Is non‐loglinear allometry a statistical artifact?

The allometric equation, y = ax^b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize the observations, thereby giving rise to what has come to be known as non‐loglinear allometry. A smooth curve for observations displayed on a log–log plot is usually interpreted to mean that the scaling exponent in the allometric equation is a continuously changing function of body size, whereas a breakpoint between two (or more) linear segments on a log–log plot is typically taken to mean that the exponent changes abruptly, coincident with some important milestone in development. I applied simple graphical and statistical procedures in re‐analyses of three well‐known examples of non‐loglinear allometry, and showed in every instance that the relationship between predictor and response can be described in the original scale by simple functions with constant values for the exponent b. In no instance does the allometric exponent change during the course of development. Transformation of data to logarithms created new distributions that actually obscured the relationships between predictor and response variables in these investigations, and led to erroneous perceptions of growth. Such confounding effects of transformation are not limited to non‐loglinear allometry but are common to all applications of the allometric method. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, ?? , ??–??.     

Sex combs,allometry, and asymmetry in Drosophila

There has been recent debate about the expected allometry of sexually‐selected traits. Although sexually‐selected traits exhibit a diversity of allometric patterns, signalling characters are frequently positively allometric. By contrast, insect genitalia tend to be negatively allometric, although the allometry of nongenital sexually‐selected characters in insects is largely unknown (with some notable exceptions). It has also been suggested that there should be a negative association between the asymmetry and size of bilaterally‐paired, sexually‐selected traits, although this claim is controversial. We assessed the allometry and asymmetry (fluctuating asymmetry, FA) of a nongenital contact–courtship structure, the sex comb, in replicate populations of three species of Drosophila (we also measured wing FA). Sex combs are sexually‐selected characters used to grasp the female's abdomen and genitalia and to spread her wings prior to and during copulation. Although species differed in the size of the sex combs, all combs were positively allometric, and comb allometry did not generally differ significantly between species or populations. Comb and wing asymmetry did vary across species, although not across populations of the same species. However, FA was trait specific and was never negatively associated with trait size. © 2011 The Linnean Society of London, Biological Journal of the Linnean Society, 2011, 103 , 923–934.     

Static allometry of unicellular green algae: scaling of cellular surface area and volume in the genus Micrasterias (Desmidiales)

The surface area‐to‐volume ratio of cells is one of the key factors affecting fundamental biological processes and, thus, fitness of unicellular organisms. One of the general models for allometric increase in surface‐to‐volume scaling involves fractal‐like elaboration of cellular surfaces. However, specific data illustrating this pattern in natural populations of the unicellular organisms have not previously been available. This study shows that unicellular green algae of the genus Micrasterias (Desmidiales) have positive allometric surface‐to‐volume scaling caused by changes in morphology of individual species, especially in the degree of cell lobulation. This allometric pattern was also detected within most of the cultured and natural populations analysed. Values of the allometric S:V scaling within individual populations were closely correlated to the phylogenetic structure of the clade. In addition, they were related to species‐specific cellular morphology. Individual populations differed in their allometric patterns, and their position in the allometric space was strongly correlated with the degree of allometric S:V scaling. This result illustrates that allometric shape patterns are an important correlate of the capacity of individual populations to compensate for increases in their cell volumes by increasing the surface area. However, variation in allometric patterns was not associated with phylogenetic structure. This indicates that the position of the populations in the allometric space was not evolutionarily conserved and might be influenced by environmental factors.     

ALLOMETRIC CONSTRAINTS AND THE EVOLUTION OF ALLOMETRY

Morphological traits often covary within and among species according to simple power laws referred to as allometry. Such allometric relationships may result from common growth regulation, and this has given rise to the hypothesis that allometric exponents may have low evolvability and constrain trait evolution. We formalize hypotheses for how allometry may constrain morphological trait evolution across taxa, and test these using more than 300 empirical estimates of static (within‐species) allometric relations of animal morphological traits. Although we find evidence for evolutionary changes in allometric parameters on million‐year, cross‐species time scales, there is limited evidence for microevolutionary changes in allometric slopes. Accordingly, we find that static allometries often predict evolutionary allometries on the subspecies level, but less so across species. Although there is a large body of work on allometry in a broad sense that includes all kinds of morphological trait–size relationships, we found relatively little information about the evolution of allometry in the narrow sense of a power relationship. Despite the many claims of microevolutionary changes of static allometries in the literature, hardly any of these apply to narrow‐sense allometry, and we argue that the hypothesis of strongly constrained static allometric slopes remains viable.     

