdispersion precluded adequacy of the logistically adapted protocol, thereby suggesting processing data through a median absolute deviation scheme aimed to remove unduly replicates. Nevertheless, achieving regularity to Huxley’s power function-like trend required the removal of many replicates, thereby questioning the integrity of a data cleaning approach. But, we managed to adapt the complexity of the error term to reliably identify Huxley’s model-like systematic part masked by variability in data. Achieving this relied on an error term conforming to a normal mixture distribution which successfully managed overdispersion in data. Compared to normal-complex allometry and data cleaning composites present arrangement delivered a coherent Q-Q normal mixture spread and a remarkable reproducibility strength of derived proxies. By keeping the analysis within Huxley’s original theory, the present approach enables substantiating nondestructive allometric proxies aimed at eelgrass conservation. The viewpoint endorsed here could also make data cleaning unnecessary.

The morphological allometry of four closely related and coexisting insect species reveals adaptation to the mean and variability of the resource size

The size of organisms may result from various, sometimes antagonistic forces operating on distinct traits, within an evolutionary framework that may also be constraining. Morphological allometry, referring to the way trait size scales with body size, has been shown to reflect ecological adaptation to the mean size of the resource exploited. We examined the allometric relationships between rostrum and body size among four insect (Curculio spp.) specialists of oak acorns. In all four species, weevil females drill a hole with their rostrum prior depositing one or a few eggs inside the seed. The four weevil species, that coexist on the same individual trees, displayed partitioned egg-laying periods in the year, thereby encountering acorns of different size and maturation stage. We found marked differences in the allometric slope among females: species laying eggs late in the season had a steeper slope, leading to increasingly longer rostrum relative to body length, along with the mean size of the growing acorns. Females of the smallest species had the longest oviposition period and also had the steepest slope, which provided them with the most variable rostrum length, thereby matching the variable size of the resource through time. Our work highlights the need to consider not only the average size but also the degree of variability in resource size to understand the adaptive value of allometric relationships.

Can Sex-Specific Metabolic Rates Provide Insight Into Patterns of Metabolic Scaling?

Females and males can exhibit striking differences in body size, relative trait size, physiology and behavior. As a consequence the sexes can have very different rates of whole-body energy use, or converge on similar rates through different physiological mechanisms. Yet many studies that measure the relationship between metabolic rate and body size only pay attention to a single sex (more often males), or do not distinguish between sexes. We present four reasons why explicit attention to energy-use between the sexes can yield insight into the physiological mechanisms that shape broader patterns of metabolic scaling in nature. First, the sexes often differ considerably in their relative investment in reproduction which shapes much of life-history and rates of energy use. Second, males and females share a majority of their genome but may experience different selective pressures. Sex-specific energy profiles can reveal how the energetic needs of individuals are met despite the challenge of within-species genetic constraints. Third, sexual selection often pushes growth and behavior to physiological extremes. Exaggerated sexually selected traits are often most prominent in one sex, can comprise up to 50% of body mass and thus provide opportunities to uncover energetic constraints of trait growth and maintenance. Finally, sex-differences in behavior such as mating-displays, long-distance dispersal and courtship can lead to drastically different energy allocation among the sexes; the physiology to support this behavior can shape patterns of metabolic scaling. The mechanisms underlying metabolic scaling in females, males and hermaphroditic animals can provide opportunities to develop testable predictions that enhance our understanding of energetic scaling patterns in nature.

Spatial variation in antler investment of Apennine red deer

Heterogeneity in resource availability and quality can trigger spatial patterns in the expression of sexually selected traits such as body mass and weaponry. While relationships between habitat features and phenotypic quality are well established at broad geographical scales, information is poor on spatial patterns at finer, intrapopulation scales. We analyzed biometric data collected on 1965 red deer Cervus elaphus males over 20 years from a nonmigratory population living on two sides of a mountainous ridge, with substantial differences in land cover and habitat quality but similar climate and population density. We investigate spatial patterns in (i) body mass, (ii) antler mass, and (iii) antler investment. We also tested for site- and age-specific patterns in allometric relationship between body mass and antler mass. Statistically significant fine-scale spatial variations in body mass, antler mass, and, to a lesser extent, antler allocation matched spatial differences in land cover. All three traits were greater in the northern slope, characterized by higher habitat heterogeneity and greater availability of open habitats, than in the southern slope. Moreover, the allometric relationship between body mass and antler mass differed among age-classes, in a pattern that was consistent between the two mountain slopes. Our results support the occurrence of spatial patterns in the expression of individual attributes also at a fine, intrapopulation scale. Our findings emphasize the role of environmental heterogeneity in shaping spatial variations of key life-history traits, with potential consequences for reproductive success.

