Biomass Accumulation and Main Stem Elongation of Durum Wheat Grown under Mediterranean Conditions

Detecting and exploiting genetic variation in biomass accumulationis of great importance for increasing wheat yield when the harvestindex is close to its upper limit. This study was undertakento analyse the pattern of biomass accumulation and main stemelongation in 25 durum wheat (Triticum turgidum L. ‘Durum’)genotypes. Field experiments were conducted over 2 years intwo environments contrasting in the amount of available water,in northeastern Spain. Plants were sampled at the main stagesof Zadoks' scale, and dry weight per plant, crop dry weight(CDW) and main stem length were measured at each stage. Measurementsfor growth traits and thermal time from sowing fitted betterto an asymmetric logistic peak curve than to the Richards logisticmodel. Four biological variables were computed from the curve.Differences among curves describing changes in biomass werefound to be greater between irrigated and rainfed sites thanbetween years. Drought stress had less effect on main stem elongationthan on biomass accumulation. Average dry weight per plant andCDW were reduced by drought by 42 and 38%, respectively, duemainly to similar reductions in the mean rate of growth of thetwo variables. In contrast, cycle length from sowing to themaximum values of dry weight per plant and CDW was only slightlymodified by drought. Copyright 2001 Annals of Botany Company Triticum turgidum L. ‘Durum’, durum wheat, biomass, crop dry weight, stem length, rate of growth, modelling, growth analysis, logistic peak curve     

The Effects of Temperature on Leaf Growth in Cyperus longus, a Temperate C4 Species

Plants of the C₄ sedge Cyperus longus L. were grown at 10, 20and 30 °C. An asymptotic growth curve, the Richards function,was fitted to growth data for successive leaves. The mean rateof leaf appearance was a linear function of temperature with0.014 leaves appearing per day for every 1 °C increase intemperature. The instantaneous relative rate of leaf extensionshowed a marked ontogenetic drift which was most rapid at 30°C and slowest at 10 °C. The mean absolute extensionrate for foliage had a temperature coefficient of 0.16 cm d^–1° C^–1 in the range from 10 to 30 °C. The durationof leaf growth was independent of leaf number at 10 and 20 °Cbut increased linearly with leaf number at 30 °C. The smalldifferences in relative growth rate at the three temperaturesresulted in large differences in foliage area produced at theend of a 30 d growth period. The final foliage areas at 20 and10 °C were 51 and 9% respectively of that at 30 °C. Cyperus longus, temperature, leaf growth, Richards function, growth analysis     

Maturing patterns of organ weights in mice selected for rapid postweaning gain

Summary Correlated responses to selection for increased 3–6 week postweaning gain in male mice were estimated for seven internal organs (testes, spleen, liver, kidneys, heart, small intestine (S intest) and stomach) weighed at specific degrees of maturity in body weight (37.5, 50.0, 62.5, 75.0, 87.5 and 100%). Correlated responses in organ weights were generally large, but the magnitude and direction of response depended upon whether 1) comparisons were made at the same age, degree of maturity or body weight and 2) absolute or proportional organ weights were used. The selected line (M16) weighed more and had larger organ weights than controls (ICR) when compared at either the same degree of maturity or the same age, indicating positive genetic correlations between body weight and the respective organ weights. Positive correlated responses were found in spleen weight/body weight at all degrees of maturity and in liver and S intest weights as a proportion of body weight at some degrees of maturity. Testes, kidneys, heart and stomach weights as a proportion of body weight had negative correlated responses, though this was consistent only for kidneys across all degrees of maturity. Correlated responses in organ weights adjusted for body weight by covariance analysis were positive for spleen, S intest and stomach and negative for testes and kidneys. Based on the constrained quadratic model, degree of maturity in organ weight relative to degree of maturity in body weight responded positively for testes, kidneys and S intest and negatively for spleen and liver. Selection for increased growth caused negative correlated responses in allometric growth of testes, kidneys, S intest and stomach.Paper No. 10,545 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, 27695-7601. The use of trade names in this publication does not imply endorsement by the North Carolina Agricultural Research Service of the products named, nor criticism of similar ones not mentioned     

Representing Cumulative Germination.: 2. The Use of the Weibull Function and Other Empirically Derived Curves