Musculoskeletal determinants of pelvic sucker function in hawaiian stream gobiid fishes: Interspecific comparisons and allometric scaling

Gobiid fishes possess a distinctive ventral sucker, formed from fusion of the pelvic fins. This sucker is used to adhere to a wide range of substrates including, in some species, the vertical cliffs of waterfalls that are climbed during upstream migrations. Previous studies of waterfall‐climbing goby species have found that pressure differentials and adhesive forces generated by the sucker increase with positive allometry as fish grow in size, despite isometry or negative allometry of sucker area. To produce such scaling patterns for pressure differential and adhesive force, waterfall‐climbing gobies might exhibit allometry for other muscular or skeletal components of the pelvic sucker that contribute to its adhesive function. In this study, we used anatomical dissections and modeling to evaluate the potential for allometric growth in the cross‐sectional area, effective mechanical advantage (EMA), and force generating capacity of major protractor and retractor muscles of the pelvic sucker (m. protractor ischii and m. retractor ischii) that help to expand the sealed volume of the sucker to produce pressure differentials and adhesive force. We compared patterns for three Hawaiian gobiid species: a nonclimber (Stenogobius hawaiiensis), an ontogenetically limited climber (Awaous guamensis), and a proficient climber (Sicyopterus stimpsoni). Scaling patterns were relatively similar for all three species, typically exhibiting isometric or negatively allometric scaling for the muscles and lever systems examined. Although these scaling patterns do not help to explain the positive allometry of pressure differentials and adhesive force as climbing gobies grow, the best climber among the species we compared, S. stimpsoni, does exhibit the highest calculated estimates of EMA, muscular input force, and output force for pelvic sucker retraction at any body size, potentially facilitating its adhesive ability. J. Morphol. 2013. © 2013 Wiley Periodicals, Inc.     

Sexual dimorphism and allometry in two seed beetles (Coleoptera: Bruchidae)

Male Callosobruchus chinensis (Coleoptera: Bruchidae) have elaborated, pectinate antennae, which are absent from conspecific females and both sexes of a congener, Callosobruchus maculatus. To begin to unravel the mechanisms producing this striking dimorphism, we examined which morphological traits best explain body size variation in bruchid beetles and quantified sexual dimorphism of antenna size through allometric analyses. Using principal component analyses, we found that elytron length and pronotum width were significantly correlated with the first principal component, which was interpreted as explaining variation in body size. Regressions of log‐transformed body size measures on log‐transformed antenna length revealed that males of both species had longer antennae than conspecific females for any given body size, although most of this effect was attributable to higher intercepts, rather than increased allometry, in males. Comparisons among heterospecific males revealed that C. maculatus males have noticeably longer antennae than C. chinensis males at large body sizes. Callosobruchus chinensis males, thus, appear to have increased the receptive area of their antennae by adding to the width of, rather than further elongating, their antennae. Finally, we found evidence for positive allometry between log‐transformed antenna length and log‐transformed antenna width in C. chinensis males. We discuss our results in the context of evidence supporting the presence of an additional, and potentially unique, sex pheromone in C. chinensis females.     

Allometric strategies for increasing respiratory surface area in the Mississippian blastoid Pentremites

The surface area of blastoid respiratory structures (hydrospires) shows positive allometry during ontogeny to offset the exponential increase in volume. Transverse cross-sections of thecae through an ontogenetic series in two blastoid species, Pentremites pyriformis and Pentremites godoni , were used to calculate surface area and volume within the hydrospires. These two congeneric species showed similar allometric change in hydrospire surface area, but this change was accomplished using different mechanisms. In P. godoni , increased hydrospire surface area was developed through an allometric increase in hydrospire length while keeping hydrospire fold count constant. In contrast, P. pyriformis , showed little allometric change in hydrospire length, but instead increased the number of hydrospire folds ontogenetically. This study shows that developmental patterns can be modified in different ways in order to solve the same functional problem.     