Positive allometry of sexually selected traits: Do metabolic maintenance costs play an important role?

Sexual selection drives the evolution of some of the most exaggerated traits in nature. Studies on sexual selection often focus on the size of these traits relative to body size, but few focus on energetic maintenance costs of the tissues that compose them, and the ways in which these costs vary with body size. The relationships between energy use and body size have consequences that may allow large individuals to invest disproportionally more in sexually selected structures, or lead to the reduced per-gram maintenance cost of enlarged structures. Although sexually selected traits can incur energetic maintenance costs, these costs are not universally high; they are dependent on the relative mass and metabolic activity of tissues associated with them. Energetic costs of maintenance may play a pervasive yet little-explored role in shaping the relative scaling of sexually selected traits across diverse taxa. Also see the video abstract here: https://youtu.be/JyuoQIeA33Q.

An effect size statistical framework for investigating sexual dimorphism in non-avian dinosaurs and other extinct taxa

Despite reports of sexual dimorphism in extinct taxa, such claims in non-avian dinosaurs have been rare over the last decade and have often been criticized. Since dimorphism is widespread in sexually reproducing organisms today, under-reporting in the literature might suggest either methodological shortcomings or that this diverse group exhibited highly unusual reproductive biology. Univariate significance testing, especially for bimodality, is ineffective and prone to false negatives. Species recognition and mutual sexual selection hypotheses, therefore, may not be required to explain supposed absence of sexual dimorphism across the grade (a type II error). Instead, multiple lines of evidence support sexual selection and variation of structures consistent with secondary sexual characteristics, strongly suggesting sexual dimorphism in non-avian dinosaurs. We propose a framework for studying sexual dimorphism in fossils, focusing on likely secondary sexual traits and testing against all alternate hypotheses for variation in them using multiple lines of evidence. We use effect size statistics appropriate for low sample sizes, rather than significance testing, to analyse potential divergence of growth curves in traits and constrain estimates for dimorphism magnitude. In many cases, estimates of sexual variation can be reasonably accurate, and further developments in methods to improve sex assignments and account for intrasexual variation (e.g. mixture modelling) will improve accuracy. It is better to compare estimates for the magnitude of and support for dimorphism between datasets than to dichotomously reject or fail to reject monomorphism in a single species, enabling the study of sexual selection across phylogenies and time. We defend our approach with simulated and empirical data, including dinosaur data, showing that even simple approaches can yield fairly accurate estimates of sexual variation in many cases, allowing for comparison of species with high and low support for sexual variation.

Variation in the ontogenetic allometry of horn length in bovids along a body mass continuum

Allometric relationships describe the proportional covariation between morphological, physiological, or life-history traits and the size of the organisms. Evolutionary allometries estimated among species are expected to result from species differences in ontogenetic allometry, but it remains uncertain whether ontogenetic allometric parameters and particularly the ontogenetic slope can evolve. In bovids, the nonlinear evolutionary allometry between horn length and body mass in males suggests systematic changes in ontogenetic allometry with increasing species body mass. To test this hypothesis, we estimated ontogenetic allometry between horn length and body mass in males and females of 19 bovid species ranging from ca. 5 to 700 kg. Ontogenetic allometry changed systematically with species body mass from steep ontogenetic allometries over a short period of horn growth in small species to shallow allometry with the growth period of horns matching the period of body mass increase in the largest species. Intermediate species displayed steep allometry over long period of horn growth. Females tended to display shallower ontogenetic allometry with longer horn growth compared to males, but these differences were weak and highly variable. These findings show that ontogenetic allometric slope evolved across species possibly as a response to size-related changes in the selection pressures acting on horn length and body mass.

Measuring Complex Morphological Traits with 3D Photogrammetry: A Case Study with Deer Antlers

The increasing availability of 3D-imaging technology provides new opportunities for measuring morphology. Photogrammetry enables easy 3D-data acquisition compared to conventional methods and here we assess its accuracy for measuring the size of deer antlers, a complex morphological structure. Using a proprietary photogrammetry software, we generated 3D images of antlers for 92 individuals from 29 species of cervids that vary widely in antler size and shape and used these to measure antler volume. By repeating the process, we found that the relative error averaged 8.5% of object size. Errors in converting arbitrary voxel units into real volumetric units accounted for 70% of the measurement variance and can therefore be reduced by replicating the conversion. We applied the method to clay models of known volume and found no indication of bias. The estimation was robust against variation in imaging device, distance and operator, but approximately 40 images per specimen were necessary to achieve good precision. We used the method to show that conventional measures of main-beam length are relatively poor estimators of antler volume. Using loose antlers of known weight, we also showed that the volume may be a relatively poor predictor of antler weight due to variation in bone density across species. We conclude that photogrammetry can be an efficient and accurate tool for measuring antlers, and likely many other complex morphological traits.

