Species adapted to arid and semiarid environments, such as forage cactus (Nopalea spp. and Opuntia spp.), can contribute to an increase of yields in the biomass of agricultural areas, improving the efficiency of use of local natural resources from such environments (^{Diniz et al. 2017} and ^{Freire et al. 2018}).
Cacti is a xerophilous plant of multiple purposes, and it is used in different countries of all continents as forage, fruit, vegetable, raw material for processing, biomass for energy production, live fence, erosion control, soil conservation, landscaping, dye production, and medicinal uses, among others (^{Inglese et al. 1995}). The forage cactus presents CAM (Crassulasean Acid Metabolism) photosynthetic metabolism, with day closure of stomata and nocturnal carbon assimilation, which provides a greater adaptive capacity to the abiotic factors (^{Santos et al. 2016} and ^{Souza Filho et al. 2016}).
Cactus presents high phytomass production in semiarid regions, with high energy value, high nonfibrous carbohydrate contents, high ruminant acceptability, high digestibility coefficient, high water reserves and easy propagation (^{Freire et al. 2018} and ^{Pereira et al. 2018}). They are highly efficient in the use of water, making it suitable to the environmental conditions with great rates of atmospheric evaporation and reduction of soil humidity (^{Winter et al. 2011} and ^{Hartzell et al. 2018}). The yield of this culture is influenced, mainly, by interception of light, which in turn, is determined by morphological characteristics (^{Pinheiro et al. 2014}).
Nopalea cochenillifera (L.) SalmDyck is cultivated in dry conditions, due to its anatomicalmorphophysiological characteristics that provide good adaptation to the climatic conditions of such region (^{Freire et al. 2018}). This species is resistant to carmine cochineal (Dactylopius opuntiae Cockerell), the main pest of this type of culture, and therefore, it gains prominence in the properties of the Brazilian semiarid (^{Pereira et al. 2018}).
In most regions where the Nopalea cochenillifera culture is crop, are dry regions where they have low water availability and high temperatures, an alternative to such adversities is the consortium with other cultures, because the consortium allows planting in the same area of crops that serve as forage material for animals, in addition to promoting greater light interception directly on the soil, which generates less plant evapotranspiration and consequently better forage development.
The growth curve describes a sequence of measurements, in function of time, of a given plant's morphological characteristic such as height, length, thickness, width, area or diameter. In general, this curve presents aspect of exponential or sigmoidal growth (^{Lucena et al. 2018a}), and the proper adjustment of the growth curves is an important tool for the adequate management of a crop (^{Leite et al. 2017}). This data generates useful information such as: morphophysiological responses, period of greater plant growth, and which is the most suitable season for fertilization and pest control (^{Lucena et al. 2018a}). In the last decades, growth curves in forage cactus have been well studied, as reported in Nopalea cochenillifera (^{Lucena et al. 2016} and ^{Freire et al. 2018}).
Although there is information about the agronomic characteristics of the specie N. cochenillifera (^{Silva et al. 2014}; ^{Lucena et al. 2018b} and ^{Freire et al. 2018}), studies focused on growth curve of this species intercropped with another culture were not detected in the literature. Therefore, the objective of this study is to define the best relation between plant height and cladode number of the forage cactus Little Sweet clone (Nopalea cochenillifera (L.) SalmDyck), with and without "pornunça" consortium.