Genotype × environment interaction in the allometry of body,genitalia and signal traits in Enchenopa treehoppers (Hemiptera: Membracidae)

Developmental plasticity may promote divergence by exposing genetic variation to selection in novel ways in new environments. We tested for this effect in the static allometry (i.e. scaling on body size) of traits in advertisement signals, body and genitalia. We used a member of the Enchenopa binotata species complex of treehoppers – a clade of plant‐feeding insects in which speciation is associated with colonization of novel environments involving marked divergence in signals, subtle divergence in body size and shape, and no apparent divergence in genitalia. We found no change in mean allometric slopes across environments, but substantial genetic variation and genotype × environment interaction (G × E) in allometry. The allometry of signal traits showed the most genetic variation and G × E, and that of genitalia showed the weakest G × E. Our findings suggest that colonizing novel environments may have stronger diversifying consequences for signal allometry than for genitalia allometry. © 2011 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, 105 , 187–196.     

Density—body size allometry does not exist in a chironomid community on Myriophyllum

SUMMARY.

• 1 It has been proposed that population density is allometrically related to body size for a wide range of animal species. An interesting question is whether this applies to species within a specific community or trophic guild, thus constituting a ‘pattern’ of community structure.
• 2 Density-body size allometry can be translated into the relationship between population biomass (B) and population density (N). If the regression coefficient b in log B = a + b log N approximates 1, there is no allometric relation between density and body size.
• 3 Data from an epiphytic chironomid community show b～1, thus indicating that no density-body size allometry exists in this community.
• 4 Biomass is less equitably distributed than numbers among species in this and many other communities.
• 5 There is some difficulty in extending the logic of the density-body size allometry to many invertebrate communities, where body size is a strongly time-dependent, variable trait.

     

Tree height–diameter allometry across the United States

The relationship between tree height and diameter is fundamental in determining community and ecosystem structure as well as estimates of biomass and carbon storage. Yet our understanding of how tree allometry relates to climate and whole organismal function is limited. We used the Forest Inventory and Analysis National Program database to determine height–diameter allometries of 2,976,937 individuals of 293 tree species across the United States. The shape of the allometric relationship was determined by comparing linear and nonlinear functional forms. Mixed‐effects models were used to test for allometric differences due to climate and floristic (between angiosperms and gymnosperms) and functional groups (leaf habit and shade tolerance). Tree allometry significantly differed across the United States largely because of climate. Temperature, and to some extent precipitation, in part explained tree allometric variation. The magnitude of allometric variation due to climate, however, had a phylogenetic signal. Specifically, angiosperm allometry was more sensitive to differences in temperature compared to gymnosperms. Most notably, angiosperm height was more negatively influenced by increasing temperature variability, whereas gymnosperm height was negatively influenced by decreasing precipitation and increasing altitude. There was little evidence to suggest that shade tolerance influenced tree allometry except for very shade‐intolerant trees which were taller for any given diameter. Tree allometry is plastic rather than fixed and scaling parameters vary around predicted central tendencies. This allometric variation provides insight into life‐history strategies, phylogenetic history, and environmental limitations at biogeographical scales.     

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.     