The Weibull, Morgan–Mercer–Flodin, Richards, Mitscherlich,Gompertz and logistic functions were each fitted to a wide rangeof cumulative germinations of non-dormant seed. The Weibullproved the most suitable for describing cumulative germinationas it provided a consistently close fit to the data and wasinsensitive to choice of starting values, thus making it fairlyeasy to fit. The others provided either an inferior fit or elsea similar fit but with a greater sensitivity to starting values. The four parameters of the Weibull function reflect maximumgermination, germination rate, the lag in the onset of germinationand the shape of the cumulative distribution. A comparison between non-linear and linear fits of the Mitscherlich,Gompertz and logistic functions showed the clear superiorityof non-linear methods. Cumulative germination, Weibull function, Richards function, Morgan–Mercer–Flodin function, Mitscherlich (monomolecular) function, Gompertz function, logistic (autocatalytic) function, modelling, simulation     

Ontogeny of sexual size dimorphism in monitor lizards: males grow for a longer period, but not at a faster rate

Monitor lizards belong to the largest and the most sexually dimorphic lizards in terms of size, making this group an ideal model for studies analyzing ontogenetic causes of sexual dimorphism. Understanding of these ontogenetic factors is essential to the current discussion concerning patterns of sexual dimorphism in animals. We examined the ontogenetic trajectories of body weight and snout-vent length to analyze the emergence of sexual size dimorphism. Experimental animals were 22 males and 13 females of mangrove-dwelling monitors (Varanus indicus) hatched at the Prague Zoo. They were regularly weighed and measured up to the age of 33-40 months, and subsequently sexed by ultrasonographic imaging. The logistic growth equation was used to describe and analyze the observed growth patterns. Our results confirm considerable sexual size dimorphism in the mangrove monitor. The mean asymptotic body weight of males was nearly three times higher than that of females. As the body size of male and female hatchlings is almost equal, and the growth rate parameter (K) of the logistic growth equation as well as the absolute growth rate up to the age of 12 months do not differ between the sexes, size differences between fully grown males and females should be attributed to timing of the postnatal growth. Males continue to grow several months after they reach the age when the growth of females is already reduced. Therefore, the sexual size dimorphism emerges and sharply increases at this period.     

Genetic-statistical analysis of growth in selected and unselected mouse lines

The growth of males sampled from two mouse lines long-term selected for over 86 generations on body weight (DU6) or on protein amount (DU6P) was analysed from birth till 120 days of age and compared to the growth of an unselected control line (DUKs). Animals from the selected lines are already approximately 40 to 50% heavier at birth than the controls. This divergence increases to about 210 to 240% at the 120 day of age. With birth weights of 2.2 and 2.4 g and weights of 78 and 89 g at the 120 day these selection lines are the heaviest known mouse lines.

The fit of three modified non-linear growth functions (Gompertz function, Logistic function, Richards function) was compared and the effect of three different data inputs elucidated. The modification was undertaken to use parameters having a direct biological meaning, for example: A: theoretical final body weight, B: maximum weight gain, C: age at maximum weight gain, D (only Richards function): determines the position of the inflection point in relation to the final weight. All three models fit the observed data very well (r² = 0.949–0.998), with a slight advantage for the Richards function. There were no substantial effects of the data input (averages, single values, fitting a curve for every animal with subsequent averaging the parameters).

The high growth of the selected mice is connected with very substantial changes in the final weight and in the maximum weight gain, whereas the changes of the age at the point of inflection were, although partially significant, relatively small and dependent on the model used.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

Phenotypic and genetic variability of estimated growth curve parameters in mice

Summary Data from 1,919 outbred ICR mice were used to examine the potential usefulness of growth curve parameters as selection criteria for altering the relationship between body weight and age. A logistic growth function was used to model growth through 12 weeks of age. Estimates of asymptotic weight (A), maximum growth rate (r) and age at point of inflection (t^*) were obtained by nonlinear least-squares. A log transformation was also used to stabilize residual variance. Phenotypic and genetic parameters were estimated for the estimated growth curve parameters and for body weights at 2, 3, 4.5, 6, 8 and 12 weeks of age. Heritabilities of estimated growth curve parameters (obtained with and without a log transformation, respectively) were: A (0.28±0.07, 0.28±0.07), r (0.35±0.07, 0.53±0.09) and t^* (0.41±0.08, 0.44±0.08). Estimated genetic correlations suggest that t^* may be useful in selecting for rapid early growth without increasing mature weight.     