Materials and Methods
The research was carried from August 2017 to January 2019 in an experimental area of the Universidade Federal Rural de Pernambuco (Serra Talhada County Campus, PE, Brazil), located at 07° 57’ 01” S, 38° 17’ 53” E, at an elevation of 431 m. According to the Köppen, the region climate classification is BSwh. Annual average rainfall, air temperature and relative humidity are 632.2 mm, 25 ºC and 60%, respectively (^{Leite et al. 2019}).
The soil of the experimental area was classified as Typical Haplic Cambisol Ta Eutrophic. The soil sample was taken at the first 20 cm, and showed the following chemical characteristics: pH(water) = 6.80; P (Mehlich) = 40.00 mg dm^{3}; K^{+} = 0.45 cmolc dm^{3}; Na^{+} = 0.06 cmolc dm^{3}; Ca^{2+} = 5.30 cmolc dm^{3}; Mg^{2+} = 1.10 cmolc dm^{3}; H + Al = 1.23 cmolc dm^{3}; S = 6,91 cmolc dm^{3}; CTC = 8.14 cmolc dm^{3}; V =84.89%; organic matter = 7.93 g kg^{1}; sand = 828.6 g kg^{1}; Silt = 148.25 g kg^{1}; clay = 23.15 g kg^{1}, and soil density of 1.45 g dm^{3}.
The experimental design was performed in randomized blocks, with single cacti (Nopalea cochenillifera (L.) SalmDyck clone Little Sweet) and "pornunça" consortium (Manihot glaziovii x Manihot esculenta), with three replicates. The experimental unit (area of 24.0 m^{2}, 5.0 m x 4.8 m) was composed of three rows of cactus, planted in August 2017, with spacing between rows of 1.60m and between plants of 0.20m, which is equivalent to the population density of 31,250 plants ha^{1}. The cladodes used in the propagation of the trial were from plants with an approximate age of three years. The planting occurred after the cure of the cladode (8 days), with the insertion of a cladode per hole, in vertical position, deep enough to bury half of the cladode.
In the consortium cacti x "pornunça", there was the addition of two rows of "pornunça" between the cacti lines. The "pornunça" was planted in October 2017, with a spacing of 1.6m x 1.0m, totaling 6,250 plants ha^{1}. The cacti and "pornunça" have grown under natural conditions of rain (dry), without the use of fertilization, and, during the conduction of the experiment, the control of spontaneous plants was carried out manually, when necessary.
For evaluations of morphogenic characteristics, thirty cactus plants per plot were selected out of the central line, resulting in ninety plants per treatment. Measures of plant height and cladode number per plant of Nopalea cochenillifera, clone Little Sweet, were gathered during one year. The first evaluation was performed in January 2017, and the others has a 30day interval between them, totaling 10 evaluations. The plant height was measured from the base (colon) to the apex of the plant, with the aid of a measuring tape, while the cladode total number per plant was measured by counting.
Results of plant height and cladode total number per plant were expressed by mean and standard error. MannWhitney test was applied in order to compare the means of the variables at 5% probability. The fittest model to predict plant height (H) of Nopalea cochenillifera in function of the days after planting (DAP) and cladode number (NC) were performed using the following regression models: linear multiple, power, Gamma, logistic, Gompertz, Weibull, tangent hyperbolic and quadratic (^{Leite et al. 2017} and ^{Lucena et al. 2018a}) (table 1). The linear multiple, power, logistic, Gompertz, tangent hyperbolic and quadratic models with normal distribution assume that the response of dependent variable is in the range (∞, ∞). The Gamma models with Gamma distribution and Weibull models with Weibull distribution assume that the response of dependent variable is in the range (0, ∞) (^{Lucena et al. 2018a} and ^{Leite et al. 2019}) (table 1).
Models  Equation of the plant height 