Modes of ontogenetic allometric shifts in crocodylian vertebrae

During postnatal ontogeny of vertebrates, allometric trends in certain morphological units or dimensions can shift drastically among isometry, positive allometry, and negative allometry. However, detailed patterns of allometric transitions in certain timings have not been explored well. Identifying the presence and nature of allometric shifts is essential for understanding the patterns of changes in relative size and shape and the proximal factors that are controlling these changes mechanistically. Allometric trends in 10 selected vertebrae (cervical 2–caudal 2) from hatchlings to very mature individuals of Alligator mississippiensis (Archosauria, Crocodylia) are reported in the present study. Allometric coefficients in 12 vertebral dimensions are calculated and compared relative to total body length, including centrum, neural spine, transverse process, zygapophysis, and neural pedicle. During the postnatal growth, positive allometry is the most common type of relative change (10 of the 12 dimensions), although the diameter of the neural canal shows a negative allometric trend. However, when using spurious breaks (i.e. allometric trends subdivided into growth stages using certain growth events, and key body sizes and/or ages), vertebral parts exhibit various pathways of allometric shifts. Based on allometric trends in three spurious breaks, separated by the end of endochondral ossification (body length: approximnately 0.9 m), sexual maturity (1.8 m), and the stoppage of body size increase (2.8 m), six types of ontogenetic allometric shifts are established. Allometric shifts exhibit a wide range from positive allometry restricted only in the early postnatal stage (Type I) to life‐long positive allometry (Type VI). This model of ontogenetic allometric shifts is then applied to interpret potential mechanisms (causes) of allometric changes, such as (1) growth itself (when allometric trend gradually decreases to isometric or negative allometric change: Type II–IV allometric shift); (2) developmental constraint (when positive allometry is limited only in the early growth stage: Type I allometric shift); and (3) functional or biomechanical drive (when positive allometry continues throughout ontogeny: Type VI allometric shift).     

A Test of Genital Allometry Using Two Damselfly Species does not Produce Hypoallometric Patterns

It is widely admitted that sexual selection is the responsible force behind genital traits. However, the particular mechanisms of genital evolution are still debated. Recently, studies of genital static allometry in insects have been used to elucidate such mechanisms. Insect genital traits are often reported to show negative allometry (i.e., a slope < 1), which has generated a number of ideas on how genital traits are selected. However, many studies that have inferred selection mechanisms have omitted consideration of the function of genital traits, used unreliable indicators of body size, and only rarely included female genitalia in their analysis. We investigated whether negative allometry operates for genitalia in two damselfly species (Protoneura cara and Ischnura denticollis). Damselflies are suitable for genital allometry tests as their genital function and body size indicators (wing length and head width) are relatively well known and established. First, we show that the aedeagus is used to physically remove sperm from both sperm storage organs (bursa and spermatheca) and that wing length and head width correlate positively with other morphological traits for the two study species. Second, we estimated genital allometry by measuring aedeagal length, vaginal length, bursal volume, and spermathecal volume. Our results indicate no consistent allometric pattern. Allometry for aedeagal length and vaginal width was not the same. Thus, there was no support for a negative allometric relationship. We urge researchers investigating allometry to look directly at how genitalia function rather than inferring function from allometric relationships only.     

Fruit size and shape: Allometry at different taxonomic levels in bird-dispersed plants

Summary The likelihood that a plant's seeds will be dispersed by fruit-eating birds may depend upon the size and shape of its fruits. Assuming that elongate fruits can be swallowed more easily than spherical fruits of equal volume and that plant fitness is enhanced by seed dispersal by many individuals and species of birds, natural selection should favour increasing fruit elongation with increasing fruit size in bird-dispersed plants. According to this view, this allometric pattern would be adaptive. Alternatively, fruit shape in bird-dispersed plants may be constrained by development or phylogeny. To determine whether there was any evidence to support the adaptive allometry hypothesis, we examined allometric relationships between length and diameter in fruits and seeds in a group of neotropical bird-dispersed plant species. Using the major axis technique, we regressed ln(diameter) on ln(length) for fruits and seeds at various taxonomic levels: (1) within individual trees ofOcotea tenera (Lauraceae) (2) among 19 trees within a population ofO. tenera, as well as among pooled fruits from multiple trees within 20 other species in the Lauraceae, (3) among 25 sympatric species within a plant family (Lauraceae) and (4) among 167 species representing 63 angiosperm families within a plant community in Monteverde, Costa Rica. At most taxonomic levels, a tendency for fruit length to increase more rapidly than fruit diameter among fruits (negative allometry) occurred more frequently than expected by chance. Estimated slopes of the regressions of fruit length on fruit diameter were < 1 within 15 of the 19 individualO. tenera trees, among tree means withinO. tenera, among pooled fruits within 16 of the 20 other species in the Lauraceae, among species means within the Lauraceae and among means of all bird-dispersed species in the lower montane forests of Monteverde. Seed allometry showed similar patterns, although for both fruits and seeds the broad confidence intervals of the slopes estimated by major axis regression overlapped 1 in many cases. Among the 63 Monteverde family means, fruit length and diameter scaled isometrically. Based on measurements of ontogenetic changes in fruit shape in a single species,O. viridifolia, we found no evidence that negative allometry in fruit shape within the Lauraceae was an inevitable consequence of developmental constraints. Instead, increasing elongation of fruits and seeds in certain plant taxa is consistent with adaptation to gape-limited avian seed dispersers. Contrary results from vertebrate-dispersed species from Malawi and Spain may reflect differences between the New and Old World in plant taxa, seed dispersers or evolutionary history.     