不同性别河北柴鸡早期生长规律及其生长曲线拟合

本研究运用Logistic、Gompertz和Bertalanffy三种非线性模型对不同性别河北柴鸡早期生长规律和生长曲线进行分析及拟合比较.结果表明,3种曲线模型拟合度均达到0.99以上,但Gompertz曲线模型在拟合度和预测极限生长量、拐点周龄和最大周增重等方面相对较好.进一步分析表明,河北柴鸡公鸡的极限体重和拐点体重均高于母鸡,拐点周龄性别间差异不大,公鸡最大周增重与实际观测值接近.本文有助于了解不同性别河北柴鸡各自的生长模式及其对营养、环境的需求,为开展规模化饲养提供参考.     

Mutual exclusion versus coexistence for discrete competitive systems

Using discrete competition models where the density dependent growth functions are either all exponential or all rational, notwithstanding the complex interactions of the species, we establish an exclusion principle. Moreover, in a 2-species discrete competition model where the growth functions are exponential and rational, an example is given illustrating coexistence when our conditions are satisfied. We obtain an exclusion principle for this 2-species model for some choice of parameters.Research partially supported by funds provided by a Science and Education Grant to the USDA-Forest Service, Southeastern Forest Experiment Station, Population Genetics of Forest Trees Research Unit, Raleigh, North Carolina     

Development of mathematical models (Logistic, Gompertz and Richards models) describing the growth pattern of Pseudomonas putida (NICM 2174)

Bacterial growth curve, which is asymptotic after a certain period, is described using three different mathematical models, namely, Logistic model, Gompertz model and Richards model. The equations for these three models are fitted by evaluating the mathematical parameters involved in these models. This is done by applying the method of partial sums to the data in Table 1 containing the optical density values for different cell mass at different time intervals. The sum of square of residuals between the expected optical density values and the experimental values is calculated for each of these models. In the cases tested, the Logistic model was found to be the best fit for the growth curve of Pseudomonas putida (NICM 2174) and was found to be easy to use. These results fit the data very well at 5% level for more than 70% of the readings.     

Modeling of the Bacterial Growth Curve

Several sigmoidal functions (logistic, Gompertz, Richards, Schnute, and Stannard) were compared to describe a bacterial growth curve. They were compared statistically by using the model of Schnute, which is a comprehensive model, encompassing all other models. The t test and the F test were used. With the t test, confidence intervals for parameters can be calculated and can be used to distinguish between models. In the F test, the lack of fit of the models is compared with the measuring error. Moreover, the models were compared with respect to their ease of use. All sigmoidal functions were modified so that they contained biologically relevant parameters. The models of Richards, Schnute, and Stannard appeared to be basically the same equation. In the cases tested, the modified Gompertz equation was statistically sufficient to describe the growth data of Lactobacillus plantarum and was easy to use.     

Characterization of commercial and biological growth curves in the Segureña sheep breed

Non-linear models were analysed to describe both the biological and commercial growth curves of the Segureña sheep, one of the most important Spanish breeds. We evaluated Brody, von Bertalanffy, Verhulst, logistic and Gompertz models, using historical data from the National Association of Segureña Sheep Breeders (ANCOS). These records were collected between 2000 and 2013, from a total of 129 610 weight observations ranging from birth to adulthood. The aim of this research was to establish the mathematical behaviour of body development throughout this breed’s commercial life (birth to slaughter) and biological life (birth to adulthood); comparison between both slopes gives important information regarding the best time for slaughter, informs dietary advice according to animals’ needs, permits economical predictions of productions and, by using the curve parameters as selection criteria, enables improvements in growth characteristics of the breed. Models were fitted according to the non-linear regression procedure of statistical package SPSS version19. Model parameters were estimated using the Levenberg–Marquardt algorithm. Candidate models were compared using the determinative coefficient, mean square error, number of iterations, Akaike information coefficient and biological coherence of the estimated parameters. The von Bertalanffy and logistic models were found to be best suited to the biological and commercial growth curves, respectively, for both sexes. The Brody equation was found to be unsuitable for studying the commercial growth curve. Differences between the parameters in both sexes indicate a strong impact of sexual dimorphism on growth. This can emphasize the value of the highest growth rate for females, indicating that they reach maturity earlier.     