Linear multiple 

Power 

Gamma 

Logistic 

Gompertz 

Weibull 

Hyperbolic T. 

Quadratic 

where, H_{i} the ith plant height of Nopalea cochenillifera; DAP_{i} the ith days after planting; CN_{i} the ith number of cladode of Nopalea cochenillifera and
The following criteria evaluated the models: Coefficient of determination of the model (R²), Akaike's Information Criterion (AIC), Sum of Square of Residuals (SSR) and the Willmott index (d).
The coefficient of determination of the model is expressed by:
The Akaike information criteria (AIC), as defined by ^{Akaike (1974)}, is given by:
The sum of square of the residuals (SSR) is the square sum of the difference between the values observed and predicted by the models, where the lowest value contributes to the choice of the best equation. The SSR for this study was defined by the following expression:
The d index defined by ^{Willmott (1981)} is given by:
where,
Results and Discussion
During the experimental period, the accumulated rainfall was 583.4 mm, where: 16.4 mm in the year 2017; 538.0 mm in 2018 and 29.0 mm in 2019 (see Figure 1). The rainfall presented high variability, which affected the soil water availability for the crops under study. It must be highlighted that the rains predominance with high intensity, as registered from March, April and December 2018 (396.0 mm), represented 67.9% of the accumulated rain in the evaluated period. This large accumulation of rainfall was possibly higher than the soil water retention capacity, which leads to a decrease in the use of rain by cacti.
The great availability of water in this period allows the plant to growth and, consequently, to produce more dry matter, generating more fodder for the animals. High temporal variability of the rainfall distribution and other climatic factors, in semiarid areas, have great ecological value, as they can limit the adaptation and growth of species.
Light radiation, soil water content and air temperature are the main environmental factors that affect plant growth and development (^{Akula and Ravishankar 2011}), because the main physiological and biochemical processes depend on these factors.
As presented in table 2, there was no difference in the plant height of Nopalea cochenillifera at 510 days after planting (DAP), either with or without the "pornunça" consortium (pvalue=0.0675). The same was observed for the cladode total number per plant (pvalue=0.1256).
The consortium with "pornunça" did not affect the growth of N. cochenillifera plants, as verified by the similarity of plant height and the total number of cladodes with or without consortium. Therefore, natural competition for light, water and nutrients of these two crops in the evaluated period, and the spacing adopted in consortium did not cause a change in the forage cactus growth pattern, making the cacti"pornunça" consortium generate greater dry matter production in the same area without the consortium, thus favoring greater production of green material and consequently greater production of forage for animals. Plant height and total number of cladodes per plant are important morphogenic characteristics of forage cactus, because they are highly correlated with phytomass production characteristics (^{Sales et al. 2013}).
Species adapted to semiarid conditions, such as forage cactus (Nopalea cochenillifera) and "pornunça" (Manihot glaziovii x Manihot esculenta), in consortium systems, can elevate the yields and quality of produced phytomass, improving the efficiency in use of production factors such as soil, water, nutrients and radiation (^{Chimonyo et al. 2018}), which provides production synergism.
These results indicate that the use of cacti"pornunça" consortium has agronomic relevance and viability, due to the great use of land, as well as the increase in the final yield of phytomass production. The use of Euphorbiaceae, such as the "pornunça", in this system consorted with cactus, besides favoring this higher performance, also minimizes soil water losses by evaporation and favors a lower air temperature and higher relative humidity, which contributes to a greater stomatal opening and, consequently, to a greater assimilation of CO_{2} by plants, which, by its turn, is converted in phytomass after the photosynthesis process.
^{Silva et al. (2015)} reported values of clone Little Sweet plant height of 68.11cm, and total number of cladodes as 35.44. When working with clone Little Sweet, ^{Cavalcante et al. (2014)}, in Frei PauloSE, verified a mean value of 32.12 cladodes per plant to 898 DAP, which is a higher value than the findings of our study, of 510 DAP. In the case of forage cactus, variables such as plant height and total number of cladode per plant may change according to species, plant age, fertilization and spacing levels, as well as environmental factors, such as availability of water in the soil and temperature (^{Sales et al. 2013}).
Through the evaluation of plant height growth of Nopalea cochenillifera without "pornunça" consortium, it was observed that the hyperbolic tangent, quadratic and Gompertz models presented the best results to estimate plant height in function of the number of days after planting (DAP) and cladode number per plant (table 3). The hyperbolic tangent model presented the highest explanatory power of 98.84%, the highest Willmott index (d=0.997), the smaller sum of squares of residuals (SSR=17.1) and Akaike information criterion (AIC=15.21), which makes it the best model to estimate the height of Nopalea cochenillifera in this situation.
Models  Equation of plant height  Criteria of adequacy of model  