Morphology and phylogenetic interpretation of a new Cambrian edrioasteroid (Echinodermata) from Spain

Abstract: The edrioasteroid, Aragocystites belli gen. nov. sp. nov. from the middle Cambrian Murero Formation of Spain, is described based on a small number of very well‐preserved specimens. Important anatomical characteristics include star‐shaped and pseudoclavate theca, rare or absent epispires, well‐developed interradially positioned oral plates and several unorganized cover plates associated with each widely exposed flooring plate. A phylogenetic analysis including several Cambro–Ordovician species shows it is more derived than Stromatocystites and Totiglobus but is a sister group to a clade comprising Cambraster and Edriodiscus. Ontogenetic observations based on juveniles of 5 mm in diameter suggest that this species changed thecal shape markedly during growth. A. belli gen. nov. sp. nov. probably lived in quiet environments where it attached directly to the sea floor on stabilized substrates.     

Shoot dynamics of the giant grass Gynerium sagittatum in Peruvian Amazon floodplains,a clonal plant that does show self-thinning

The giant rhizomatous grass Gynerium sagittatum is an early successional species that forms dense monocultures in Peruvian Amazon floodplains. We studied the shoot population structures by recording shoot densities and shoot heights. Leaf areas and stem volumes were allometrically estimated. Stands of two varieties of G. sagittatum were examined that differ in height and in the degree of shoot branching. In stands of increasing age, marked decreases in shoot densities were accompanied with an increase in mean shoot size. Self-thinning was indicated by the negative correlation between log stem volume per unit ground area and log shoot density, significant at least for one of the two varieties. The difference in thinning slope between the varieties could be largely accounted for by their different shoot geometry, as was revealed by calculations based on the allometric model of Weller (1987b). The relationship between log leaf area per shoot and log shoot density was significantly negative with slopes close to –1. Shoot size inequalities decreased with increasing mean stem volume per shoot, probably as a result of density-dependent mortality of the smaller shoots. All of these results accord with expectations for shoot self-thining. Gynerium sagittatum is the first clear example of a clonal plant species that exhibits self-thining in natural monospecific stands. It is argued that self-thinning occurs in this giant tropical grass because its shoots are perennial and do not experience seasonal die-back (periodic density-independent mortality), in contrast to many of the clonal plant species that have been studies so far.     

Morphology and biomechanical implications of isolated discocystinid plates (Edrioasteroidea, Echinodermata) from the Carboniferous of North America

Detailed examination of isolated thecal plates belonging to three discocystinid edrioasteroids, Spiraclavus nacoensis Sumrall, Hypsiclavus kinsleyi Sumrall, and Giganticlavus bennisoni Sumrall and Bowsher, reveals striking similarity in morphology among these species. Stereom observed in the ambulacral floor plates indicates that ligamentous connective tissue opened the ambulacral cover plates and muscle tissue closed them. The ambulacral floor plates are interpreted as rigid supports for the oral surface with the interambulacral areas acting as flexible integuments of plates. The aboral surface is interpreted as flexible and highly contractile. All discocystinid thecal openings are consistent in morphology with adaptations for thecal pressurization. Extension and contraction of the theca was accomplished by pumping water in and out of an inflatable sac associated with the periproctial opening. The pedunculate zone is interpreted as passively expanding and contracting by relaxing of mutable collagenous tissue and stiffening when the theca was in the desired position. All of these features illustrate that discocystinid edrioasteroids have highly–evolved morphology and function.     