Age, growth and food composition of Scorpaena porcus (Linnaeus, 1758) in the southeastern Black Sea

Samples of 525 Scorpaena porcus (Linnaeus, 1758) were collected from the southeastern Black Sea between January 2002 and May 2003 for information on age, growth, length–weight relationships and stomach contents. Total length of sampled fish ranged from 4.6 to 22.9 cm and total weight from 1.3 to 220 g. The sex ratio (1 : 1.61) was biased toward females (P < 0.05). Positive allometric growth was determined in the collected samples. The length–weight relationship for all individuals was described by the parameters a  = 0.0149 and b  = 3.09, with r ² = 0.99. The sample was composed of five age-classes (0–7 years). Three growth models were used to identify the growth characteristics: von Bertalanffy,     ; logistic,     ; and Schnute and Richards,     . Among these, the Schnute and Richards model was best fitted to the data as     The diet was composed of Carcinus mediterraneus (18.2%), Crangon crangon (12.1%), unidentified crab species (10.6%), Gobius sp. (4.6%), Palaemon sp. (4.5%), Macropipus sp. (4.5%), Engraulis encrasicholus (1.5%), Gastropoda (1.5%), and unidentified remains (34.9%).     

Plant Viability as a Function of Temperature Stress (The Richards Function Applied to Data from Freezing Tests of Growing Shoots)

Frost resistance of growing Salix viminalis L. shoots was determined by rating mortality percentage under two commonly used freezing conditions: a condition in which plants were encased in crushed ice and another in which plants were moistened with tap water prior to freezing. The mortality-temperature data were fitted with a logistic function (having a fixed inflection point halfway between the asymptotes) and with a Richards function, which is a double asymptotic sigmoid function with a variable inflection point. Different frost resistance curves were obtained, depending on the freezing conditions used. However, conditions were inadequate for efficient ice nucleation under either condition. This implies that the applied freezing conditions are not suitable when the purpose is to induce and duplicate early ice crystal formation conditions. The Richards derivatives were negatively skewed in the one case and positively skewed in the other case, giving inflection points, as a function of the upper asymptote, situated at 0.37 when shoots were frosted in the presence of ice and at 0.81 when shoots were frozen in the presence of added moisture. These values differed significantly from 0.50, through which the logistic function would have forced the curves. Because of the significant asymmetry in these frost-resistance curves, the Richards function led to a more accurate reflection of the temperature-mortality course of growing Salix stems than the logistic function. The Richards function possesses the flexibility needed to describe plant injury response in terms of physical and plant physiological mechanisms. Therefore, the Richards function is recommended rather than the logistic function for the assessment of frost resistance.     

Logistic growth curve of chickens: a comparison of techniques to estimate parameters

Parameters of a mathematical function of growth, fit to the body weight curve of two randombred control populations of each sex of chickens from hatching through 45 weeks of age, were estimated. The logistic function was chosen from among growth formulae that express rate of gain as a function of weight at a given time and gain to be made. Two logistic parameters, growth-rate constant and age at the point of inflection, were estimated by the methods of sample quantiles and nonlinear regression from weekly mean body weights of 225 males and 281 females of the Rhode Island Red (RIR) line, and 164 males and 239 females of the White Leghorn (WL) line. Males had a larger growth-rate constant than females of the same line. The RIR line had a larger rate constant than the WL line, for each sex. Age at the point of inflection was similar for males and females in the RIR line, but smaller for males than females in the WL line. Sample quantiles yielded larger, less precise estimates of the growth-rate constant than nonlinear regression. Estimates of age at the point of inflection were usually smaller using sample quantiles.     

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.     

Investigation of growth phases for bottlenose dolphins using a Bayesian modeling approach

The Gompertz function is the most commonly used growth function for cetacean studies. However, this function cannot represent multiple phases of growth. In this study, we present a Bayesian framework fitting parameters of a triple-logistic growth function to describe multiple phases of growth for bottlenose dolphins ( Tursiops truncatus ), simultaneously fitting and comparing all growth parameters between South Carolina (SC), Mississippi Sound (MSS), and Indian River Lagoon (IRL) cohorts. The fitted functions indicated a preliminary early, rapid growth phase, followed by a second phase of slower growth, and then a moderate growth spurt later in life. Growth parameters between geographic cohorts did not show obvious differences, although asymptotic length for SC dolphins was lower than MSS and IRL dolphins and significantly lower between females from SC and the IRL. Growth rate velocities between the sexes showed females exceed males initially (<1 yr), followed by males gaining an advantage around the ages of 3–4 yr until the age of around 15 yr when growth rates for both sexes approached zero (asymptotic length). This study demonstrates age-related changes in growth rates between bottlenose dolphin sexes and evidence of at least some differences ( i.e. , asymptotic length) across geographic cohorts.