Crops ›› 2019, Vol. 35 ›› Issue (2): 128-135.doi: 10.16035/j.issn.1001-7283.2019.02.020

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A Mathematical Model of Density Coupling and Its Optimization in Maize-Soybean Intercropping

Xixi Dai1,Heming Zhan1,Xinghong Cui2,Yinyue Zhao1,Dandan Shan1,Liang Zhang1,Tiejun Wang1   

  1. 1 Institute of Food Crops, Yunnan Academy of Agricultural Sciences, Kunming 650205, Yunnan, China
    2 Huize Agricultural Technology Extension Center, Qujing 654200, Yunnan,China
  • Received:2018-09-17 Revised:2018-12-25 Online:2019-04-15 Published:2019-04-12
  • Contact: Tiejun Wang

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Abstract:

Maize and soybean intercropping planting is mainly composed of soybean planting in Yunnan Province. In order to achieve high yield of maize and soybean intercropping system, we must have suitable planting density. The effect of planting density on population grain yield and economic output value was studied with the design of the double saturated D-optimal regression, which was tested at three sites including Songming County (A), Huize County (B) and Ludian County (C), respectively. And a binary quadratic mathematical model was established, in which the planting densities of maize and soybean were independent variables, and population grain yield and economic output value were dependent variables. The results showed that the planting density of maize and soybean on population grain yield and economic output value of intercropping group effect significantly, and the change curve was a convex parabola. The population yield and economic output increased with density under the low level of density. According to model analysis, each test site could reach the highest population grain yield with the optimized combination of maize 64 110 plant/hm 2+ soybean 147 013 plant/hm 2 (A), maize 63 068 plant/hm 2+soybean 147 116 plant/hm 2 (B) and maize 64 059 plant/hm 2+soybean 145 077 plant/hm 2 (C), respectively. Each test site could reach the highest economic output value with the optimized combination of maize 62 909 plant/hm 2+soybean 149 852 plant/hm 2 (A), maize 61 499 plant/hm 2+soybean 151 807 plant/hm 2 (B) and maize 62 762 plant/hm 2+soybean 147 108 plant/hm 2 (C), respectively. According computer simulation, the optimum combination of densities of maize and soybean were 59 251-66 437 plant/hm 2 for maize and 140 075-161 495 plant/hm 2 for soybean (A), 58 927-65 366 plant/hm 2 for maize and 144 159-169 203 plant/hm 2 for soybean (B), 58 821-66 703 plant/hm 2 for maize and 139 315-154 886 plant/hm 2 for soybean (C) in order to obtain population grain yield greater than 12 270kg/hm 2 and economic output value greater 24 000 yuan/hm 2 under the condition of this experiment. A reasonable density collocation can effectively increase population yield and economic output value.

Key words: Maize, Soybean, Intercropping, Planting density coupling, Mathematics model, Optimization scheme

Table 1

Level coding value and field actual value of experiment factors"

处理
Treatment		 水平编码值
Level coding value		 田间实际值(株/hm2)
Field actual value (plant/hm2)
玉米
Maize		 大豆
Soybean		 玉米
Maize		 大豆
Soybean
T1 -1 -1 44 978 89 955
T2 -1 -1 71 964 89 955
T3 -1 -1 44 978 194 903
T4 -0.1315 -0.1315 56 696 135 528
T5 -0.3944 -1 63 793 194 903
T6 -1 -0.3944 71 964 163 124

Table 2

Statistics of grain yield in three experimental places kg/hm2"

处理
Treatment		 嵩明Songming 会泽Huize 鲁甸Ludian
玉米
Maize		 大豆
Soybean		 群体
Total		 玉米
Maize		 大豆
Soybean		 群体
Total		 玉米
Maize		 大豆
Soybean		 群体
Total
T1 9 499.3a 1 201.3de 10 700.6d 9 778.8c 1 078.1d 10 856.9d 9 960.5d 1 084.5e 11 045.0d
T2 10 655.5b 1 144.7e 11 800.2b 10 984.5ab 888.7e 11 873.2b 10 967.0c 1 011.5f 11 978.5b
T3 9 373.4a 1 338.5bc 10 711.9d 9 713.1c 1 194.0c 10 907.1d 9 835.6e 1 186.4d 11 022.0d
T4 10 651.3b 1 419.2b 12 070.5a 10 991.5ab 1 318.6b 12 310.1a 10 909.6c 1 298.4b 12 207.9a
T5 10 822.6ab 1 293.4cd 12 116.0a 11 121.1a 1 175.9c 12 297.0a 11 108.0b 1 121.4e 12 229.4a
T6 10 669.9b 1 363.2bc 12 033.1ab 10 893.5b 1 240.4c 12 133.9a 10 880.6c 1 236.9c 12 164.9a
SS - 3 582.2a 3 582.2e - 3 568.2a 3 568.2e - 3 647.2a 3 647.2e
MM 11 023.7a - 11 023.7c 11 155.9a - 11 155.9c 11 223.4a - 11 223.4c

Table 3

Statistics of economic output value in all experimental places 元/hm2 yuan/hm2"

