Mathematical Modeling of Broccoli Cultivation and Growth Period and Yield of Flower Heads

: This study was carried out to evaluate the effects of transplant age, planting density and the level of nitrogen fertilizer on growth period and yield of broccoli. Two mathematical models were constructed for these parameters, they are 1 y =75.1322+1.6361 1 x -0.0001 2 x +0.0014 3 x ( R =0.8888 ** ), and 2 y =14999.1852+404.0006 1 x -0.1986 2 x +3.6196 3 x ( R =0.6058 ** ), where 1 y and 2 y are growth period and yield, and 1 x , 2 x ， 3 x are transplant age, planting density and dose of nitrogen fertilizer, respectively. These models can be used for the optimization of cultivation techniques to achieve the goal of developing an high yield and high profit competitive broccoli production system.


Mathematical modeling of broccoli cultivation and growth period and yield of flower heads
Liu Wei-ming 1,a , Wang En-guo 2

Introduction
Transplant age, planting density and the amount of nitrogen fertilizer are important cultivation factors influencing quality of broccoli and profit of the crop. Previously, these three factors were evaluated for their effects on growth period and yield of broccoli [1][2][3][4][5][6][7][8][9], but no mathematical model systems were established. This study aimed at developing a simulation modeling system for providing scientific information for decision making in secure high yield and high profit production of broccoli.

Construction of mathematical models and simulation
Units used in this study were as follows: transplant age (Days), planting density (Plants/hm2), and nitrogen fertilizer doses: kg/hm2; growth period (days); yield (kg/hm2).
Data were subjected to polynomial regression analysis using DPS software [10] to build models for transplant age, planting density, nitrogen fertilizer levels, and their relationship with growth period, and yield. These models were used to simulate the cultivation system for broccoli.

The effects of transplant ages on growth period and yield of broccoli
The growth periods of broccoli varied greatly among those transplant age treatments, it was from 113 to 162 days. In general when using older transplants, the number of days increased during the period from transplanting to floral bud emergence, the duration of vegetative growth post transplanting, floral head growth period, the duration from transplanting to harvest, and total growth period. Statistical analysis identified a significant linear correlation between transplant age ( 1 x ) and growth period ( However, the density of florets on flower heads, the ratio of marketable flower heads and yield of flower head showed a "low-high-low" trend in responses to increasing transplant age. The regression mathematical model for transplant age ( 1 x ) and flower head yield (

Effects of planting density on growth period and yield of broccoli
Results indicate that increasing planting density led to decreases in plants height, canopy width, and the number of leaves per plant. Floral buds emerged prematurely, which resulted in shortened growth period. Statistical analysis indicated that planting density (  x +2.8151 2 x -21740, ( r =0.9571**).

Effects of nitrogen fertilizer level on growth period and yield of broccoli
Results in Table 2 show that when nitrogen fertilizer was provided at 600 kg/hm2,, plants produced the flower heads of the biggest diameter (l3.3cm), fresh weight per head ( 476.85g) and total yield (1440.7 kg/667m2). However, these values started to decline at increasing nitrogen level. Florets became thinner at higher nitrogen level. The width of florets in the high nitrogen level treatments was significantly lower than those supplied with lower doses of nitrogen fertilizer. The highest ratio of marketable flower heads was found in the 375 kg/hm2 of nitrogen fertilizer treatment, it was 89.6%. Nitrogen fertilizer applied at below of above this dose both resulted in reduced ratio of marketable flower heads. The lowest ratio of 76.2% was found in the treatment of 150kg/hm2. Statistical analysis revealed a binomial curve changes in flower head yield ( y ) with nitrogen fertilizer level ( 3 x ). The regression model is:

Construction of mathematical models
Data of growth periods and yield of broccoli from treatments of transplant age, planting density, nitrogen fertilizer doses were analyzed using stepwise regression analysis. The polynomial regression models for total growth period ( 1

The application of the polynomial decision making mathematical model
Mathematical models can be used to simulate and select the optimum combinations of parameters according to market demand and goals of production. For example, in case of a planting scheme comprised of 130-140 day growth season with targeted yield of 24000-26000 kg/hm 2 , the planting density, seedling age and nitrogen fertilizer (pure N) combination should be 35-40 day transplants, at a density of 33000-36750 plants / hm 2 , and with the use of 375-600 kg/hm 2 of nitrogen fertilizer. The priority goal of growing broccoli is profit. Because profit is affected by market price, there are differences between yield and profit during some seasons. These modules can be used to arrange production schedule by taking into considerations of predicted market demands and unit price of broccoli. This study indicates that the growth period of broccoli plants varied significantly under different transplant ages and planting densities. The growth period changed from 113 to 161 days. In general, when older transplants were used, it lengthened the duration from transplanting to floret emergence, the vegetative growth period in the field, flower head development stage, days from transplanting to harvest, and the duration of growth period. There was an extremely significant correlation between treatments of transplant age and the duration of those growth period. The growth period was 126-142 days in the planting density treatments. Under high planting density, florets emerged soon after transplanting, and the whole growth period became short.
These information are very important when making recommendation for a secure production strategy of broccoli.

The yield of flower heads has a binominal curve relationship with transplant age, planting density and nitrogen fertilizer level
Results indicate that the yield of flower heads followed a non-linear "low-high-low" trend in response to increasing levels of transplant age, planting density and nitrogen fertilizer level treatments. Therefore, in cultivation of broccoli crop, it is very important to pay attention to appropriate combinations of transplant age, planting density and the amount of nitrogen fertilizer.

The polynomial mathematical models can provide information to decision making in designing schemes for the production of broccoli
The polynomial regression models constructed in this study can simulate the effects of transplant age, planting density, nitrogen fertilizer doses on growth period and yield. These models are very useful in designing schemes in response to market demands with specific goals, and guide production management of the broccoli crop.