On the analysis of non‐linear allometries

Abstract 1. Non‐linear allometries are those where a log–log scatterplot of trait size against body size deviates from simple linearity. These are found in many insects, including the horns of beetles, the forceps of earwigs, and the heads of certain castes of ant. 2. Non‐linear allometries are often associated with polyphenism that is itself related to behaviour: for example, the alternative mating tactics displayed by many species of beetle are widely associated with dimorphisms in horn size. 3. This paper critically reviews the current techniques used to analyse these datasets. 4. Recommendations include the use of scatterplots and assessment of the goodness of fit of simple linear models as an initial screen for non‐linear allometry. The use of recently developed algorithms for ‘segmented' regression to analyse continuous allometric relationships, and a pragmatic approach to the analysis of discontinuous relationships that recognises that there is no simple way to distinguish between morphs in some cases, and that all of the proposed methods for doing so have some drawbacks. 5. Worked examples of the analysis of two sets of data from animals that have been the subject of controversy regarding the nature of their allometric relationships are given: further worked examples are provided as online Supporting Information.     

Application of Landscape Allometry to Restoration of Tidal Channels

Oligohaline tidal channels (sloughs) in the Pacific Northwest were shown to have allometric form with respect to outlet width and depth, channel length, perimeter, and surface area. In contrast, an artificial slough, excavated to mitigate port improvements, did not conform to natural slough allometry, resulting in high retention of allochthonous inputs and sediment accumulation. Additionally, intertidal sedge habitat abundance was related to slough size for smaller sloughs, but larger sloughs did not fit this allometric pattern. This suggests that sedge habitat in large sloughs has been destroyed due to extensive log storage and transportation from the 1890s to the 1970s. Finally, the abundance of salmonid prey of terrestrial origin—aphids and adult flies—in slough surface waters was correlated with slough perimeter and, for aphids, with the amount of intertidal sedge habitat. An allometric perspective on landscape form and function has several implications for habitat restoration and mitigation: (1) Size‐related constraints on replication for landscape‐scale studies are loosened (e.g., rather than requiring reference sites that are similar in size to experimental sites, analysis of covariance can be used to control size effects); (2) physical processes, such as sedimentation and erosion, affect landscape form, whereas landscape form can affect ecological processes, so design of restoration or mitigation projects should conform to allometric patterns to maximize physical and ecological predictability; (3) landscape allometry may provide insight into undocumented human disturbances; and (4) allometric patterns suggest design goals and criteria for success.     

Zero-sum allocational strategies determine the allometry of specific leaf area

? Premise of the study: Specific leaf area (SLA) is a critical component of the leaf economics spectrum, and many functional leaf traits have been empirically demonstrated to covary with SLA. However, a complete understanding of how change in leaf size influences SLA has not yet emerged. ? Methods: To help develop a more complete understanding of the determinants of variability in SLA, we present a covariation model of leaf allometry that predicts a zero-sum interdependence of leaf thickness, density, and surface area on leaf mass. We test the model's predictions on measurements of 900 leaves from 44 angiosperm species. ? Key results: We observe that "diminishing returns," the negative allometry (slope < 1) of surface area versus mass, does not hold universally across species. Rather, the scaling of SLA is linked to the relative allocation to thickness and density. Specifically, diminishing returns are observed when leaves grow thicker, more than their density decreases, with increasing mass. Finally, we confirm model predictions that the allometric dependence of area, thickness, and density on mass can be well approximated by a zero-sum allocational process. ? Conclusions: Our work adds to the growing body of evidence that allometric covariation is a hallmark of the scaling behavior of complex plant and leaf traits. Moreover, because our model makes predictions based on allocational constraints, it provides a foundation to understand how deviations from zero-sum tradeoffs in allocation to leaf thickness, density, or area determine the allometry of SLA and, ultimately, underlie adaptive strategies within and across plant species.