Journal of Ecological Engineering Volume 18, Issue 1, Jan. 2017, pages 125–131 DOI: 10.12911/22998993/66245 Research Article APPLICATION OF A LOGISTIC FUNCTION TO DESCRIBE THE GROWTH OF FODDER GALEGA Katarzyna Rymuza1, Antoni Bombik1 1 University of Natural Sciences and Humanities in Siedlce, Prusa 14, 08-110 Siedlce, Poland, e-mail: katarzyna. [email protected] Received: 2016.07.20 ABSTRACT Accepted: 2016.10.23 It was attempted in the paper to describe the growth of oriental goat’s rue (Galega Published: 2017.01.01 orientalis Lam.) growth by means of a logistic function. The studies demonstrated that the function describes empirical data well as the coefficients of determination obtained ranged between 97 to 98%, depending on the growing season. The estimated function parameters indicate that both the plant height and growth rate were condi- tioned by thermal conditions and moisture during the growing season. Plants had the longest stems in the year 2010 (127.9 cm), and the shortest in 2008 (105.9 cm). The maximum growth rate ranged from 2.12 to 3.97 cm per day in, respectively, 2009 and 2008. Also the inflection point, the point at which plants grew the fastest, depended on the conditions of the growing season. Keywords: oriental goat’s rue, logistic function, growth equation, growth rate equa- tion, growth rate. INTRODUCTION goat’s rue, conducted by Raig [2001], Ignac- zak [2010] have demonstrated that the develop- Oriental goat’s rue (Galega orientalis Lam.) ment of the plant is a complex process. During is a plant of the family Fabaceae Lindl., subfam- the growing season, leaf rosettes are formed on ily Fabaoideae, tribe Galegae and subtribe Gale- root necks in early spring. When the minimum ginae. It is only recently that scientists world- temperature exceeds 5oC, plants start to produce wide [Šlepetys, 2003; Baležentienė, 2009, 2011; aerial stems from root necks that have overwin- Egamberdieva et al., 2010; Mikkonen et al., 2011] tered, and one-year-old underground roots. Dur- and in Poland [Deska, 2009; Deska et al., 2012, ing blossoming and pod maturation, cluster tips Kozłowski and Zielewicz, 2013] have started to wither and shrink if rainfall is insufficient. Under be interested in the plant. Indigenous to the north- the climatic conditions of Poland, pods in main ern Caucasus, the plant can be found in the wild clusters mature in late July. However, if late July in forested regions of Georgia, southern Arme- is rainy, oriental goat’s rue’s plants develop sec- nia, south-western Azerbaijan and sometimes in ondary and tertiary stems which produce flowers the Crimea. Due to the fact that the plant’s green [Deska and Wyrzykowska, 2005; Deska, 2009; parts are rich in protein (the content ranges be- Deska et al., 2012]. tween 16–27%), oriental goat’s rue is commonly Similarly to any other plant, the development grown in some EU Baltic countries (Estonia, Fin- of oriental goat’s rue can be described mathe- land, Latvia) and in Russia. In Poland the species matically by means of indices and functions. The is relatively unpopular, just like in France, Can- functions that describe the relationship: age-time, ada, Croatia, Belarus and Serbia where the first are called growth curves and make it possible to attempts are being made to introduce the plant on include many measurements into one equation a wider scale. Studies on the biology of growth based on several parameters which can be in- and morphological development of the oriental terpreted from a biological standpoint. They are 125 Journal of Ecological Engineering Vol. 18(1), 2017 sigmoid curves consisting of three stages: “expo- MATERIALS AND METHODS nental” characterized by slow but constantly in- creasing growth, “linear” when the growth is the Analysis of data was based on measurements fastest till maximum is reached, and the stage of of stem lengths of oriental goat’s rue (Galega ori- “growing old” when the growth slows [Gregorc- entalis Lam.) cv. Gale which were taken during zyk, 1991; Wesołowska-Janczarek, 1993; Kara- a four-year period as part of a long-term experi- davut et al., 2008]. Analysis of plant growth fre- ment set up in 2007 at the University of Natu- quently employs the logistic function – a function ral Sciences and Humanities experimental site which is continuous and differentiable at each (52º10’N, 22º17’E Poland). point in its domain, with an inflection point with From 2007 to 2010, 70 plants were randomly respect to which the graph is symmetrical [Tsou- selected every seven days from the start of growth laris, 2001; Tsoularis and Wallace, 2002]. to the stage of full maturity