A flexible sigmoid function of determinate growth

A new empirical equation for the sigmoid pattern of determinate growth, 'the beta growth function', is presented. It calculates weight (w) in dependence of time, using the following three parameters: t(m), the time at which the maximum growth rate is obtained; t(e), the time at the end of growth; and w(max), the maximal value for w, which is achieved at t(e). The beta growth function was compared with four classical (logistic, Richards, Gompertz and Weibull) growth equations, and two expolinear equations. All equations described successfully the sigmoid dynamics of seed filling, plant growth and crop biomass production. However, differences were found in estimating w(max). Features of the beta function are: (1) like the Richards equation it is flexible in describing various asymmetrical sigmoid patterns (its symmetrical form is a cubic polynomial); (2) like the logistic and the Gompertz equations its parameters are numerically stable in statistical estimation; (3) like the Weibull function it predicts zero mass at time zero, but its extension to deal with various initial conditions can be easily obtained; (4) relative to the truncated expolinear equation it provides more reasonable estimates of final quantity and duration of a growth process. In addition, the new function predicts a zero growth rate at both the start and end of a precisely defined growth period. Therefore, it is unique for dealing with determinate growth, and is more suitable than other functions for embedding in process-based crop simulation models to describe the dynamics of organs as sinks to absorb assimilates. Because its parameters correspond to growth traits of interest to crop scientists, the beta growth function is suitable for characterization of environmental and genotypic influences on growth processes. However, it is not suitable for estimating maximum relative growth rate to characterize early growth that is expected to be close to exponential.     

The strain energy function in axial plant growth

A model for axial plant growth is formulated based on conservation of energy. The model derivation assumes that a strain energy function exists to describe the dissipation of potential energy associated with water uptake, mechanical deformation, and biosynthesis during growth. The derivation does not, however, make any further assumption on the mathematical form of this constitutive relation. The model is employed to investigate possible forms of the strain energy function as applied to steady root growth. Solutions of the nonlinear partial differential equations governing growth are given for cases when the third derivative of the strain energy function is >, <, or =0. These three cases encompass a multitude of mathematical forms of the strain energy function. The resulting solutions are compared with the realization of steady axial root growth. The results of this analysis indicate that a quadratic form of the strain energy function best described steady growth. This conclusion is consistent with previous assumptions on the form of constitutive relations for growth, and allows further interpretation on the water relations, mechanical, and biosynthetic energies associated with plant growth.Research support provided by state and federal funds appropriated to the OSU/OARDC. Journal article no. 12–88     

Use of monocharacteristic growth forms and phenological phases to describe and differentiate plant communities in Mediterranean-type ecosystems

The ecomorphological and phenological study was carried out within aMediterranean vegetation context, in Quercus suberforests,which have been substituted by shrublands of Cistus sppwithin two Natural Parks in the south of the Iberian Peninsula. Theecomorphological characters that show meaningful differences between both typesof vegetation are: location of renewal buds, spinescence, stratification,maximum height of the vegetation, organs periodically shed, leaf consistency,leaf tomentosity, leaf size, and life duration of leaves, plant duration,vegetative regeneration after fire, main vegetative growth season, mainflowering season and fruit type. The phenological phases also help to discernbetween forest and shrubland, specially flower bud formation, fruiting, seeddispersal, and the existence of brachyblast vegetative growth and brachyblastleaf shedding. We propose three new indexes based on phenological phases:active period of the species (APS), active period of thecommunity (APC) and reproductive/vegetative activity of thespecies (RVA). The results of their application, in combination with theecomorphological characters, have proved promising in describing vegetation andin clearly differentiating communities. The results also show the existence ofdifferent ecomorphological groups of plants at community level, with consequentecological, historical, phytocoenological and adaptive implications     

Use of linear and non-linear functions to describe the growth of young sport- and race-horses born in Normandy

The objectives of this study were to establish standards for growth and to model the evolution of wither height (WH) between birth and adult age in different breeds of sport- and race-horses. Therefore, 398 foals, then yearlings of three different breeds, were measured regularly between birth and 18 months of age. Linear and non-linear functions were compared for describing the growth in each breed group. The monomolecular, Gompertz, logistic and cubic models correctly estimated WH in the three breeds during the first 2 years (R2 = 0.99, s.e. 3.9 to 4.5) and better than the cubic and quadratic models (R2 = 0.93, s.e. = 4.7 to 5.3). The logarithmic and power model seemed better in the last part of the growth period (2 to 6 years, R2 = 0.85, s.e. = 5.6 to 5.9). The linear model did not fit with data on most of the growth period. Comparison of the growth in the three breeds using these models confirmed that race-horses had an intense growth in their first months whereas sport-horses had a more regular growth prolonged in their first years of life.     

