Preston’s equation is a general model describing the egg shape of birds. The parameters of Preston’s equation are usually estimated after re-expressing it as the Todd-Smart equation and scaling the egg’s actual length to two. This method assumes that the straight line through the two points on an egg’s profile separated by the maximum distance (i.e., the longest axis of an egg’s profile) is the mid-line. It hypothesizes that the photographed egg’s profile is perfectly bilaterally symmetrical, which seldom holds true because of photographic errors and placement errors. The existing parameter estimation method for Preston’s equation considers an angle of deviation for the longest axis of an egg’s profile from the mid-line, which decreases prediction errors to a certain degree. Nevertheless, this method cannot provide an accurate estimate of the coordinates of the egg’s center, and it leads to sub-optimal parameter estimation. Thus, it is better to account for the possible asymmetry between the two sides of an egg’s profile along its mid-line when fitting egg-shape data. In this paper, we propose a method based on the optimization algorithm (optimPE) to fit egg-shape data and better estimate the parameters of Preston’s equation by automatically searching for the optimal mid-line of an egg’s profile and testing its validity using profiles of 59 bird eggs spanning a wide range of existing egg shapes. We further compared this method with the existing one based on multiple linear regression (lmPE). This study demonstrated the ability of the optimPE method to estimate numerical values of the parameters of Preston’s equation and provide the theoretical egg length (i.e., the distance between two ends of the mid-line of an egg’s profile) and the egg’s maximum breadth. This provides a valuable approach for comparing egg shapes among conspecifics or across different species, or even different classes (e.g., birds and reptiles), in future investigations.

/pdf/a-new-program-to-estimate-the-parameters-of-preston-s-3t7vww9s.pdf

A New Program to Estimate the Parameters of Preston's Equation, a General Formula for Describing the Egg Shape of Birds

The inequality in leaf and fruit size distribution per plant can be quantified using the Gini index, which is linked to the Lorenz curve depicting the cumulative proportion of leaf (or fruit) size against the cumulative proportion of the number of leaves (or fruits). Prior researches have predominantly employed empirical models-specifically the original performance equation (PE-1) and its generalized counterpart (GPE-1)-to fit rotated and right-shifted Lorenz curves. Notably, another potential performance equation (PE-2), capable of generating similar curves to PE-1, has been overlooked and not systematically compared with PE-1 and GPE-1. Furthermore, PE-2 has been extended into a generalized version (GPE-2). In the present study, we conducted a comparative analysis of these four performance equations, evaluating their applicability in describing Lorenz curves related to plant organ (leaf and fruit) size. Leaf area was measured on 240 culms of dwarf bamboo (Shibataea chinensis Nakai), and fruit volume was measured on 31 field muskmelon plants (Cucumis melo L. var. agrestis Naud.). Across both datasets, the root-mean-square errors of all four performance models were consistently smaller than 0.05. Paired t-tests indicated that GPE-1 exhibited the lowest root-mean-square error and Akaike information criterion value among the four performance equations. However, PE-2 gave the best close-to-linear behavior based on relative curvature measures. This study presents a valuable tool for assessing the inequality of plant organ size distribution. 

Comparison of four performance models in quantifying the inequality of leaf and fruit size distribution

The Lorenz curve is used to describe the relationship between the cumulative proportion of household income and the number of households of an economy. The extent to which the Lorenz curve deviates from the line of equality (i.e., y = x) is quantified by the Gini coefficient. Prior models are based on the simulated and empirical data of income distributions. In biology, the Lorenz curves of cell or organ size distributions tend to have similar shapes. When the Lorenz curve is rotated by 135 degrees counterclockwise and shifted to the right by a distance of 2, a three-parameter performance equation (PE), and its generalized version with five parameters (GPE), accurately describe this rotated and right-shifted curve. However, in prior studies, PE and GPE were not compared with the other Lorenz equations, and little is known about whether the skewness of the distribution could influence the validity of these equations. To address these two issues, simulation data from the beta distributions with different skewness values and six empirical datasets of plant (organ) size distributions were used to compare PE and GPE with three other Lorenz equations in describing the rotated and right-shifted plant (organ) size distributions. The root-mean-square error and Akaike information criterion were used to assess the validity of the two performance equations and the three other Lorenz equations. PE and GPE were both validated in describing the rotated and right-shifted simulation and empirical data of plant (organ) distributions. Nevertheless, GPE worked better than PE and the three other Lorenz equations from the perspectives of the goodness of fit, and the trade-off between the goodness of fit and the model structural complexity. Analyses indicate that GPE provides a powerful tool for quantifying size distributions across a broad spectrum of organic entities and can be used in a variety of ecological and evolutionary applications. Even for the simulation data from hypothetical extreme skewed distribution curves, GPE still worked well.

