Determination of a point sufficiently close to the asymptote in nonlinear growth functions

Abstract

Growth functions with upper horizontal asymptote do not have a maximum point, but we frequently question from which point growth can be considered practically constant, that is, from which point the curve is sufficiently close to its asymptote, so that the difference can be considered non-significant. Several methods have been employed for this purpose, such as one that verifies the significance of the difference between the curve and its asymptote using a t-test, and that of Portz et al. (2000), who used segmented regression. In the present work, we used logistic growth function, which has horizontal asymptote and one inflection point, and applied a new method consisting in the mathematical determination of a point in the curve from which the growth acceleration asymptotically tends to zero. This method showed the advantage to have biological meaning besides leading to a point quite close to those obtained using the beforementioned methods. Key words: nonlinear regression, logistic model, critical point of growth Determinação de um ponto suficientemente próximo à assíntota em funções de crescimento não lineares RESUMO: Em funções de crescimento que apresentam uma assíntota horizontal superior à curva, frequentemente surge a questão sobre quando se pode considerar o crescimento como praticamente constante, isto é, quando a curva está suficientemente próxima à sua assíntota, de modo que se possa declarar a diferença como sendo não-significativa. Vários métodos têm sido empregados, entre eles o que verifica através do teste t a significância da diferença entre a curva e sua assíntota. O uso de regressão segmentada, como em Portz et al., também tem esse objetivo, isto é, a determinação de um ponto de início de crescimento praticamente constante. Utilizou-se a função logística de crescimento, a qual possui assíntota horizontal e ponto de inflexão, e aplicou-se um novo método, que consiste na determinação matemática de um ponto da curva a partir do qual a aceleração do crescimento tende assintoticamente a zero. Este método, além de ter um significado biológico, conduz a um ponto bastante próximo aos obtidos pelos métodos anteriormente citados. Palavras-chave: regressão não linear, modelo logístico, ponto crítico de crescimento

Soluble sugars and germination of Annona emarginata (Schltdl.) H. Rainer seeds submitted to immersion in GA3 up to different water contents

“…With the water acquisition curve it was possible to characterize Phases I, II and III of germination of A. emarginata seeds, in accordance with the triphasic pattern proposed by Bewley andBlack in 1994 (FERREIRA et al, 2006) (Figure 1). Phase I lasted 109 hours, from the start of imbibition, finishing when the seeds reached 35.2% of water, as described for endospermatic seeds (CARVALHO; NAKAGAWA, 2000).…”

“…Periodical weights for the calculation of seed water uptake were performed and by the stabilization of the water intake, the seeds were removed from the water and placed in 'germitest' paper, with the weighing being held until the beginning of phase III, when the emission of the primary root occurs. To study the variation in the seed water content, in phase I and the beginning of phase II, the Brody model, or monomolecular model, was adjusted to the equation y=α [1-β exp (-c x)], with y = moisture degree (%), x = time, α, β e c = model parameter (MISCHAN et al, 2011).…”

“…The germination process starts with water uptake by the seed, leading to a number of metabolic changes which culminate in radicle protrusion (NONOGAKI et al, 2010). Thus, determination of water acquisition curve helps setting times for immersion treatments of seeds in aqueous solutions, as performed with plant growth regulators (SOCOLOWSKI; CICERO, 2011) or priming (VARIER et al, 2010), where the duration of treatments has been defined according to the duration of Phase I.…”

Inflection and stability points of diphasic logistic analysis of growth

“…The stability points of the logistic function, mono or diphasic can be determined by various methods (Passos et al, 2012). The method that equates the fourthorder derivative of the function y to zero is employed here (Mischan et al, 2011): , X = exp(-b -γx) (13) which gives the points…”

Critical points on growth curves in autoregressive and mixed models

“…The critical points of the functions, such as the infl ection point (pi) and deceleration asymptotic point (pda), are determined using the method developed in Mischan et al (2011) for the model with independent errors, model I, for the logistic function. However, autoregressive models, model II, and mixed model, model III, lead to estimated values that are discrete variables defi ned only for the observed values of x i , i = 1, ..., n. Therefore we propose a new approach, which consists in the determination of the critical points through the function (15) the estimate of F(x i , θ) in (2) and (4), obtained by adjusting the models.…”