R² (%)  SSR  AIC  d  
Linear multiple  H=36.21+0.087DAP+1.47CN  85.81  162.48  64.26  0.970 
Power  H=17.64DAP^{0.2226} CN^{0.109}  95.21  73.51  28.87  0.987 
Gamma  H=35.37+0.097DAP+1.41CN  88.56  168.37  64.19  0.970 
Logistic  H=80.67/(1+exp^{0.02266DAP+0.044CN})  94.39  82.65  7.94  0.984 
Gompertz  H=80.67exp^{exp(0.0217DAP+0.0296CN)}  98.16  27.09  5.44  0.995 
Weibull  H=exp^{(3.615+0.001DAP+0.035CN)}  80.00  299.30  19.4  0.950 
Hyperbolic T.  H=80.67*tanh(0.038*DAP^{0.88} CN^{0.202})  98.84  17.10  15.21  0.997 
Quadratic  H=31.14+0.34DAP0.0007DAP^{2}+0.57CN  98.66  19.64  45.13  0.997 
The increase of plant height of Nopalea cochenillifera consorted with "pornunça" is best estimated by the hyperbolic tangent and quadratic models (table 4). The hyperbolic tangent model showed to be the best one to estimate plant height in function of cladode number and number of days after planting. This model presented the best criteria for model adequacy: higher explanatory power (R²=99.17%), higher Willmott index (d=0.998), and smaller sum of squares of residuals (SSR=12.49) and Akaike information criterion (AIC=23.59).
Models  Equation of plant height  criteria of adequacy of model  

R² (%)  SSR  AIC  d  
Linear multiple  H=33.06+0.082DAP+2.068CN  90.82  138.80  62.68  0.975 
Power  H=15.96DAP^{0.22}CN^{0.156}  93.29  101.43  32.66  0.984 
Gamma  H=31.15+0.091DAP+2.12CN  90.35  146.06  61.92  0.976 
Logistic  H=80.78/(1+exp^{0.0147DAP+0.0061CN})  94.19  87.86  3.53  0.983 
Gompertz  H=80.78exp^{exp(0.0136DAP0.017CN)}  84.19  239.01  3.12  0.964 
Weibull  H=exp^{(3.53+0.0011DAP+0.05CN)}  78.15  330.25  20.5  0.949 
Hyperbolic T.  H=80.78*tanh(0.059*DAP^{0.67}CN^{0.052})  99.23  11.70  22.30  0.998 
Quadratic  H=28.1+0.3DAP0.0006DAP^{2}+1.1CN  98.90  16.57  43.43  0.997 
^{Lucena et al. (2016)} adjusted the growth of cladode length from Nopalea cochenillifera using an hyperbolic tangent model with high precision power (R²=99.9%) and low values of SSR=0.076 and AIC=23.155. ^{Lucena et al. (2018a)} estimated the cladode length of Nopalea cochenillifera in function of the cladode fractionation, through the use of a power model with R² greater than 99%, sum of squares of residues less than 2 and Akaike information index lower than 30. ^{Freire et al. (2018)} adjusted the quadratic model for cladode length from Nopalea cochenillifera with low power of explanation (R²=54.05%) and linear model for cladode number (R²=67.03%) in function of different salinity levels. ^{Silva et al. (2014)} adjusted the logistic model to estimate cladode area index from the Nopalea cochenillifera in function of the number of days after planting, with precision power of 96.0%.
After defining the hyperbolic tangent model as the most adequate to explain the plant height of Nopalea cochenillifera, with and without the consortium of "pornunça", residuals analysis was performed (figures 2 and 3).
The residuals analysis showed that the hyperbolic tangent models for treatments with and without consortium of "pornunça" are well adjusted, because there was no point of leverage or influence, and all residuals were within the confidence interval of the normal quantilequantile graph (figures 2 and 3).
The Nopalea cochenillifera consortium with "pornunça" (Manihot glaziovii x Manihot esculenta) is promising, and it indicates agronomic viability and advantages when consorted. The adoption of the hyperbolic tangent model H=80.78*tanh(0.059*DAP^{0.67}CN^{0.052}) allows the explanation of the plant height of Nopalea cochenillifera, in a nondestructive way, with high precision, speed and low cost, depending on the age of the plant and total number of cladode per plant. The use of mathematical models is an important tool for animal production, since these models accurately estimate dry matter production.