The longer the better: evidence that narwhal tusks are sexually selected

Once thought to be the magical horn of a unicorn, narwhal tusks are one of the most charismatic structures in biology. Despite years of speculation, little is known about the tusk's function, because narwhals spend most of their lives hidden underneath the Arctic ice. Some hypotheses propose that the tusk has sexual functions as a weapon or as a signal. By contrast, other hypotheses propose that the tusk functions as an environmental sensor. Since assessing the tusks function in nature is difficult, we can use the morphological relationships of tusk size with body size to understand this mysterious trait. To do so, we collected morphology data on 245 adult male narwhals over the course of 35 years. Based on the disproportional growth and large variation in tusk length we found, we provide the best evidence to date that narwhal tusks are indeed sexually selected. By combining our results on tusk scaling with known material properties of the tusk, we suggest that the narwhal tusk is a sexually selected signal that is used during male-male contests.

Assessment of a Takagi-Sugeno-Kang fuzzy model assembly for examination of polyphasic loglinear allometry

Background

The traditional allometric analysis relies on log- transformation to contemplate linear regression in geometrical space then retransforming to get Huxley’s model of simple allometry. Views assert this induces bias endorsing multi-parameter complex allometry forms and nonlinear regression in arithmetical scales. Defenders of traditional approach deem it necessary since generally organismal growth is essentially multiplicative. Then keeping allometry as originally envisioned by Huxley requires a paradigm of polyphasic loglinear allometry. A Takagi-Sugeno-Kang fuzzy model assembles a mixture of weighted sub models. This allows direct identification of break points for transition between phases. Then, this paradigm is seamlessly appropriate for efficient allometric examination of polyphasic loglinear allometry patterns. Here, we explore its suitability.

Methods

Present fuzzy model embraces firing strength weights from Gaussian membership functions and linear consequents. Weights are identified by subtractive clustering and consequents through recursive least squares or maximum likelihood. Intersection of firing strength factors set criterion to estimate breakpoints. A multi-parameter complex allometry model follows by adapting firing strengths by composite membership functions and linear consequents in arithmetical space.

Results

Takagi-Sugeno-Kang surrogates adapted complexity depending on analyzed data set. Retransformation results conveyed reproducibility strength of similar proxies identified in arithmetical space. Breakpoints were straightforwardly identified. Retransformed form implies complex allometry as a generalization of Huxley’s power model involving covariate depending parameters. Huxley reported a breakpoint in the log–log plot of chela mass vs. body mass of fiddler crabs (Uca pugnax), attributed to a sudden change in relative growth of the chela approximately when crabs reach sexual maturity. G.C. Packard implied this breakpoint as putative. However, according to present fuzzy methods existence of a break point in Huxley’s data could be validated.

Conclusions

Offered scheme bears reliable analysis of zero intercept allometries based on geometrical space protocols. Endorsed affine structure accommodates either polyphasic or simple allometry if whatever turns required. Interpretation of break points characterizing heterogeneity is intuitive. Analysis can be achieved in an interactive way. This could not have been obtained by relying on customary approaches. Besides, identification of break points in arithmetical scale is straightforward. Present Takagi-Sugeno-Kang arrangement offers a way to overcome the controversy between a school considering a log-transformation necessary and their critics claiming that consistent results can be only obtained through complex allometry models fitted by direct nonlinear regression in the original scales.

Biological interpretations of the biphasic model of ontogenetic brain–body allometry: a reply to Packard

Allometry is a description of organismal growth. Historically, a simple power law has been used most widely to describe the rate of growth in phenotypic traits relative to the rate of growth in overall size. However, the validity of this standard practice has repeatedly been criticized. In an accompanying opinion piece, Packard reanalysed data from a recent study on brain–body ontogenetic allometry and claimed that the biphasic growth model suggested in that study was an artefact of logarithmic transformation. Based on the model selection, Packard proposed alternative hypotheses for brain–body ontogenetic allometry. Here, I examine the validity of these models by comparing empirical data on body sizes at two critical neurodevelopmental events in mammals, i.e. at birth and at the time of the peak rate of brain growth, with statistically inferred body sizes that are supposed to characterize neurodevelopmental processes. These analyses support the existence of two distinct phases of brain growth and provide weak support for Packard's uniphasic model of brain growth. This study demonstrates the importance of considering alternative models in studies of allometry, but also highlights that such models need to respect the biological theoretical context of allometry.