Rotated Lorenz Curves of Biological Size Distributions Follow Two Performance Equations

biogeom: Biological Geometries

Habitat fragmentation threatens the sustainability of ecological restoration. Understanding the variation in intraspecific traits helped to reveal the functional resource-use strategies of plants in response to environmental changes. We sampled different landscape types of forest configurations, where the most widespread species was Robinia pseudoacacia. From each plot, from two to five R. pseudoacacia individuals were selected for further examinations. Plant development and leaf traits—leaf area (LA), specific leaf area (SLA), leaf dry matter content (LDMC), and leaf thickness (Lth)—were measured in 135 individuals in total. The effects of plant development and landscape fragmentation on R. pseudoacacia leaf traits were assessed using linear mixed-effects models. The environmental factors explained the changes in leaf traits of R. pseudoacacia individuals, and the effect of stand type was the most significant. Compared with continuous forests, R. pseudoacacia individuals in fragmented forests adopted a more conservative resource-use strategy, with smaller LA and SLA and larger Lth and LDMC values. With an increase in landscape heterogeneity, SLA increased and LDMC decreased. In conclusion, the occurrence of landscape fragmentation plays a substantial role in inducing changes in leaf characteristics. The restoration of fragmented forests to continuous forests requires the appropriate addition of land-use types and systematic adjustment of landscape configurations.

Environmental Drivers of Landscape Fragmentation Influence Intraspecific Leaf Traits in Forest Ecosystem

Egg geometry can be described using Preston's equation, which has seldom been used to calculate egg volume (V) and surface area (S) to explore S versus V scaling relationships. Herein, we provide an explicit re‐expression of Preston's equation (designated as EPE) to calculate V and S, assuming that an egg is a solid of revolution. The side (longitudinal) profiles of 2221 eggs of six avian species were digitized, and the EPE was used to describe each egg profile. The volumes of 486 eggs from two avian species predicted by the EPE were compared with those obtained using water displacement in graduated cylinders. There was no significant difference in V using the two methods, which verified the utility of the EPE and the hypothesis that eggs are solids of revolution. The data also indicated that V is proportional to the product of egg length (L) and maximum width (W) squared. A 2/3‐power scaling relationship between S and V for each species was observed, that is, S is proportional to (LW2)2/3. These results can be extended to describe the shapes of the eggs of other species to study the evolution of avian (and perhaps reptilian) eggs.

A simple way to calculate the volume and surface area of avian eggs

Quantifying α-diversity as a continuous function of location—a case study of a temperate forest

The proportions in the size of the avian egg albumen, yolk, and shell are crucial for understanding bird survival and reproductive success, because their relationships with volume and surface area can affect ecological and life history strategies. Prior studies have focused on the relationship between the albumen and the yolk, but little is known about the scaling relationship between eggshell mass and shape, and the mass of the albumen and the yolk. Toward this end, 691 eggs of six precocial species were examined, and their 2-D egg profiles were photographed and digitized. The explicit Preston equation, which assumes bilateral symmetrical geometry, was used to fit the 2-D egg profiles and to calculate surface areas and volumes based on the hypothesis that eggs can be treated as solids of profile revolution. The scaling relationships of eggshell mass (Ms), albumen mass (Ma), and yolk mass (My), as well as the surface area (S), volume (V), and total mass (Mt) were determined. The explicit Preston equation was validated in describing the 2-D egg profiles. The scaling exponents of Ma vs. Ms, My vs. Ms, and My vs. Ma were smaller than unity, indicating that increases in Ma and My fail to keep pace with increases in Ms, and that increases in My fail to keep pace with increases in Ma. Therefore, increases in unit nutrient contents (i.e., the yolk) involve disproportionately larger increases in eggshell mass and disproportionately larger increases in albumen mass. The data also revealed a 2/3-power scaling relationship between S and V for each species, i.e., simple Euclidean geometry is obeyed. These findings help to inform our understanding of avian egg construction and reveal evolutionary interspecific trends in the scaling of egg shape, volume, mass, and mass allocation.