Aquatic plants are open flexible structures – a reply to Sukhodolov

1. Aquatic plant stands are flexible, mesh‐like open structures that undergo modification in shape and experience a cascade of declining flow velocities and micro‐scale Reynolds numbers with increasing distance into the stands. It is not possible to define or measure the frontal area of this open flexible plant structure. Total wetted area was used as a reference area for drag because it can be measured with high accuracy and it was the most suitable measure of plant size impeding the flow and absorbing light for photosynthetic production. What is important is that it is made absolutely clear which type of reference area that is used. 2. Numerous important relationships in biology are open to discussion because of differences in dimensions between variables. Relating dimensionless drag coefficients to dimensionless Reynolds numbers resolve such discussions, but defining Reynolds numbers cause other problems. Relating drag coefficients to macro‐scale Reynolds numbers would result in exactly the same form of relationship as to water velocity because macro‐scale Reynolds numbers changed in direct proportion to water velocity in the experiments, while kinematic viscosity and characteristic length within species remained constant.     

Analysis of logistic growth models

A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most predictive models are shown to be based on variations of the classical Verhulst logistic growth equation. We review and compare several such models and analyse properties of interest for these. We also identify and detail several associated limitations and restrictions.A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.     

Modelling logistic growth by a new diffusion process: Application to biological systems

The present paper introduces a new diffusion process for the purpose of modelling logistic-type behaviour patterns. Unlike other processes in the same context, this one verifies that its mean function is a logistic curve. In addition, its transition density can be found explicitly, which allows to analyse inference from the discrete sampling of trajectories. The main features of the process will be analysed and the maximum likelihood estimation of parameters will be carried out through discrete sampling. Regarding the numerical problems found to solve the likelihood equations, several strategies are developed for obtaining initial solutions for the usual numerical procedures. Such strategies are compared by means of a simulation example. Also, another simulation study is carried out in order to compare the estimation in this process to that developed by means of continuous sampling in the logistic diffusion model considered by Giovanis and Skiadas (1999). Finally an example is given for the growth of a microorganism culture. This example illustrates the predictive possibilities of the new process, as well as its ability to study time variables formulated as first-passage-times.     

Contribution to the logistic theory of growth: temporal structure and growth potential

The analysis of a structured population according to three (juvenile, mature and senescent) cellular states is carried out within the framework of Delattre's transformation systems theory. Growth in number, with the dissymmetry of cell divisions, is determined by an autocatalysis process under the constraint of the availability of a source. Two models are presented: their dynamics results in a growth of the exponential type or of the sigmoidal type, respectively. In the sigmoidal case, the logistic equation (Richards-Nelder's function with adjunction of a lower asymptote Y not equal to 0) fits satisfactorily the simulated data of the total cell number Y. The growth potential is defined as the instantaneous capacity of autocatalysis, which is expressed in relation to the present 'mitotic resources' (source + non-senescing mature cells). The acceleration variations d2Y/dt2 are in close agreement with the growth potential gradient. The analysis is then generalized to other population structuring. As a result, the logistic equation can be interpreted in terms of a formal model of growth of a structured population submitted to autocatalysis and competition.     

Use of transgenic plants and mutants to study the regulation and function of lipid composition

Mutants and transgenic plants with altered expression of genes implicated in lipid metabolism are providing fresh insights into the regulation and function of lipid composition. To date, several genes encoding fatty acid desaturases, acyltransferases, a thioesterase, a lipid transfer protein and an isoform of acyl-carrier protein have been introduced into transgenic plants. Despite the fact that some of these transgenic plants had large alterations in lipid composition, they showed surprisingly little phenotypic variation from wild-type plants. Although detailed analyses of these plants are just beginning, several theories regarding the roles of particular genes in various plant processes, such as cold tolerance and transfer of lipids between membranes, have been either substantiated or discarded on the basis of the data already obtained. In addition, constructs that contain the promoter regions of genes implicated in lipid metabolism fused to reporter genes have been introduced into transgenic plants and are providing some clues as to how lipid composition is regulated.     

Use of the ureide technique to describe the nitrogen economy of field-grown soybeans

Nitrogen fixation was estimated in `Bragg,' `Forrest,' and `Bethel' soybean (Glycine max [L.] Merrill) from seven locations northwest of New South Wales, Australia, by relating ureide and nitrate contents of plant parts sampled at regular intervals during growth to standard curves derived under controlled nitrate regimes. Estimates were combined with data on crop growth and mineral N contents of soils to (a) determine the total requirements for N by the crops, (b) determine the contributions of N₂ fixation to crop growth, and (c) relate symbiotic dependence ([N₂ fixed/total plant N] × 100) of the crops to levels of mineral N in the soil at sowing. At two locations, Myall Vale and Glenara, levels of ureides in the shoot axes and roots of unnodulated seedlings were surprisingly high at the first time of sampling, perhaps reflecting effects of uptake of ammonium-N by the soybeans or breakdown and remobilization of cotyledonary protein. Ureides in plant parts declined significantly by the second (V5 to V7 growth stage) sampling. Subsequently, ureide contents increased whereas levels of nitrate in plant parts decreased. The relative abundance of ureides ([ureide-N/ureide-N + nitrate-N] × 100) in the shoot axes and nodulated roots of both crops increased linearly from almost zero during mid-vegetative growth (V5 to V7) to virtually 100% during late reproductive growth (R4 to R5, Myall Vale and R6, Glenara). The data suggest a steady transition in soybeans at both locations from dependence upon mineral N for early growth to complete reliance upon fixed N during late reproductive growth. Estimates of seasonal N₂ fixation for soybeans at the seven locations ranged from 73 to 288 kilograms per hectare N (shoot axes ureides) and from 147 to 337 kilograms per hectare N ha (nodulated roots ureides).     