Comparison of linear model and artificial neural network using antler beam diameter and length of white-tailed deer (Odocoileus virginianus) dataset

Evaluation of harvest data remains one of the most important sources of information in the development of strategies to manage regional populations of white-tailed deer. While descriptive statistics and simple linear models are utilized extensively, the use of artificial neural networks for this type of data analyses is unexplored. Linear model was compared to Artificial Neural Networks (ANN) models with Levenberg–Marquardt (L-M), Bayesian Regularization (BR) and Scaled Conjugate Gradient (SCG) learning algorithms, to evaluate the relative accuracy in predicting antler beam diameter and length using age and dressed body weight in white-tailed deer. Data utilized for this study were obtained from male animals harvested by hunters between 1977–2009 at the Berry College Wildlife Management Area. Metrics for evaluating model performance indicated that linear and ANN models resulted in close match and good agreement between predicted and observed values and thus good performance for all models. However, metrics values of Mean Absolute Error and Root Mean Squared Error for linear model and the ANN-BR model indicated smaller error and lower deviation relative to the mean values of antler beam diameter and length in comparison to other ANN models, demonstrating better agreement of the predicted and observed values of antler beam diameter and length. ANN-SCG model resulted in the highest error within the models. Overall, metrics for evaluating model performance from the ANN model with BR learning algorithm and linear model indicated better agreement of the predicted and observed values of antler beam diameter and length. Results of this study suggest the use of ANN generated results that are comparable to Linear Models of harvest data to aid in the development of strategies to manage white-tailed deer.

Nutrition and ontogeny influence weapon development in a long-lived mammal

Selection in male cervids should optimize allocation of nutritional resources to the competing demands of body growth versus weapon development. We investigated allocation decisions of growing and mature male white-tailed deer (Odocoileus virginianus (Zimmerman, 1780)) from three regions of low, moderate, and high diet quality. We tested (i) if deer under greater nutritional limitations would allocate proportionally less to antler growth, (ii) if antler and body mass became less variable with age, and (iii) if antler size consistently exhibited positive allometry with body mass across age classes and nutritional planes. Greater nutrition increased antler allocation in 2.5 to 4.5 year olds but not in yearlings or prime-aged males. Variability of antler mass decreased with age and was generally less in more fertile regions, but body mass was equally variable across all ages and regions. Antler mass was positively allometric with body mass for all combinations of age class and region but exhibited age- and region-related differences. Our results suggest that accruing body mass is more important to lifetime reproductive success than increasing weapon size. Reduced allometric coefficients in older males likely stem from increasing use of skeletal mineral reserves, selective pressures favoring greater body mass, and possible selection for optimal weapon strength and structure.

How sexual selection can drive the evolution of costly sperm ornamentation

Post-copulatory sexual selection (PSS), fuelled by female promiscuity, is credited with the rapid evolution of sperm quality traits across diverse taxa. Yet, our understanding of the adaptive significance of sperm ornaments and the cryptic female preferences driving their evolution is extremely limited. Here we review the evolutionary allometry of exaggerated sexual traits (for example, antlers, horns, tail feathers, mandibles and dewlaps), show that the giant sperm of some Drosophila species are possibly the most extreme ornaments in all of nature and demonstrate how their existence challenges theories explaining the intensity of sexual selection, mating-system evolution and the fundamental nature of sex differences. We also combine quantitative genetic analyses of interacting sex-specific traits in D. melanogaster with comparative analyses of the condition dependence of male and female reproductive potential across species with varying ornament size to reveal complex dynamics that may underlie sperm-length evolution. Our results suggest that producing few gigantic sperm evolved by (1) Fisherian runaway selection mediated by genetic correlations between sperm length, the female preference for long sperm and female mating frequency, and (2) longer sperm increasing the indirect benefits to females. Our results also suggest that the developmental integration of sperm quality and quantity renders post-copulatory sexual selection on ejaculates unlikely to treat male-male competition and female choice as discrete processes.

Early life expenditure in sexual competition is associated with increased reproductive senescence in male red deer

The evolutionary theories of senescence predict that investment in reproduction in early life should come at the cost of reduced somatic maintenance, and thus earlier or more rapid senescence. There is now growing support for such trade-offs in wild vertebrates, but these exclusively come from females. Here, we test this prediction in male red deer (Cervus elaphus) using detailed longitudinal data collected over a 40-year field study. We show that males which had larger harems and thereby allocated more resources to reproduction during early adulthood experienced higher rates of senescence in both harem size and rut duration. Males that carried antlers with more points during early life did not show more pronounced declines in reproductive traits in later life. Overall, we demonstrate that sexual competition shapes male reproductive senescence in wild red deer populations and provide rare empirical support for the disposable soma theory of ageing in males of polygynous vertebrate species.