Scaling relationships among the mass of eggshell, albumen, and yolk in six precocial birds.

Abstract Sexual dimorphism in body size has been observed for many insect species. However, whether dimorphism influences the flight performance for closely related insects or between the genders of conspecifics has seldom been examined. Wing loading (i.e., body mass/total wing area, WL = M/A) is an important indicator of flight efficiency and maneuverability. We examined the potential for sexual dimorphism and interspecific differences in wing loading for three cicada species (Cryptotympana atrata, Meimuna mongolica, and Platypleura kaempferi) and tested the hypothesis that females have higher WLs than males possibly because they must carry the extra weight of eggs. There were significant dimorphic differences in body mass and WL for each of the three species. However, there were no significant dimorphic differences in total wing area for C. atrata and M. mongolica, whereas P. kaempferi females had larger mean total wing area than males. Among the three cicada species, males had lower WLs (and by inference, greater maneuverability) than females. In addition, the M vs. A scaling exponent (i.e., α) was numerically significantly greater than unity (i.e., α > 1). Thus, WL should scale with respect to M with α > 1, which accounts for why females have a larger WL than males. We interpret these results to indicate that gender dimorphism in body size and flight performance likely reflect selection on traits affecting reproductive success that, in turn, likely depend on flight maneuverability.

Sexual Dimorphism in Body Size and Wing Loading for Three Cicada Species

Leaf ontogeny and maturation are attended by a transition from importing to exporting photosynthates, which has important implications to plant growth and reproductive success. However, the influence of leaf age on leaf size has been seldom studied, although ontogenetic growth trajectories plotted against time typically exhibit sigmoidal curves, which can be approximated by corresponding sigmoidal functions. The three-parameter logistic equation (LE) is a classical sigmoidal function that assumes a symmetrical growth rate curve (i.e., the derivative of the LE) when plotted against time. LE has been verified as a general tool for describing ontogenetic growth such as seen in leaf maturation. However, there are other sigmoidal functions that assume that growth rate curves are skewed bell-shaped. Whether leaf ontogeny follows the LE, or some other sigmoidal functions is unknown. To explore the relationship between leaf area and leaf age, we recorded the leaf length and width of 61 leaves of Camptotheca acuminata Decne at 21 time-points during leaf expansion, and used three sigmoidal functions to fit leaf area (represented by 2/3 × leaf length × leaf width) when plotted against leaf age, i.e., the LE, the beta sigmoidal equation (BSE), and the Lobry-Rosso-Flandrois sigmoidal equation (LRFSE). LE had the best close-to-linear behavior and the lowest Akaike information criterion (AIC) among the three sigmoidal functions for the pooled data of 61 leaves. For 49 of the 61 leaves (80.3 %), the AICs of the LE were smaller than those of the BSE, and for 52 out of 61 leaves (85.2 %), the AICs of the LE were smaller than those of the LRFSE. In addition, the BSE provided incorrect predictions for the starting time of leaf growth. Thus, the data indicate that (1) the LE is the best sigmoidal function for describing ontogeny at the individual leaf level and for the combined data, and (2) the growth rate of C. acuminata leaves tends to be a symmetrical function rather than an asymmetrical function of leaf age. These findings provide insights into the ontogenetic importing-exporting transition attending leaf maturation. 

Comparison of three sigmoidal functions describing the leaf growth of Camptotheca acuminata Decne

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