处理
Treatment		 嵩明Songming 会泽Huize 鲁甸Ludian
玉米
Maize		 大豆
Soybean		 群体
Total		 玉米
Maize		 大豆
Soybean		 群体
Total		 玉米
Maize		 大豆
Soybean		 群体
Total
T1 15 198.8c 6 607.1de 21 805.9c 15 646.1c 5 929.4d 21 575.5d 15 936.8d 5 964.5e 21 901.3f
T2 17 048.8b 6 296.0e 23 344.7b 17 575.2ab 4 887.8e 22 463.0c 17 547.2c 5 563.2f 23 110.4d
T3 14 997.3c 7 361.9bc 22 359.2c 15 541.0c 6 566.7c 22 107.8c 15 736.9e 6 525.2d 22 262.2e
T4 17 042.2b 7 805.4b 24 847.6a 17 586.4ab 7 252.6b 24 839.0a 17 455.3c 7 140.9b 24 596.2a
T5 17 316.1ab 7 113.6cd 24 429.7a 17 793.7a 6 467.5c 24 261.3b 17 772.7b 6 167.9e 23 940.6c
T6 17 071.8b 7 497.6bc 24 569.4a 17 429.7b 6 822.1c 24 251.8b 17 484.9c 6 802.8c 24 287.7b
SS - 19 699.4a 19 699.4d - 19 625.2a 19 625.2e - 20 059.5a 20 059.5g
MM 17 638.4a - 17 638.4e 17 849.5a - 17 849.5f 17 957.4a - 17 957.4h

Table 4

Regression equations between experiment factors and population grain yield (Y)"

回归方程Regression equation
YA=12163.87+570.1X1+25.9X2-688.03X12-199.59X22+20.25X1X2（P<0.01,R=1.000）
YB=12400.3+519.56X1+36.51X2-769.15X12-229.58X22+11.41X1X2（P<0.01,R=1.000）
YC=12285.96+486.93X1+8.68X2-592.45X12-173.09X22+20.18X1X2（P<0.01,R=1.000）

Table 5

Regression equations between experiment factors and economic output value (Z)"

回归方程 Regression equation
ZA=25022.86+770.11X1+277.36X2-1179.07X12-991.13X22+0.71X1X2（P<0.01,R=1.000）
ZB=25010.89+581.26X1+403.66X2-1363.3X12-1224.68X22+137.51X1X2（P<0.01,R=1.000）
ZC=24736.32+610.88X1+186.78X2-968.45X12-1075.29X22+6.33X1X2（P<0.01,R=1.000）

Fig.1

Analysis of maize planting density on population yield and economic output value"

Fig.2

Analysis of soybean planting density on population yield and economic output value"

Fig.3

Interactive effects of planting density on population grain yield and economic output values in Songming"

Fig.4

Interactive effects of planting densities on population grain yield and economic output values in Huize"

Fig.5

Interactive effects of planting densities on population grain yield and economic output values in Ludian"

Table 6

Related parameters of optimization scheme on three sites"

优化目标
Optimal target value		 相关参数
Relevant parameter		 嵩明Songming 会泽Huize 鲁甸Ludian
X1 X2 X1 X2 X1 X2
群体产量≥12 270kg/hm2
Population grain yield ≥12 270kg/hm2		 优化方案数
Optimization scheme		 14 18 19
平均值Mean 0.3270 0.0751 0.2397 0.0584 0.2797 0.1080
标准误Standard error 0.1132 0.1198 0.0962 0.1472 0.0994 0.1478
95%的置信区间
95% confidence interval		 0.0824~0.5716 -0.1837~0.3339 0.0368~0.4426 -0.2521~0.3690 0.0709~0.4885 -0.2025~0.4185
种植密度(株/hm2)
Planting density
(plant/hm2)		 59 612~66 216 132 855~160 032 58 997~64 475 129 263~161 871 59 457~65 095 131 868~164 470
经济产值≥24 000元/hm2
Economic output value ≥24000 yuan/hm2		 优化方案数
Optimization scheme		 15 15 12
平均值Mean 0.2701 0.2701 0.2701 0.2701 0.3157 0.0876
标准误Standard error 0.1112 0.1112 0.1112 0.1112 0.1326 0.0674
95%的置信区间
95% confidence interval		 0.0316~0.5086 0.0316~0.5086 0.0316~0.5086 0.0316~0.5086 0.0238~0.6077 -0.0607~0.2359
种植密度(株/hm2)
Planting density
(plant/hm2)		 58 927~65 366 144 159~169 203 58 927~65 366 144 159~169 203 58 821~66 703 139 315~154 886

Table 7

Frequency analysis of optimal combination with population grain yield≥12 270kg/hm2, economic output value≥24 000 yuan/hm2"

变量
Variable		 嵩明Songming 会泽Huize 鲁甸Ludian
X1 X2 X1 X2 X1 X2
次数
Times		 频率
Frequency		 次数
Times		 频率
Frequency		 次数
Times		 频率
Frequency		 次数
Times		 频率
Frequency		 次数
Times		 频率
Frequency		 次数
Times		 频率
Frequency
-1 0 0 0 0 0 0 0 0 0 0 0 0
-0.1315 3 0.2308 4 0.3077 4 0.2667 4 0.2667 3 0.25 4 0.3333
0 3 0.2308 4 0.3077 4 0.2667 4 0.2667 3 0.25 4 0.3333
0.3944 4 0.3077 4 0.3077 4 0.2667 4 0.2667 3 0.25 4 0.3333
1 3 0.2308 1 0.0769 3 0.2000 3 0.2000 3 0.25 0 0
合计 Total 13 1 13 1 15 1 15 1 12 1 12 1
平均值 Mean 0.3218 0.1578 0.2701 0.2701 0.3157 0.0876
标准误 Standard error 0.1222 0.0936 0.1112 0.1112 0.1326 0.0674
95%置信区间
95% confidence interval		 0.0556~0.5879 -0.0462~0.3618 0.0316~0.5086 0.0316~0.5086 0.0238~0.6077 -0.0607~0.2359
种植密度(株/hm2)
Plant density (plant/hm2)		 59 251~66 437 140 075~161 495 58 927~65 366 144 159~169 203 58 821~66 703 139 315~154 886
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