of pods. The begin- The mathematical description of a completed ning of growth was assumed to be the moment process may contribute to enhanced knowledge of forming leaf buds on root necks of the plants of the biology of plant development, indicate which overwintered. Thermal conditions and moments of optimum agrotechnological actions, rainfall during the growing seasons are presented help to forecast yields as well as indicate mor- in Figures 1 and 2. phological and developmental differences of in- The temperature varied in all the study years. dividuals and the whole population. Relatively cool spring of 2007 saw a sudden in- The objective of the work was to examine crease in temperature in late April, and high tem- the growth of oriental goat’s rue Galega orien- peratures at the end of the growing season. In the talis Lam.) during a 4-year period by means years 2008 and 2010 the beginning of the grow- of a logistic function. ing season was characterized by quite low tem- 2007 2008 2009 2010 1987 - 2010 30,0 25,0 20,0 C) o 15,0 10,0 5,0 temperature ( temperature 0,0 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 -5,0 III IV V VI VII months and ten-day temperature Figure 1. Ten-day and monthly temperatures (oC) in the study years 80,0 2007 2008 2009 2010 1987 - 2010 70,0 60,0 50,0 40,0 30,0 20,0 10,0 precipitation (mm) 0,0 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 III IV V VI VII months and ten-day periods Figure 2. Ten day and monthly precipitation (mm) in the study years 126 Journal of Ecological Engineering Vol. 18(1), 2017 peratures, around 5oC, but later they increased Moreover, the equation of growth rate was steadily. The temperature in the year 2009 was calculated in addition to the theoretical initial the lowest in the spring and was still around 1oC value of growth and inflection points, according as late as in early April. Then a sudden increase in to the formulae [Gregorczyk, 1991]: temperature was observed. • equation of growth rate: The worst hydrological conditions were re- dy k corded in 2007 and 2009 when the precipitation y(t)(a y(t)) dt a did not exceed 200 cm during the growing sea- son. Moreover, it was unevenly distributed over • theoretical initial value of growth: time. With this respect, the best year was 2010 a y0 when the high rainfall was also evenly distributed 1 b and, as a result, met the needs of plants. • coordinates of the inflection point of the The following function was used to describe function: the growth of oriental goat’s rue [Richards, 1969]: ln b a y(t) = t i , y(ti ) 0,5 a 1+ b ⋅ e −k⋅t k where: y(t) – denotes plant’s height on day t, In order to compare the growth patterns in a – the value of asymptote, individual years, the function was linearized fol- b – an integration constant and lowing the formula: lnb – k×t. Next, the hypothe- k – the index of growth rate. sis was checked whether or not the slopes of strait The terms a, k and b are somehow interdepend- lines are equal so as to find out if the course of the straight lines examined is the same. The hypoth- able. If k values are high, a values are low and de- = = = = scribe small plants. By contrast, low k values gener- esis H0 k1 k2 k3 k4 was checked using ate high values, which is typical of high plants. Val- Tukey’s test following Elandt [1964]. ues of the parameter b depend on initial values and, All the calculations were performed using in general, higher plants have got lower b values. Statistica 9.0 and the Excel spreadsheet. Equations of functions of average growth were estimated for each year separately. The cri- RESULTS terion of goodness of fit was the determination 2 coefficient R , which is frequently used for this Figure 1 shows the growth of goat’s rue in the purpose, calculated following the formula [Seber study years. It demonstrates that the growth was and Wild, 1989; Ryan, 1997]: the best in 2010 when precipitation distribution SSE R 2 = 1− coincided with the water needs of plants. Also SST function parameters, describing growth equations in which: and growth rate equations, confirm this finding n n 2 (Table 1, 2 and Figure 3). In 2010 the plants had SSE = (y − f ) 2 , SST = (y − y) , ∑ i i ∑ i the highest theoretical asymptotic value (param- i=1 i=1 eter a) and low values of coefficients k and b. where: y – experimental data; i In 2010 oriental goat’s rue plants as early f – values of the function fitted corre- i as after 42 days of growth reached the inflec- sponding to the values y , i tion points at the stem length of 64 cm and the y – arithmetical mean of the experimen- maximum growth rate of 3.45 cm ·day-1.