Comparison of non-linear growth models to describe the growth curve in West African Dwarf sheep

The objectives of this study were to compare the goodness of fit of four non-linear growth models, i.e. Brody, Gompertz, Logistic and Von Bertalanffy, in West African Dwarf (WAD) sheep. A total of 5274 monthly weight records from birth up to 180 days of age from 889 lambs, collected during 2001 to 2004 in Betecoucou breeding farm in Benin were used. In the preliminary analysis, the General Linear Model Procedure of the Statistical Analysis Systems Institute was applied to the dataset to identify the significant effects of the sex of lamb (male and female), type of birth (single and twin), season of birth (rainy season and dry season), parity of dam (1, 2 and 3) and year of birth (2001, 2002, 2003 and 2004) on the observed birth weight and monthly weight up to 6 months of age. The models parameters (A, B and k), coefficient of determination (R²), mean square error (MSE) were calculated using language of technical computing package Matlab®, 2006. The mean values of A, B and k were substituted into each model to calculate the corresponding Akaike’s Information Criterion (AIC). Among the four growth functions, the Brody model has been selected for its accuracy of fit according to the higher R², lower MSE and AIC. Finally, the parameters A, B and k were adjusted in Matlab®, 2006 for the sex of lamb, year of birth, season of birth, birth type and the parity of ewe, providing a specific slope of the Brody growth curve. The results of this study suggest that Brody model can be useful for WAD sheep breeding in Betecoucou farm conditions through growth monitoring.     

A general bilinear model to describe growth or decline time profiles

Linear models are widely used because of their unrivaled simplicity, but they cannot be applied for data that have a turning-or rate-change-point, even if the data show good linearity sufficiently far from this point. To describe such bilinear-type data, a completely generalized version of a linearized biexponential model (LinBiExp) is proposed here to make possible smooth and fully parametrizable transitions between two linear segments while still maintaining a clear connection with the linear models. Applications and brief conclusions are presented for various time profiles of biological and medical interest including growth profiles, such as those of human stature, agricultural crops and fruits, multicellular tumor spheroids, single fission yeast cells, or even labor productivity, and decline profiles, such as age-effects on cognition in patients who develop dementia and lactation yields in dairy cattle. In all these cases, quantitative model selection criteria such as the Akaike and the Schwartz Bayesian information criteria indicated the superiority of the bilinear model compared to adequate less parametrized alternatives such as linear, parabolic, exponential, or classical growth (e.g., logistic, Gompertz, Weibull, and Richards) models. LinBiExp provides a versatile and useful five-parameter bilinear functional form that is convenient to implement, is suitable for full optimization, and uses intuitive and easily interpretable parameters.     

Evaluation of mathematical models to describe testicular growth in Blackbelly ram lambs

The primary objective was to compare various mathematical models to describe scrotal circumference (SC) and paired testis volume development in Blackbelly ram lambs. The study was conducted in the state of Querétaro, México (20° 43' N, 100° 15' W). Spring-born Blackbelly ram lambs (n = 41) were housed outdoors and fed alfalfa hay and concentrate. Body weight, SC, and testis length, diameter, and volume were recorded every 2 wk from 24 to 172 d of age (June 18 to November 3). The following mathematical functions were used to model SC-age and testis volume-age relationship: Von Bertalanffy, Brody, Gompertz, Logistic, and Richards. The suitability of the models was evaluated based on parameter values and standard errors, residual mean square, the coefficient of determination (R²), and the average prediction error (APE). All models, except for Brody's, had good fit to SC (R² > 0.98) and testis volume (R² > 0.95), and produced similar growth curves in the range of ages studied. The logistic model predicted SC at maturity quite well, 33.6 ± 0.6 cm as compared with 33.9 ± 0.5 cm observed in adult animals; all models had APE's smaller than ±7% between 56 and 168 d of age. The Bertalanffy model predicted testis volume at maturity quite well, 513 ± 22 cm³ as compared with 488 ± 20 cm³ calculated for adult animals. The logistic model had a good fit to testis volume during the period of study, but underestimated the volume at maturity by 28%. All models, except for Brody's, had APE's smaller than ±14% between 98 and 168 d of age. The logistic and Bertalanffy models predicted the inflection point for SC at 83 and 59 d of age, and testis volume at 116 and 109 d of age, respectively. In conclusion, all models, except for Brody's, had good fit to actual SC and testis volume data in the range of age evaluated, whereas the logistic and Bertalanffy's models made the best predictions for adult SC and testis volume, respectively.     

Note on combining risks using a logistic discriminant function approach

A simple method for combining risks based on repeated observations on an individual is presented. Implications of this model are explored for several special cases, and an example is presented.